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research-article

Integrated radial basis functions (IRBFs) to simulate nonlinear advection–diffusion equations with smooth and non-smooth initial data

Authors:

Ali Ebrahimijahan,

Mostafa AbbaszadehAuthors Info & Claims

Engineering with Computers, Volume 38, Issue 2

Pages 1071 - 1106

https://doi.org/10.1007/s00366-020-01039-2

Published: 01 April 2022 Publication History

Abstract

In this article, a meshfree method for the numerical solution of conversation law equations is considered. Some problems which have shock such as advection problems are not properly solved by radial basis function collocation meshfree method. Therefore, we use the integrated radial basis function (IRBF) method for some of these problems. In the current study, the governing models have been discretized by IRBF technique in the spatial direction and by finite difference approximation for time variable. This converts the main problem to a system of nonlinear ordinary differential equations (ODEs). Furthermore, the obtained ODEs will be solved by Runge–Kutta technique. This is the meshless method of lines technique. Numerical examples indicate the acceptable accuracy, proficiency and easy implementation of the presented method.

References

[1]

Abbaszadeh M and Dehghan M The reproducing kernel particle Petrov–Galerkin method for solving two-dimensional nonstationary incompressible Boussinesq equations Eng Anal Bound Elem 2019 106 300-308

[2]

Abbaszadeh M, Dehghan M (2019) A meshless numerical investigation based on the RBF-QR approach for elasticity problems. AUT J Math Comput (AJMC) (in press)

[3]

Abbaszadeh M and Dehghan M The interpolating element-free Galerkin method for solving Korteweg-de Vries-Rosenau-regularized long-wave equation with error analysis Nonlinear Dyn 2019 96 1345-1365

[4]

Abedian R, Adibi H, and Dehghan M A high-order weighted essentially non-oscillatory (WENO) finite difference scheme for nonlinear degenerate parabolic equations Comput Phys Commun 2013 184 1874-1888

[5]

Anderson D, Tannehill JC, and Pletcher RH Computational fluid mechanics and heat transfer 2016 London CRC Press

[6]

Benkhaldoun F, Sari S, and Seaid M A family of finite volume Eulerian-Lagrangian methods for two-dimensional conservation laws J Comput Appl Math 2015 285 181-202

[7]

Dag I, Canivar A, and Sahin A Taylor–Galerkin and Taylor–Collocation methods for the numerical solutions of Burgers’ equation using B-splines Commun Nonlinear Sci Numer Simul 2011 16 2696-2708

[8]

Dehghan M and Abbaszadeh M The use of proper orthogonal decomposition (POD) meshless RBF-FD technique to simulate the shallow water equations J Comput Phys 2017 351 478-510

[9]

Dehghan M and Shokri A A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions Math Comput Simul 2008 79 3 700-715

[10]

Dehghan M and Abbaszadeh M The space-splitting idea combined with local radial basis function meshless approach to simulate conservation laws equations Alex Eng J 2018 57 2 1137-1156

[11]

Dehghan M and Jazlanian R On the total variation of a third-order semi-discrete central scheme for 1D conservation laws J Vib Cont 2010 17 9 1348-1358

[12]

Dehghan M and Jazlanian R A high-order non-oscillatory central scheme with non-staggered grids for hyperbolic conservation laws Comput Phys Commun 2011 182 1284-1294

[13]

Dehghan M and Shokri A A numerical method for two-dimensional Schrodinger equation using collocation and radial basis functions Comput Math Appl 2007 54 136-146

[14]

Dehghan M Numerical solution of the three-dimensional advection–diffusion equation Appl Math Comput 2004 150 1 5-19

[15]

Driscoll TA and Heryudono ARH Adaptive residual subsampling methods for radial basis function interpolation and collocation problems Comput Math Appl 2007 53 6 927-939

[16]

Duijn CJV, Peletier LA, and Pop IS A new class of entropy solutions of Buckley–Leverett equation SIAM J Math Anal 2007 39 2 507-536

[17]

Fasshauer GE and McCourt M Stable evaluation of Gaussian RBF interpolants SIAM J Sci Comput 2012 34 2 737-762

[18]

Fasshauer GE Meshfree approximation methods with MATLAB 2007 Singapore World Scientific

[19]

Fornberg B and Lehto E Stabilization of RBF-generated finite difference methods for convective PDEs J Comput Phys 2011 230 2270-2285

[20]

Fornberg B, Lehto E, and Powell C Stable calculation of Gaussian-based RBF-FD stencils Comput Math Appl 2013 65 627-637

[21]

Ho PL, Le CV, and Tran-Cong T Limit state analysis of reinforced concrete slabs using an integrated radial basis function based mesh-free method Appl Math Model 2018 53 1-11

[22]

Ho PL, Le CV, and Tran-Cong T Displacement and equilibrium mesh-free formulation based on integrated radial basis functions for dual yield design Eng Anal Bound Elem 2016 71 92-100

[23]

Hardy RL Multiquadric equations of topography and other irregular surfaces J Geophys Res 1971 76 1905-1915

[24]

Lin J, Zhang C, Sun L, and Lu J Simulation of seismic wave scattering by embedded cavities in an elastic half-plane using the novel singular boundary method Adv Appl Math Mech 2018 10 2 322-342

[25]

Lin J, Reutskiy YS, and Lu J A novel meshless method for fully nonlinear advection–diffusion–reaction problems to model transfer in anisotropic media Appl Math Comput 2018 339 459-476

[26]

Lin J, Xu Y, Zhang Y (2020) Simulation of linear and nonlinear advection-diffusion-reaction problems by a novel localized scheme. Appl Math Lett 99. Article ID 106005

[27]

Lin J, Reutskiy S, (2020) A cubic B-spline semi-analytical method for 3D steady-state convection–diffusion–reaction problems. Appl Math Comput 371. Article ID 124944

[28]

Li XK and Zienkiewicz OC A numerical model for immiscible two-phase fluid flow in a porous medium and its time domain Int J Numer Meth Eng 1990 30 6 1195-1212

[29]

Li XK, Zhang S, Wang Y, and Chen H Analysis and application of the element-free Galerkin method for nonlinear sine-Gordon and generalized sinh-Gordon equations Comput Math Appl 2016 71 1655-1678

[30]

Lin SY, Wu TM, and Chin YS Upwind finite-volume method with a triangular mesh for conservation laws J Comput Phys 1993 107 324-337

[31]

Kansa EJ, Aldredge RC, and Ling L Numerical simulation of two-dimensional combustion using mesh-free methods Eng Anal Bound Elem 2009 33 940-950

[32]

Jiwari R A hybrid numerical scheme for the numerical solution of the Burgers’ equation Comput Phys Commun 2015 188 59-67

[33]

Powell MJD (1992) The theory of radial basis function approximation in 1990. Adv Numer Anal 105–210

[34]

Mai-Duy N and Tanner RI A collocation method based on one-dimensional RBF interpolation scheme for solving PDEs Int J Numer Meth H 2007 17 2 165-186

[35]

Mai-Duy N and Tran-Cong T A multidomain integrated radial basis function collocation method for elliptic problems Numer Meth Part Differ Equ 2008 24 5 1301-1320

[36]

Mai-Duy N Compact approximation stencils based on integrated flat radial basis functions Eng Anal Bound Elem 2017 74 79-87

[37]

Mai-Duy N A compact 9 point stencil based on integrated RBFs for the convection-diffusion equation Appl Math Model 2014 38 4 1495-1510

[38]

Mai-Duy N and Tran-Cong T Compact local integrated-RBF approximations for second-order elliptic differential problems J Comput Phys 2011 230 12 4772-4794

[39]

Muller F, Jenny P, and Meyer DW Multilevel Monte Carlo for two phase flow and Buckley–Leverett transport in random heterogeneous porous media J Comput Phys 2013 250 685-702

[40]

Sarra SA A local radial basis function method for advection diffusion reaction equations on complexly shaped domains Appl Math Comput 2012 218 9853-9865

[41]

Sarra SA Regularized symmetric positive definite matrix factorizations for linear systems arising from RBF interpolation and differentiation Eng Anal Bound Elem 2014 44 76-86

[42]

Sarra SA Radial basis function approximation methods with extended precision floating point arithmetic Eng Anal Bound Elem 2011 35 68-76

[43]

Sarra SA Adaptive radial basis function methods for time dependent partial differential equations Appl Numer Math 2005 54 79-94

[44]

Sarra SA, Kansa EJ (2009) Multiquadric radial basis function approximation methods for the numerical solution of partial differential equations. Adv Comput Mech 2(2)

[45]

Sarra SA Integrated multiquadric radial basis function approximation methods Comput Math Appl 2006 51 1283-1296

[46]

Sethian JA, Chorin AJ, and Concus P Numerical solution of the Buckley–Leverett equations 1983 California SPE Reservoir Simulation Symposium, 15–18 November

[47]

Seydaoglu M, Erdogan U, and Ozis T Numerical solution of Burgers equation with high order splitting methods J Comput Appl Math 2016 291 410-421

[48]

Shu C and Wu YL Integrated radial basis functions-based differential quadrature method and its performance Int J Numer Meth Fluids 2007 5 6 969-984

[49]

Wang Y and Kao CY Central schemes for the modified Buckley–Leverett equation J Comput Sci 2013 4 12-23

[50]

Wazwaz AM Multiple-front solutions for the Burgers equation and the coupled Burgers equations Appl Math Comput 2007 190 1198-1206

[51]

Wu YS, Pruess K, Chen ZX (1990) Buckley–Leverett flow in composite porous media. SPE Advanced Technology Series

[52]

Zhang L, Ouyang J, Wang X, and Zhang X Variational multiscale element-free Galerkin method for 2D Burgers equation J Comput Phys 2010 229 7147-7161

Cited By

Jiwari R(2022)Local radial basis function-finite difference based algorithms for singularly perturbed Burgers’ modelMathematics and Computers in Simulation10.1016/j.matcom.2022.02.024198:C(106-126)Online publication date: 1-Aug-2022
https://dl.acm.org/doi/10.1016/j.matcom.2022.02.024

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Integrated radial basis functions (IRBFs) to simulate nonlinear advection–diffusion equations with smooth and non-smooth initial data

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Information & Contributors

Information

Published In

cover image Engineering with Computers

Engineering with Computers Volume 38, Issue 2

Apr 2022

979 pages

ISSN:0177-0667

EISSN:1435-5663

Issue’s Table of Contents

© Springer-Verlag London Ltd., part of Springer Nature 2020.

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 April 2022

Accepted: 06 May 2020

Received: 10 February 2020

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Cited By

Jiwari R(2022)Local radial basis function-finite difference based algorithms for singularly perturbed Burgers’ modelMathematics and Computers in Simulation10.1016/j.matcom.2022.02.024198:C(106-126)Online publication date: 1-Aug-2022
https://dl.acm.org/doi/10.1016/j.matcom.2022.02.024

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