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• Cited by 34

Phaseless Imaging by Reverse Time Migration: Acoustic Waves

• Optics, electromagnetic theory - General
• Zhiming Chen , Guanghui Huang
• Published online by Cambridge University Press: 20 February 2017 , pp. 1-21
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We propose a reliable direct imaging method based on the reverse time migration for finding extended obstacles with phaseless total field data. We prove that the imaging resolution of the method is essentially the same as the imaging results using the scattering data with full phase information when the measurement is far away from the obstacle. The imaginary part of the cross-correlation imaging functional always peaks on the boundary of the obstacle. Numerical experiments are included to illustrate the powerful imaging quality

• Cited by 30

Towards Textbook Efficiency for Parallel Multigrid

• Björn Gmeiner , Ulrich Rüde , Holger Stengel , Christian Waluga , Barbara Wohlmuth
• Published online by Cambridge University Press: 03 March 2015 , pp. 22-46

In this work, we extend Achi Brandt's notion of textbook multigrid efficiency (TME) to massively parallel algorithms. Using a finite element based geometric multigrid implementation, we recall the classical view on TME with experiments for scalar linear equations with constant and varying coefficients as well as linear systems with saddle-point structure. To extend the idea of TME to the parallel setting, we give a new characterization of a work unit (WU) in an architecture-aware fashion by taking into account performance modeling techniques. We illustrate our newly introduced parallel TME measure by large-scale computations, solving problems with up to 200 billion unknowns on a TOP-10 supercomputer.

• Cited by 29

Numerical Solution to the Multi-Term Time Fractional Diffusion Equation in a Finite Domain

• Partial differential equations, initial value and time-dependent initial-boundary value problems
• Miscellaneous topics - Partial differential equations
• Gongsheng Li , Chunlong Sun , Xianzheng Jia , Dianhu Du
• Published online by Cambridge University Press: 20 July 2016 , pp. 337-357
• You have access Access

This paper deals with numerical solution to the multi-term time fractional diffusion equation in a finite domain. An implicit finite difference scheme is established based on Caputo's definition to the fractional derivatives, and the upper and lower bounds to the spectral radius of the coefficient matrix of the difference scheme are estimated, with which the unconditional stability and convergence are proved. The numerical results demonstrate the effectiveness of the theoretical analysis, and the method and technique can also be applied to other kinds of time/space fractional diffusion equations.

• Cited by 25

Multidimensional Iterative Filtering Method for the Decomposition of High–Dimensional Non–Stationary Signals

• Numerical methods in Fourier analysis
• Multivariate analysis
• Stochastic processes
• Numerical approximation and computational geometry
• Antonio Cicone , Haomin Zhou
• Published online by Cambridge University Press: 09 May 2017 , pp. 278-298

Iterative Filtering (IF) is an alternative technique to the Empirical Mode Decomposition (EMD) algorithm for the decomposition of non–stationary and non–linear signals. Recently in [3] IF has been proved to be convergent for any L 2 signal and its stability has been also demonstrated through examples. Furthermore in [3] the so called Fokker–Planck (FP) filters have been introduced. They are smooth at every point and have compact supports. Based on those results, in this paper we introduce the Multidimensional Iterative Filtering (MIF) technique for the decomposition and time–frequency analysis of non–stationary high–dimensional signals. We present the extension of FP filters to higher dimensions. We prove convergence results under general sufficient conditions on the filter shape. Finally we illustrate the promising performance of MIF algorithm, equipped with high–dimensional FP filters, when applied to the decomposition of two dimensional signals.

• Cited by 22

Adaptive Mixed GMsFEM for Flows in Heterogeneous Media

• Partial differential equations, boundary value problems
• Ho Yuen Chan , Eric Chung , Yalchin Efendiev
• Published online by Cambridge University Press: 17 November 2016 , pp. 497-527

In this paper, we present two adaptive methods for the basis enrichment of the mixed Generalized Multiscale Finite Element Method (GMsFEM) for solving the flow problem in heterogeneous media. We develop an a-posteriori error indicator which depends on the norm of a local residual operator. Based on this indicator, we construct an offline adaptive method to increase the number of basis functions locally in coarse regions with large local residuals. We also develop an online adaptive method which iteratively enriches the function space by adding new functions computed based on the residual of the previous solution and special minimum energy snapshots. We show theoretically and numerically the convergence of the two methods. The online method is, in general, better than the offline method as the online method is able to capture distant effects (at a cost of online computations), and both methods have faster convergence than a uniform enrichment. Analysis shows that the online method should start with a certain number of initial basis functions in order to have the best performance. The numerical results confirm this and show further that with correct selection of initial basis functions, the convergence of the online method can be independent of the contrast of the medium. We consider cases with both very high and very low conducting inclusions and channels in our numerical experiments.

• Cited by 21

Fast Linearized Augmented Lagrangian Method for Euler's Elastica Model

• Communication, information
• Jun Zhang , Rongliang Chen , Chengzhi Deng , Shengqian Wang
• Published online by Cambridge University Press: 20 February 2017 , pp. 98-115

Recently, many variational models involving high order derivatives have been widely used in image processing, because they can reduce staircase effects during noise elimination. However, it is very challenging to construct efficient algorithms to obtain the minimizers of original high order functionals. In this paper, we propose a new linearized augmented Lagrangian method for Euler's elastica image denoising model. We detail the procedures of finding the saddle-points of the augmented Lagrangian functional. Instead of solving associated linear systems by FFT or linear iterative methods (e.g., the Gauss-Seidel method), we adopt a linearized strategy to get an iteration sequence so as to reduce computational cost. In addition, we give some simple complexity analysis for the proposed method. Experimental results with comparison to the previous method are supplied to demonstrate the efficiency of the proposed method, and indicate that such a linearized augmented Lagrangian method is more suitable to deal with large-sized images.

• Cited by 18

Numerical Approaches for Linear Left-invariant Diffusions on SE (2), their Comparison to Exact Solutions, and their Applications in Retinal Imaging

• Stochastic analysis
• Jiong Zhang , Remco Duits , Gonzalo Sanguinetti , Bart M. ter Haar Romeny
• Published online by Cambridge University Press: 15 February 2016 , pp. 1-50

Left-invariant PDE-evolutions on the roto-translation group SE (2)(and their resolvent equations) have been widely studied in the fields of cortical modeling and image analysis. They include hypo-elliptic diffusion (for contour enhancement) proposed by Citti & Sarti, and Petitot, and they include the direction process (for contour completion) proposed by Mumford. This paper presents a thorough study and comparison of the many numerical approaches, which, remarkably, are missing in the literature. Existing numerical approaches can be classified into 3 categories: Finite difference methods, Fourier based methods (equivalent to SE (2)-Fourier methods), and stochastic methods (Monte Carlo simulations). There are also 3 types of exact solutions to the PDE-evolutions that were derived explicitly (in the spatial Fourier domain) in previous works by Duits and van Almsick in 2005. Here we provide an overview of these 3 types of exact solutions and explain how they relate to each of the 3 numerical approaches. We compute relative errors of all numerical approaches to the exact solutions, and the Fourier based methods show us the best performance with smallest relative errors. We also provide an improvement of Mathematica algorithms for evaluating Mathieu-functions, crucial in implementations of the exact solutions. Furthermore, we include an asymptotical analysis of the singularities within the kernels and we propose a probabilistic extension of underlying stochastic processes that overcomes the singular behavior in the origin of time-integrated kernels. Finally, we show retinal imaging applications of combining left-invariant PDE-evolutions with invertible orientation scores.

A Stochastic Galerkin Method for the Boltzmann Equation with Multi-Dimensional Random Inputs Using Sparse Wavelet Bases

• Equations of mathematical physics and other areas of application
• Numerical analysis: Ordinary differential equations
• Ruiwen Shu , Jingwei Hu , Shi Jin
• Published online by Cambridge University Press: 09 May 2017 , pp. 465-488

We propose a stochastic Galerkin method using sparse wavelet bases for the Boltzmann equation with multi-dimensional random inputs. Themethod uses locally supported piecewise polynomials as an orthonormal basis of the random space. By a sparse approach, only a moderate number of basis functions is required to achieve good accuracy in multi-dimensional random spaces. We discover a sparse structure of a set of basis-related coefficients, which allows us to accelerate the computation of the collision operator. Regularity of the solution of the Boltzmann equation in the random space and an accuracy result of the stochastic Galerkin method are proved in multi-dimensional cases. The efficiency of the method is illustrated by numerical examples with uncertainties from the initial data, boundary data and collision kernel.

• Cited by 14

Deferred Correction Methods for Forward Backward Stochastic Differential Equations

• Probabilistic methods, simulation and stochastic differential equations
• Tao Tang , Weidong Zhao , Tao Zhou
• Published online by Cambridge University Press: 09 May 2017 , pp. 222-242

The deferred correction (DC) method is a classical method for solving ordinary differential equations; one of its key features is to iteratively use lower order numerical methods so that high-order numerical scheme can be obtained. The main advantage of the DC approach is its simplicity and robustness. In this paper, the DC idea will be adopted to solve forward backward stochastic differential equations (FBSDEs) which have practical importance in many applications. Noted that it is difficult to design high-order and relatively “clean” numerical schemes for FBSDEs due to the involvement of randomness and the coupling of the FSDEs and BSDEs. This paper will describe how to use the simplest Euler method in each DC step–leading to simple computational complexity–to achieve high order rate of convergence.

High Order Mass-Lumping Finite Elements on Simplexes

• Tao Cui , Wei Leng , Deng Lin , Shichao Ma , Linbo Zhang
• Published online by Cambridge University Press: 09 May 2017 , pp. 331-350

This paper is concerned with the construction of high order mass-lumping finite elements on simplexes and a program for computing mass-lumping finite elements on triangles and tetrahedra. The polynomial spaces for mass-lumping finite elements, as proposed in the literature, are presented and discussed. In particular, the unisolvence problem of symmetric point-sets for the polynomial spaces used in mass-lumping elements is addressed, and an interesting property of the unisolvent symmetric point-sets is observed and discussed. Though its theoretical proof is still lacking, this property seems to be true in general, and it can greatly reduce the number of cases to consider in the computations of mass-lumping elements. A program for computing mass-lumping finite elements on triangles and tetrahedra, derived from the code for computing numerical quadrature rules presented in [7], is introduced. New mass-lumping finite elements on triangles found using this program with higher orders, namely 7, 8 and 9, than those available in the literature are reported.

A Few Benchmark Test Cases for Higher-Order Euler Solvers

• Liang Pan , Jiequan Li , Kun Xu
• Published online by Cambridge University Press: 12 September 2017 , pp. 711-736

There have been great efforts on the development of higher-order numerical schemes for compressible Euler equations in recent decades. The traditional test cases proposed thirty years ago mostly target on the strong shock interactions, which may not be adequate enough for evaluating the performance of current higher-order schemes. In order to set up a higher standard for the development of new algorithms, in this paper we present a few benchmark cases with severe and complicated wave structures and interactions, which can be used to clearly distinguish different kinds of higher-order schemes. All tests are selected so that the numerical settings are very simple and any higher order scheme can be straightforwardly applied to these cases. The examples include highly oscillatory solutions and the large density ratio problem in one dimensional case. In two dimensions, the cases include hurricane-like solutions; interactions of planar contact discontinuities with asymptotic large Mach number (the composite of entropy wave and vortex sheets); interaction of planar rarefaction waves with transition from continuous flows to the presence of shocks; and other types of interactions of two-dimensional planar waves. To get good performance on all these cases may push algorithm developer to seek for new methodology in the design of higher-order schemes, and improve the robustness and accuracy of higher-order schemes to a new level of standard. In order to give reference solutions, the fourth-order gas-kinetic scheme (GKS) will be used to all these benchmark cases, even though the GKS solutions may not be very accurate in some cases. The main purpose of this paper is to recommend other CFD researchers to try these cases as well, and promote further development of higher-order schemes.

A Multilevel Correction Method for Steklov Eigenvalue Problem by Nonconforming Finite Element Methods

• Acceleration of convergence
• Xiaole Han , Yu Li , Hehu Xie
• Published online by Cambridge University Press: 05 August 2015 , pp. 383-405

In this paper, a multilevel correction scheme is proposed to solve the Steklov eigenvalue problem by nonconforming finite element methods. With this new scheme, the accuracy of eigenpair approximations can be improved after each correction step which only needs to solve a source problem on finer finite element space and an Steklov eigenvalue problem on the coarsest finite element space. This correction scheme can increase the overall efficiency of solving eigenvalue problems by the nonconforming finite element method. Furthermore, as same as the direct eigenvalue solving by nonconforming finite element methods, this multilevel correction method can also produce the lower-bound approximations of the eigenvalues.

• Cited by 13

Numerical Solution of Stochastic Ito-Volterra Integral Equations using Haar Wavelets

• Theoretical approximation of solutions
• Fakhrodin Mohammadi
• Published online by Cambridge University Press: 20 July 2016 , pp. 416-431

This paper presents a computational method for solving stochastic Ito-Volterra integral equations. First, Haar wavelets and their properties are employed to derive a general procedure for forming the stochastic operational matrix of Haar wavelets. Then, application of this stochastic operational matrix for solving stochastic Ito-Volterra integral equations is explained. The convergence and error analysis of the proposed method are investigated. Finally, the efficiency of the presented method is confirmed by some examples.

Spectral Method Approximation of Flow Optimal Control Problems with H 1 -Norm State Constraint

• Existence theories
• Yanping Chen , Fenglin Huang
• Published online by Cambridge University Press: 20 June 2017 , pp. 614-638

In this paper, we consider an optimal control problem governed by Stokes equations with H 1 -norm state constraint. The control problem is approximated by spectral method, which provides very accurate approximation with a relatively small number of unknowns. Choosing appropriate basis functions leads to discrete system with sparse matrices. We first present the optimality conditions of the exact and the discrete optimal control systems, then derive both a priori and a posteriori error estimates. Finally, an illustrative numerical experiment indicates that the proposed method is competitive, and the estimator can indicate the errors very well.

• Cited by 12

A Hybrid Spectral Element Method for Fractional Two-Point Boundary Value Problems

• General theory in ordinary differential equations
• Miscellaneous topics of analysis in the complex domain
• Real functions
• Changtao Sheng , Jie Shen
• Published online by Cambridge University Press: 09 May 2017 , pp. 437-464

We propose a hybrid spectral element method for fractional two-point boundary value problem (FBVPs) involving both Caputo and Riemann-Liouville (RL) fractional derivatives. We first formulate these FBVPs as a second kind Volterra integral equation (VIEs) with weakly singular kernel, following a similar procedure in [16]. We then design a hybrid spectral element method with generalized Jacobi functions and Legendre polynomials as basis functions. The use of generalized Jacobi functions allow us to deal with the usual singularity of solutions at t = 0. We establish the existence and uniqueness of the numerical solution, and derive a hptype error estimates under L 2 ( I )-norm for the transformed VIEs. Numerical results are provided to show the effectiveness of the proposed methods.

Linear Stability of Hyperbolic Moment Models for Boltzmann Equation

• Partial differential equations on manifolds; differential operators
• Time-dependent statistical mechanics (dynamic and nonequilibrium)
• Partial differential equations
• Yana Di , Yuwei Fan , Ruo Li , Lingchao Zheng
• Published online by Cambridge University Press: 09 May 2017 , pp. 255-277

Grad's moment models for Boltzmann equation were recently regularized to globally hyperbolic systems and thus the regularized models attain local well-posedness for Cauchy data. The hyperbolic regularization is only related to the convection term in Boltzmann equation. We in this paper studied the regularized models with the presentation of collision terms. It is proved that the regularized models are linearly stable at the local equilibrium and satisfy Yong's first stability condition with commonly used approximate collision terms, and particularly with Boltzmann's binary collision model.

Positivity-Preserving Runge-Kutta Discontinuous Galerkin Method on Adaptive Cartesian Grid for Strong Moving Shock

• Shock waves and blast waves
• Jianming Liu , Jianxian Qiu , Mikhail Goman , Xinkai Li , Meilin Liu
• Published online by Cambridge University Press: 15 February 2016 , pp. 87-110

In order to suppress the failure of preserving positivity of density or pressure, a positivity-preserving limiter technique coupled with h -adaptive Runge-Kutta discontinuous Galerkin (RKDG) method is developed in this paper. Such a method is implemented to simulate flows with the large Mach number, strong shock/obstacle interactions and shock diffractions. The Cartesian grid with ghost cell immersed boundary method for arbitrarily complex geometries is also presented. This approach directly uses the cell solution polynomial of DG finite element space as the interpolation formula. The method is validated by the well documented test examples involving unsteady compressible flows through complex bodies over a large Mach numbers. The numerical results demonstrate the robustness and the versatility of the proposed approach.

• Cited by 11

Bootstrap Algebraic Multigrid: Status Report, Open Problems, and Outlook

• Achi Brandt , James Brannick , Karsten Kahl , Ira Livshits
• Published online by Cambridge University Press: 03 March 2015 , pp. 112-135

This paper provides an overview of the main ideas driving the bootstrap algebraic multigrid methodology, including compatible relaxation and algebraic distances for defining effective coarsening strategies, the least squares method for computing accurate prolongation operators and the bootstrap cycles for computing the test vectors that are used in the least squares process. We review some recent research in the development, analysis and application of bootstrap algebraic multigrid and point to open problems in these areas. Results from our previous research as well as some new results for some model diffusion problems with highly oscillatory diffusion coefficient are presented to illustrate the basic components of the BAMG algorithm.

Selected Recent Applications of Sparse Grids

• Benjamin Peherstorfer , Christoph Kowitz , Dirk Pflüger , Hans-Joachim Bungartz
• Published online by Cambridge University Press: 03 March 2015 , pp. 47-77

Sparse grids have become a versatile tool for a vast range of applications reaching from interpolation and numerical quadrature to data-driven problems and uncertainty quantification. We review four selected real-world applications of sparse grids: financial product pricing with the Black-Scholes model, interactive exploration of simulation data with sparse-grid-based surrogate models, analysis of simulation data through sparse grid data mining methods, and stability investigations of plasma turbulence simulations.

A Multistep Scheme for Decoupled Forward-Backward Stochastic Differential Equations

• Weidong Zhao , Wei Zhang , Lili Ju
• Published online by Cambridge University Press: 24 May 2016 , pp. 262-288

Upon a set of backward orthogonal polynomials, we propose a novel multi-step numerical scheme for solving the decoupled forward-backward stochastic differential equations (FBSDEs). Under Lipschtiz conditions on the coefficients of the FBSDEs, we first get a general error estimate result which implies zero-stability of the proposed scheme, and then we further prove that the convergence rate of the scheme can be of high order for Markovian FBSDEs. Some numerical experiments are presented to demonstrate the accuracy of the proposed multi-step scheme and to numerically verify the theoretical results.

Numerical Methods and Applications

9th International Conference, NMA 2018, Borovets, Bulgaria, August 20-24, 2018, Revised Selected Papers

• Conference proceedings
• © 2019
• Geno Nikolov   ORCID: http://orcid.org/0000-0001-5608-2488 0 ,
• Natalia Kolkovska 1 ,
• Krassimir Georgiev   ORCID: http://orcid.org/0000-0001-5277-2887 2

Sofia University “St. Kliment Ohridski”, Sofia, Bulgaria

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Bulgarian Academy of Sciences, Sofia, Bulgaria

Part of the book series: Lecture Notes in Computer Science (LNCS, volume 11189)

Part of the book sub series: Theoretical Computer Science and General Issues (LNTCS)

Included in the following conference series:

• NMA: International Conference on Numerical Methods and Applications

Conference proceedings info: NMA 2018.

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About this book

This book constitutes the thoroughly refereed post-conference proceedings of the 9th International Conference on Numerical Methods and Applications, NMA 2018, held in Borovets, Bulgaria, in August 2018.

The 56 revised regular papers presented were carefully reviewed and selected from 61 submissions for inclusion in this book. The papers are organized in the following topicalsections: numerical search and optimization; problem-driven numerical method: motivation and application, numerical methods for fractional diffusion problems; orthogonal polynomials and numerical quadratures; and Monte Carlo and Quasi-Monte Carlo methods.

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Numerical Methods

A Generalized Framework for Direct Discontinuous Galerkin Methods for Nonlinear Diffusion Equations

• numerical methods
• approximation techniques
• numerical linear algebra
• finite element methods
• finite difference methods
• Monte Carlo methods
• mathematical modeling
• non-linear problems
• metaheuristic algorithms
• high performance computing
• fractional diffusion problems
• signal processing
• differential equations
• time domain anaylsis
• finite difference
• wireless telecommunication systems
• wireless networks
• genetic alogrithms
• graph theory
• algorithm analysis and problem complexity

Table of contents (56 papers)

Front matter, invited papers, new stabilized discretizations for poroelasticity equations.

• Francisco J. Gaspar, Carmen Rodrigo, Xiaozhe Hu, Peter Ohm, James Adler, Ludmil Zikatanov

A Class of Staggered Schemes for the Compressible Euler Equations

• Raphaele Herbin, Jean-Claude Latché

Numerical Search and Optimization

An effective guided fireworks algorithm for solving ucav path planning problem.

• Adis Alihodzic, Damir Hasic, Elmedin Selmanovic

Cuckoo Search Algorithm for Parameter Identification of Fermentation Process Model

• Maria Angelova, Olympia Roeva, Tania Pencheva

Optimization of String Rewriting Operations for 3D Fractal Generation with Genetic Algorithms

• Todor Balabanov, Janeta Sevova, Kolyu Kolev

Quantifying the Effectiveness of First-Hop Redundancy Protocols in IP Networks

• Paul Bourne, Neville Palmer, Jan Skrabala

Derivation of a Coordinate System of Three Laser Triangulation Sensors in a Plane

• Ján Buša, Miroslav Dovica, Lukáš Kačmár

Optimisation Techniques in Wildfire Simulations. Test Case Kresna Fire August 2017

• Nina Dobrinkova, Momchil Panayotov, Peter Boyvalenkov

Subtraction of Two 2D Polygons with Some Matching Vertices

• Georgi Evtimov, Stefka Fidanova

InterCriteria Analysis of Different Variants of ACO Algorithm for Wireless Sensor Network Positioning

• Stefka Fidanova, Olympia Roeva

Description of Dynamics of Ellipsoidal Estimates of Reachable Sets of Nonlinear Control Systems with Bilinear Uncertainty

• Tatiana F. Filippova

Evaluation of Serial and Parallel Shared-Memory Distance-1 Graph Coloring Algorithms

• Lukas Gnam, Siegfried Selberherr, Josef Weinbub

Solving Function Approximation Problems Using the \(L^2\) -Norm of the Log Ratio as a Metric

• Ivan D. Gospodinov, Stefan M. Filipov, Atanas V. Atanassov

Application of Parallel and Hybrid Metaheuristics for Graph Partitioning Problem

• Zbigniew Kokosiński, Marcin Pijanowski

Monte Carlo Approach for Modeling and Optimization of One-Dimensional Bimetallic Nanostructures

• Vladimir Myasnichenko, Nickolay Sdobnyakov, Leoneed Kirilov, Rossen Mikhov, Stefka Fidanova

Factors for Search Methods Scalability

• Kalin Penev

The Statistic Analysis of Conjunctive Adverbs Used in the First Bulgarian School Books in Mathematics (from the First Half of XIX c.)

• Velislava Stoykova

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Editors and affiliations.

Geno Nikolov

Natalia Kolkovska, Krassimir Georgiev

Bibliographic Information

Book Title : Numerical Methods and Applications

Book Subtitle : 9th International Conference, NMA 2018, Borovets, Bulgaria, August 20-24, 2018, Revised Selected Papers

Editors : Geno Nikolov, Natalia Kolkovska, Krassimir Georgiev

Series Title : Lecture Notes in Computer Science

DOI : https://doi.org/10.1007/978-3-030-10692-8

Publisher : Springer Cham

eBook Packages : Computer Science , Computer Science (R0)

Copyright Information : Springer Nature Switzerland AG 2019

Softcover ISBN : 978-3-030-10691-1 Published: 18 January 2019

eBook ISBN : 978-3-030-10692-8 Published: 21 January 2019

Series ISSN : 0302-9743

Series E-ISSN : 1611-3349

Edition Number : 1

Number of Pages : XX, 500

Number of Illustrations : 144 b/w illustrations, 114 illustrations in colour

Topics : Numeric Computing , Math Applications in Computer Science , Artificial Intelligence , Algorithm Analysis and Problem Complexity , Computer Systems Organization and Communication Networks , Discrete Mathematics in Computer Science

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Numerical Mathematics

Theory, methods and applications.

Numerical Mathematics: Theory, Methods and Applications (NMTMA) published quarterly. It is an international journal covering numerical methods, analysis and applications. The journal publishes original research papers in all branches of modern computational mathematics such as numerical linear algebra, numerical optimization, numerical differential equations and computational statistics. Papers containing new ideas, creative approaches and/or innovative applications as well as invited reviews are expected to appear regularly in the journal. NMTMA has been accepted for the Science Citation Index-Expanded (SCIE). NMTMA's paper starting from volume 1(1) 2008 will be included in SCIE.

ISSN #: 1004-8979

Editor-in-Chief:

Zhiming Chen

• © 2007, Dept. of Math. at Nanjing University
• NM   (Chinese Series)

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Bibliometrics & citations, view options, recommendations, numerical solution of boundary value problems for the eikonal equation in an anisotropic medium.

A Dirichlet problem is considered for the eikonal equation in an anisotropic medium. The nonlinear boundary value problem (BVP) formulated in the present work is the limit of the diffusion–reaction problem with a diffusion parameter ...

MultiStencils Fast Marching Methods: A Highly Accurate Solution to the Eikonal Equation on Cartesian Domains

A wide range of computer vision applications require an accurate solution of a particular Hamilton-Jacobi (HJ) equation known as the Eikonal equation. In this paper, we propose an improved version of the fast marching method (FMM) that is highly ...

A boundary-only meshless method for numerical solution of the Eikonal equation

The radial basis function (RBF) collocation methods for the numerical solution of partial differential equation have been popular in recent years because of their advantage. For instance, they are inherently meshless, integration free and highly ...

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Numerical study has proven to be one of the core tools for exploring the emergent properties of many-body quantum systems. A wide array of numerical techniques have been developed for the analysis of quantum many-body physics, some of which rely on the specific physics in question (for example, matrix product state methods whose accuracy relies on the relevant physics having low entanglement) and some of which can treat more general cases.

Dynamite is a numerical package that uses massively parallel Krylov subspace algorithms for quantum dynamics and eigensolving. It has a straightforward Python interface for user-specified Hamiltonians and states. The backend is highly optimized for modern supercomputing environments, including distributed memory parallelism with MPI and hand-tuned computational kernels for GPUs. Other features include symmetry subspaces such as charge conservation, and “matrix-free” methods to drastically reduce memory usage. For the documentation, see  https://dynamite.readthedocs.io/ . 

Density Matrix Truncation

One method [ https://github.com/Jack-Kemp/dmt ] the group uses for calculating long-time quantum dynamics is density matrix truncation or DMT, originally proposed by [1]. Unlike the exact Krylov methods discussed above, this algorithm is approximate, but the truncation method is chosen to preserve local observables, making it particularly well-suited for calculating hyrdrodynamic properties such as transport coefficients. The group has used DMT in various setting from Floquet heating [2] to universal Kardar-Parisi-Zhang dynamics in integrable spin chains [3].

Neural Quantum States

In the last few years AI has brought significant advances to multiple branches of science and technology. Some of those AI methods (such as neural networks) have been recently used to variationally study classical simulation of quantum many body systems. Concrete tasks include solving for the system’s ground state or simulating its time dynamics. In our group we are trying to utilize recent advances from the ML community to perform accurate, interpretable and large-scale classical simulations of challenging quantum many-body physics problems.

For systems that exhibit interesting physical behavior in excited states, such as the phenomenon of many-body localization, interrogating novel physics requires solving for eigenvalues and eigenvectors in the middle of a Hamiltonian’s spectrum. This is generally challenging for large-scale iterative numerical methods, because the transformations required use large amounts of computer memory, ultimately limiting the system sizes that can be studied. We have approached the eigenproblem with a different spectral transform for which the memory usage is dramatically decreased, while still achieving good convergence by using a cutting-edge eigensolver called LOBPCG (Locally Optimal Block Preconditioned Conjugate Gradient) [4, 5].

• White, C.D., Zaletel, M., Mong R.S.K., et al . Quantum dynamics of thermalizing systems. Phys. Rev. B 97, 035127 (2018). https://doi.org/10.1103/PhysRevB.97.035127
• Ye, B., Machado, F., White, C.D., et al. Emergent Hydrodynamics in Nonequilibrium Quantum Systems. Phys. Rev. Lett. 125, 030601 (2020). https://doi.org/10.1103/PhysRevLett.125.030601
• Ye, B., Machado, F., Kemp, J., et al. Universal Kardar-Parisi-Zhang Dynamics in Integrable Quantum Systems. Phys. Rev. Lett. 129, 230602 (2022). https://doi.org/10.1103/PhysRevLett.129.230602
• Van Beeumen, R., Kahanamoku-Meyer, G.D., Yao, N.Y., and Yang, C. A Scalable Matrix-Free Iterative Eigensolver for Studying Many-Body Localization. In Proceedings of the International Conference on High Performance Computing in Asia-Pacific Region (HPCAsia2020). Association for Computing Machinery, New York, NY, USA, 179–187 (2020). https://doi.org/10.1145/3368474.3368497
• Van Beeumen, R., Ibrahim, K.Z., Kahanamoku–Meyer, G.D., Yao, N.Y., Yang, C. Enhancing scalability of a matrix-free eigensolver for studying many-body localization.  The International Journal of High Performance Computing Applications 36(3):307-319 (2022). doi: 10.1177/10943420211060365

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• Published: 19 June 2024

Free convection in a square wavy porous cavity with partly magnetic field: a numerical investigation

• Amirmohammad Mirzaei 1 ,
• Bahram Jalili 2 ,
• Payam Jalili 2 &
• Davood Domiri Ganji 3  

Scientific Reports volume  14 , Article number:  14152 ( 2024 ) Cite this article

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• Fluid dynamics
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Natural convection in a square porous cavity with a partial magnetic field is investigated in this work. The magnetic field enters a part of the left wall horizontally. The horizontal walls of the cavity are thermally insulated. The wave vertical wall on the right side is at a low temperature, while the left wall is at a high temperature. The Brinkman-Forchheimer-extended Darcy equation of motion is utilized in the construction of the fluid flow model for the porous media. The Finite Element Method (FEM) was used to solve the problem’s governing equations, and the current study was validated by comparing it to earlier research. On streamlines, isotherms, and Nusselt numbers, changes in the partial magnetic field length, Hartmann number, Rayleigh number, Darcy number, and number of wall waves have been examined. This paper will show that the magnetic field negatively impacts heat transmission. This suggests that the magnetic field can control heat transfer and fluid movement. Additionally, it was shown that heat transfer improved when the number of wall waves increased.

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Introduction.

Natural convection is vital in various engineering applications, including heat transfer in electronic devices, energy-efficient building design, and thermal management in industrial processes. To design and optimize such systems, it is important to understand and predict the fluid flow and heat transfer properties in complex geometries 1 , 2 , 3 , 4 , 5 , 6 .

Several studies have focused on the influences of magnetic fields on heat transfer and fluid flow in square wavy cavities. Researchers have examined the influence of different parameters, for instance, magnetic field strength, orientation, and boundary conditions, on the convective heat transfer process. Sreedevi et al. 7 examined the behavior of nanofluid in a square cavity considering magnetic field and thermal radiation. The finite difference approach was used to solve the governing equations. They found that the maximum average value of the Nusselt number can be obtained with higher values of the Rayleigh number. To study the natural convection in a cavity where an inclined oval heater is exposed to a magnetic field, Dogonchi et al. 8 performed a numerical investigation using the CVFEM method. According to their findings, improving the Hartmann number reduced the average Nusselt at a constant Rayleigh number. Li et al.’s research 9 concentrated on convective heat transfer in an alumina water nanofluid-filled square cavity that was angled toward the horizon. In addition to convective heat transfer, they also considered radiative heat transfer. The study also analyzed entropy production. A hot circular baffle was placed inside the cavity under a fixed horizontal magnetic field. They understood that with the increase of the Rayleigh number from 103 to 106, the pace of heat transfer improved 4.5 times, and the entropy produced also increased with the development of the Rayleigh number. Furthermore, as the magnetic field strength (Ha) increased, there was a notable decrease in both the average Nusselt number (Nu) and the amount of entropy generated. Specifically, the Nu decreased by 45%, while the generated entropy diminished by 35%. In order to study the entropy generation and heat transfer in a porous corrugated cavity filled with an integrated nanofluid under the influence of natural convection, Dogonchi et al. 10 conducted a numerical study using the finite element method (FEM). This study shows that raising the Rayleigh number (Ra), and Darcy number (Da) is necessary to raise the thermal-natural convection rate. The results showed that complete irreversibility is established with Ra and Da, while it is reduced with Ha. Also, this analysis provides valuable insights into entropy production and convective heat transfer characteristics in such complex systems. Jalili et al. 11 , 12 , 13 , 14 , 15 , 16 , 17 did several works using the FEM method in this field. Mahmoudi 18 conducted theoretical and numerical research to investigate natural convection heat transfer in a porous cavity exposed to a magnetic field. This study used the scale analysis method to create a correlation to predict the heat transfer. A numerical study was also conducted using the Lattice Boltzmann method to validate the theoretical findings. The numerical simulations’ results were consistent with the theoretical predictions, indicating the proposed correlation’s correctness and reliability. This research contributes to a better understanding of heat transfer in porous cavities with magnetic field effects. Izadi et al. 19 examined the influence of two non-constant magnetic sources on the natural convection of a magnetic nanofluid in a porous medium. They discovered that as the intensity ratio between the two magnetic sources grew along with a rise in the Rayleigh (Ra) numbers, the Nusselt number initially dropped. This suggests that the convection rate decreases when the magnetic field strength becomes more asymmetric. It highlights the complex interaction between magnetic fields and natural convection in porous media and provides further insights into the behavior of such systems. Massoudi et al. 20 studied the physical properties of natural convection and radiation heat transfer in a nonagon cavity with a variable magnetic field length in a porous medium with nanofluid exposed to a uniform magnetic field. In order to find out how metal foam and rotation angle affected the natural convection of nanofluids in a hollow exposed to a magnetic field, Qi et al. 21 conducted an experiment. Their research demonstrated that a horizontal magnetic field lessens the cavity’s heat transfer, but a vertical direction improves it. The horizontal magnetic field with more force brings down the number of Nusselt, and the vertical magnetic field with higher strength enhances heat transfer, resulting in a larger Nusselt number. The numerical examination of natural convection in an F-shaped cavity with a horizontal periodic magnetic field containing non-Newtonian silver nanofluid in a porous medium was conducted by AK Hussein et al. 22 . It was noticed that heat transfer improves with increasing solid volume fraction (φ), Darcy number (Da), and Rayleigh number (Ra). Izadi et al. 23 numerically studied nanofluid’s natural convection in a porous medium influenced by a nonuniform magnetic field. Buoyancy forces, Lorentz forces, and magnetism influence the hybrid nanofluid. They found that in a porous medium at high Da, the Nusselt number falls as the porosity coefficient rises, but the Nusselt number is unaffected by the porosity coefficient of the porous medium at low and high values of Da and Ra, respectively. At low Rayleigh numbers, with the change of porosity and permeability coefficient, natural convection heat transfer change is not noticeable. Hashemi et al. 24 studied the natural convection of micropolar copper–water nanofluid in a porous chamber that produces heat. Their investigation focused on the thermal and dynamic characteristics of the nanofluid in a square chamber, where heat is produced in both the fluid and solid phases of the porous medium. They employed the Galerkin finite element method with a nonuniform structured grid to solve the governing equations and the Darcy model to simulate the flow dynamics. Finally, they showed the effect of various dimensionless parameters on velocity, temperature, and rotation. Fenghua Li et al. 25 explored the flow behavior of hybrid nanomaterials inside a permeable cavity under the influence of magnetic force. They considered the effects of permeability and external force in the Navier–Stokes equations to simulate the free convection of hybrid nanomaterials. They also evaluated the effect of various factors such as Darcy number, Hartmann number, and radiation on the fluid flow. The steady-state flow of magnetized nanofluid in the wavy cavity with radiation was examined by Nong et al. 26 . The control volume finite element (CVFE) technique is used to estimate numerical modeling. The behavior of streamlines, average Nusselt number, and isotherms was shown to be affected by magnetic force, Rayleigh number, porosity coefficient, and shape factors of nanoparticles according to them. The natural convection of a micropolar fluid was examined by Nikita et al. 27 . They reviewed their study in the wave cavity. They discussed how the flow patterns, temperature fields, and average Nusselt number in the hot corrugated wall were affected by various characteristics, including the Rayleigh number, Prandtl number, wave number, and vortex viscosity parameter. Their work is based on partial differential equations constructed in non-dimensional variables and solved with second-order precision using a finite difference approach. Mahmoud et al. 28 focused on investigating fluid flow and heat transfer in a porous media when subjected to a magnetic field. They investigated the effect of different parameters such as Darcy number, porosity parameter, radiation parameter, and Richardson number on heat transfer and flow characteristics. They applied numerical techniques and the Galerkin weighted residual finite element approach to solve the governing equations. According to their findings, there is a positive correlation between these factors and heat transfer, while an increase in the slant angle of the magnetic field causes a slight increase in velocity. In an effort to improve heat transfer, Tusi et al. 29 examined the natural motion of a water fluid containing nanocopper particles in a square cavity that was partially filled with porous media. They utilized the Darcy-Brinkman-Forchheimer relationship for fluid flow via porous media and the two-phase mixture model for simulating nanofluid flow. Additionally, they looked into how fluid flow and heat transfer were affected by the concentration of nanoparticles, Rayleigh and Darcy numbers, and the thickness ratio of the porous layer. Geridonmez et al. 30 , 31 , 32 investigated the mathematical analysis of natural convection flow in a square cavity that is exposed to a constant magnetic field. The RBF method for spatial derivatives and the backward Euler method for time derivatives are used to discretize the governing dimensionless equations. The findings demonstrate that the Lorentz force significantly reduces fluid flow and heat transfer in the affected area. In summary, this study provides significant findings regarding the aforementioned aspects. Convective heat transfer is decreased, and fluid flows more slowly as the influence area grows because of the increase in Lorentz force. As they concluded, the applied magnetic field is capable of controlling fluid flow and heat transfer. In an effort to enhance heat transfer, Several investigations were carried out to examine a cavity under various circumstances when a magnetic field was present 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 . Alsabery et al. 48 , 49 , 50 did several works in the field of heat transfer inside the porous cavity.

This paper explores the natural convection in a square, wavy, porous cavity with a partial magnetic field. Our objective is to analyze the fluid flow patterns and heat transfer behavior within the cavity through numerical simulations and mathematical modeling. This study supports existing knowledge by providing information on the influence of external factors, such as magnetic fields and porous media, on natural convection phenomena.

Problem formulation

A cavity with porous material and a partially imposed magnetic field is being considered for a laminar, incompressible natural convection flow. The configuration of the problem can be seen in Fig.  1 . The cavity’s top and bottom walls are thermally insulated, while the left wall is the heat source (T = T h ), and the right wavy wall is the cold boundary (T = T c ). The cavity’s length (L) and height (H) are in unity. The values of the parameters related to the right wall are: a = 0.9, b = 0.1, and \(k{\prime}\) is the wave number. The porous material inside the cavity is uniform and has isotropic properties. The fluid and porous material reach a local thermal equilibrium. The current analysis does not account for the effects of radiation effects, viscous dissipation, Joule heating, or the induced magnetic field.

Definition of problem geometry.

Darcy’s law alone may not be sufficient in various scenarios, such as when dealing with more porous materials, high velocities, or high Reynolds number effects. In 1901, Forchheimer introduced the concept of a quadratic drag term (also known as the inertial or Forchheimer term) to Darcy’s law. Later, Brinkmann further expanded on the model to incorporate larger porosity and account for viscous effects. It’s worth mentioning that this study excludes certain factors, such as induced magnetic fields and Joule heating.

Brinkman-Forchheimer-extended Darcy’s model is utilized in this study to analyze the porous material within the cavity. The problem’s configuration involves a laminar, incompressible natural convection flow with a partially applied magnetic field. This study’s governing equations are the continuity, momentum, and energy expressed in the u-v-p-T form. These nonlinear equations capture the complex dynamics of the system and provide insights into the fluid flow, pressure distribution, and temperature distribution within the porous material-filled cavity, as follows 31 :

In which \(\mu \) corresponds to the dynamic viscosity of the fluid, the effective dynamic viscosity is \({\mu }_{e}\) , the fluid’s density is \({\rho }_{f}\) , the norm of the velocity vector \(\sqrt{({u}^{2}+{v}^{2})}\) is \(\left|\mathbf{u}\right|\) , \(g\) is the gravitational acceleration, the thermal expansion coefficient is \(\beta \) , the pressure is defined by \(p\) , the porosity of a porous medium is measured by \({\epsilon }_{p}\) , the magnitude of the applied magnetic field is referred to as \({B}_{0}\) , the fluid’s electrical conductivity is defined as σ, the effective thermal diffusivity is \({{\alpha }}_{e}=\frac{{k}_{e}}{{({\rho }_{f}{c}_{p})}_{f}}\) , the effective thermal conductivity is \({k}_{e}={\epsilon }_{p}{k}_{f}+(1-{\epsilon }_{p}){k}_{s}\) , the specific heat at constant pressure is \({c}_{p}\) , the coefficient of form is \({c}_{F}=\frac{1.75(1-{\epsilon }_{p})}{{d}_{p}{{\epsilon }_{p}}^{3}}\) , \(K=\frac{{{d}_{p}}^{2}{{\epsilon }_{p}}^{3}}{150{(1-{\epsilon }_{p})}^{2}}\) refers to the permeability of the porous medium, \({d}_{p}\) is the size of a solid particle in a porous medium, and the heat capacity ratio \({\upsigma }_{h}=\frac{{\epsilon }_{p}{({\rho }_{f}{c}_{p})}_{f} +(1-{\epsilon }_{p}){({\rho }_{f}{c}_{p})}_{s}}{{({\rho }_{f}{c}_{p})}_{f}}\) , is considered one in this model. Additionally, the fluid \((f)\) and solid \((s)\) are assumed to have the same thermal conductivity and thermal diffusivity. That means, \({k}_{e}={k}_{f}={k}_{s}\) and \({\alpha }_{e}={\alpha }_{f}=\alpha \) . In addition, \({\mu }_{e}=\mu \) is assumed. Air fluid was used for the present work. The physical properties of air are given in Table 1 .

The boundary conditions are outlined below:

The dimensionless quantities are defined below:

With the help of non-dimensional quantities, they are replaced in the original Eqs. ( 1 )-( 4 ), and assuming the elimination of prime symbols, the following dimensionless equations are obtained.

where, \({c}_{g}=\frac{ 1.75}{\sqrt{150}{{\epsilon }_{p}}^{3/2}}\) . The dimensionless parameters Prandtl, Darcy, Rayleigh, and Hartmann numbers are as follows:

where \(\Delta T={T}_{h}-{T}_{c}\) . By setting the stream function \(\psi \) as \(u=\partial \psi /\partial y, v=-\partial \psi /\partial x\) , it eliminates the continuity equation due to satisfying the continuity condition, and the pressure term is removed by using the vorticity definition \(\omega =\nabla \times \mathbf{u}\) in the momentum equations. Non-dimension equations are deduced from the effects of stream and vorticity functions:

In the case of the incoming magnetic field, \({\delta }_{B}\) , is defined as:

The reduced boundary conditions are as follows:

The average Nusselt number for the warm wall is determined as follows:

Method of solution and numerical results

The Eqs. ( 12 )-( 14 ) have been solved using the Finite Element Method with boundary conditions (16). The reliability of this method is a result of its high strength and flexibility. Once the initial meshing is done, the solution continues continuously. To achieve acceptable accuracy, it may also alter the mesh structure. This process continues until the convergence condition is met. The convergence condition for the present work is to reach 10 –5 accuracy. The Finite Element Method (FEM) operates on a fundamental principle: breaking down a complex problem domain into discrete sub-regions that are called finite elements. Each of these elements possesses a distinct geometry and is characterized mathematically through a set of equations that describe the behavior of the system within that particular section. This approach simplifies the solution of complex governing equations, which would be arduous or nearly impossible to solve manually. Shape functions are used in FEM to interpolate element-level solutions based on nodal values. In this study, FEM serves as a numerical tool for solving the governing equations related to fluid flow and heat transfer due to natural convection of a square wavy cavity with a magnetic field and porous media. To improve understanding of the numerical solution technique, Fig.  2 displays the flowchart of the numerical method. Through FEM, a more precise and detailed solution for the natural convection of a square wavy cavity with a magnetic field and porous media can be obtained. Figure  3 shows a good match between the streamlines and isotherm lines compared to the work of Geridonmez et al. 31 .

The flowchart of the numerical method.

Comparison of streamlines (on the left) and isotherm lines (on the right) at Ra = 10 5 , Da = 0.01, ϵ p  = 0.9, Ha = 50, L B  = 0.7.

In order to obtain a mesh-independent solution, mesh independence is investigated. The grid-independent solution was examined by providing a solution for natural convection in a square cavity with one side having a cosine wave with a magnetic field applied to a portion of the square side. Five of the grids have been tested. As can be seen in Table 2 , with the number of cells 3006, the solution becomes independent and reaches an accuracy of 10 –5 . The optimal mesh is shown in Fig.  4 .

Grid of geometry.

Results and discussions

Figure  5 shows the impact of the entranced magnetic field length on the left wall. As the magnetic field length increases, the main vortex is compressed towards the lower left nook of the cavity, but the upper part of the vortex is stretched towards the upper right nook of the cavity because of the increased Lorentz force area introduced by the magnetic field. The vortex’s strength decreases with the magnetic field’s length. On the other hand, the buoyancy force slows down. The temperature difference in the isotherms decreases. In addition, the isotherm lines tend to flatten, indicating the magnetic force’s inhibition of natural convection.

Impact of magnetic field length on streamlines and isotherm lines with Ra = 10 5 , Da = 0.01, ϵ p  = 0.9, Ha = 50, \(k{\prime}\) = 2; from top to bottom Nu = 5.127, 4.245, 3.247, respectively.

The velocity and temperature distribution based on the variable magnetic field length can be seen in Fig.  6 . It is concluded that by raising the magnetic field length, the velocity decreases, and the temperature increases from the bottom of the cavity to its center, but this process is the opposite for the upper half. The maximum velocity u belongs to L B  = 0.5.

Velocity and temperature distribution based on magnetic field length change with Ra = 10 5 , Da = 0.01, ϵ p  = 0.9, Ha = 50, \(k{\prime}\) = 2; ( a ) u at y = 0.5, ( b ) v at x = 0.5, ( c ) θ at x = 0.5.

Figure  7 shows the effect of the Rayleigh number on streamlines and isothermal lines. L B  = 0.5, as defined, means that the magnetic field affects the left wall of the cavity from the top to the center. Therefore, the streamlines show the natural convection at the bottom of the cavity, while the upper part of the cavity shows the lagging impact of the Lorentz force. Because of the dominance of Lorentz force over the buoyancy force, the second vortex is seen in the streamlines at Ra = 10 4 , while it is not observed at Ra = 10 5 or 10 6 due to the high buoyancy force. The isotherm lines are approximately 90 degrees to the upper wall for Ra = 10 4 , 10 5 . With the increase of Rayleigh, the isotherm lines tend to bend more from the flat state, and heat penetrates more in the upper half.

Influence of Rayleigh number on streamlines and isotherm lines with Da = 0.01, ϵ p  = 0.9, Ha = 100, L B  = 0.5, \(k{\prime}\) = 2; from top to bottom Nu = 1.465, 3.6, 8.944, respectively.

The velocity and temperature distribution based on the Rayleigh number change can be seen in Fig.  8 . Raising the Rayleigh number lowered the temperature in the cavity’s lower region while increasing the maximum velocity, but this trend changes for the upper part of the cavity, and the temperature increases.

Velocity and temperature distribution based on Rayleigh number change with Da = 0.01, ϵ p  = 0.9, Ha = 100, L B  = 0.5, \(k{\prime}\) = 2; ( a ) u at y = 0.5, ( b ) v at x = 0.5, ( c ) θ at x = 0.5.

Figure  9 shows the effect of the Hartmann number on streamlines and isothermal lines. In case there’s no magnetic field due to the non-existence of Lorentz force delay, the streamlines show natural convection in all parts of the cavity. With the augmentation of the Hartmann number, the vortex tends to the lower side of the cavity. Also, the increase in Hartmann reduces the strength of the vortex.

Impact of Hartmann number on streamlines and isotherm lines with Da = 0.01, ϵ p  = 0.9, Ra = 10 5 , L B  = 0.5, \(k{\prime}\) = 2; from top to bottom Nu = 5.973, 3.6, 3.342, respectively.

The isotherm lines are almost perpendicular to the top wall for Ra = 10 5 , 10 6 . As the Rayleigh increases, the isotherm lines tend to bend less from the flat state, and heat penetrates less in the upper half, indicating that the magnetic field was controlling the heat transfer and fluid flow.

The velocity and temperature distribution due to the Hartmann number change is shown in Fig.  10 . In the absence of a magnetic field, u has the lowest value, and with the augmentation of the Hartmann number, u and v decrease. With increasing the Hartmann number, the temperature trend for the cavity’s lower half increases, but this trend is reversed for the upper half, and the temperature decreases.

Velocity and temperature distribution based on Hartmann number change with Da = 0.01, ϵ p  = 0.9, Ra = 10 5 , L B  = 0.5, \(k{\prime}\) = 2; ( a ) u at y = 0.5, ( b ) v at x = 0.5, ( c ) θ at x = 0.5.

The influence of Darcy number on streamlines and isotherm lines was examined, and it is shown as a contour in Fig.  11 . At the Darcy number close to zero, the vorticity tends to the center, and the isothermal lines are almost parallel to each other because the Lorentz force loses its effect at a low Darcy number. With the increase of Darcy, the penetration of heat increases, and as a result, the fluid moves at a higher speed, and the vortex is transferred downwards.

Impact of Darcy number on streamlines and isotherm lines with Ra = 10 5 , ϵ p  = 0.9, Ha = 50, L B  = 0.5, \(k{\prime}\) = 2; from top to bottom Nu = 1.234, 2.404, 4.245, respectively.

The velocity and temperature distribution due to the Darcy number change can be seen in Fig.  12 . With increasing the Darcy number, the magnitude of u and v increase, and it can be said that the temperature also has an increasing trend, and in Darcy 10 –4 , the temperature changes are insignificant.

Velocity and temperature distribution based on Darcy number change with Ra = 10 5 , Ha = 50, ϵ p  = 0.9, L B  = 0.5, \(k{\prime}\) = 2; ( a ) u at y = 0.5, ( b ) v at x = 0.5, ( c ) θ at x = 0.5.

The impact of wave number on streamlines and isotherm lines was studied, and the outcomes are shown in Fig.  13 . As the wave number increases, the eddy tends to rise less, and the fluid moves with a higher partial velocity. The number of waves has no noticeable effect on isothermal lines. Nonetheless, there is a certain increase in the Nusselt number as the wave number rises. The behavior of velocity in horizontal and vertical, as well as temperature distribution in the vertical case, is shown. See Fig.  14 for better understanding.

Impact of wave number on streamlines and isotherm lines with Ra = 10 5 , Da = 0.01, ϵ p  = 0.9, Ha = 50, L B  = 0.5; from top to bottom Nu = 3.978, 4.093, 4.245, 4.243, respectively.

Velocity and temperature distribution based on Wave number change with Ra = 10 5 , Ha = 50, Da = 0.01, ϵ p  = 0.9, L B  = 0.5; a) u at y = 0.5, b) v at x = 0.5, c) θ at x = 0.5.

Average Nusselt is influenced by magnetic field length, Rayleigh, Hartmann, and Darcy number, as shown in Fig.  15 . The average Nusselt is proportional to the Rayleigh and Darcy numbers and increases with the increase of Rayleigh and Darcy. At the same time, it has an inverse relationship with the Hartmann number and the length of the magnetic field. Figures  16 and 17 show the simultaneous effect of magnetic field length, Rayleigh number, and Hartmann and Darcy numbers on the average Nusselt. The average Nusselt number reached its highest point when there was no magnetic field, and the Rayleigh and Darcy numbers were high.

Impact of magnetic field length, Rayleigh, Hartmann, and Darcy numbers on Average Nusselt with Ra = 10 5 , Da = 0.01, ϵ p  = 0.9, Ha = 50, L B  = 0.5, \(k {\prime}\) = 2.

Simultaneous effect of magnetic field length and Rayleigh number on average Nusselt number with Da = 0.01, ϵ p  = 0.9, Ha = 50, \(k\) = 2.

Simultaneous impact of Hartmann and Darcy numbers on average Nusselt number with Ra = 10 5 , ϵ p  = 0.9, L B  = 0.5, \(k\) = 2.

In summary, the set of average Nusselt number and maximum psi value is given in Table 3 .

This paper presents the results of a numerical study on natural convection in a square wavy cavity in the presence of a partial magnetic field under effective factors such as magnetic field length (L B ), Rayleigh number (Ra), Hartmann number (Ha), Darcy number (Da) and wave number ( \(k{\prime}\) ). The effective parameters were investigated in L B  = 0.3–0.7, Ra = 10 4 –10 6 , Da = 0.0001–0.01, \(k{\prime}\) = 0–3 intervals. In addition, the distribution of temperature, average Nusselt, streamlines, and isotherm lines were shown for the cavity. The most important results obtained are as follows:

As the magnetic field’s length increases, heat transfer decreases. As the length of the magnetic field increases from 0.3 to 0.7, the Nusselt number decreases by 36.6%, and the value of maximum psi decreases by 34%.

As the Rayleigh number increases, the Nusselt number also increases. In fact, with the increase of the Rayleigh number, the buoyancy force increases, and the buoyancy force increases the heat transfer and fluid velocity. As the Rayleigh number increases from 10 4 to 10 6 , the Nusselt number increases more than five times, and the value of maximum psi increases more than seven times.

In the lower Rayleigh, the second vortex is formed due to the dominance of the Lorentz force over the buoyancy force.

The effect of Lorentz force on fluid flow and heat transfer increases at a high Rayleigh number.

Nusselt number decreases with increasing Hartmann number. As the Hartmann number increases, the Lorentz force increases. Lorentz force reduces heat transfer and fluid velocity. As the Hartmann number increases from 0 to 200, the Nusselt number decreases by 44%, and the value of maximum psi decreases by 38.8%.

By increasing the permeability in the porous medium, heat transfer is improved. As the Darcy number increases from 0.0001 to 0.01, the Nusselt number increases more than two times, and the value of maximum psi increases more than nine times.

At low Darcy numbers, the magnetic field becomes ineffective.

As the wave number of the right wall increases, the Nusselt number increases due to the increase of the heat transfer area. As the wave number increases from 0 to 2, the Nusselt number increases by 6.71%, and the value of maximum psi increases by 1.3%.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Abbreviations

Prandtl number

Rayleigh number

Hartmann number

Darcy number

Specific heat, J.kg -1 .K -1

Thermal conductivity, W.m −1 .K −1

Permeability,m 2

Temperature, K

Gravitational acceleration

Length of the cavity, m

Height of the cavity, m

Wave number

Magnetic field length

Average Nusselt number

Velocity, m.s -1

Dimensionless velocity

Dimensionless time

Pressure, pa

Dimensionless pressure

Magnitude of the magnetic field, N.A -1 .m 2

Cartesian coordinates,m

Dimensionless cartesian coordinates

Dimensionless temperature

Density, kg.m -3

Dynamic viscosity, Pa.S

Thermal expansion coefficient, K -1

Thermal diffusivity, m 2 .s

Dimensionless stream function

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Mirzaei, A., Jalili, B., Jalili, P. et al. Free convection in a square wavy porous cavity with partly magnetic field: a numerical investigation. Sci Rep 14 , 14152 (2024). https://doi.org/10.1038/s41598-024-64850-7

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