Adaptive radial basis function for time dependent partial differential equations | [Journal of Prime Research in Mathematics • 2017]

Author(s):

1. SYEDA LAILA NAQVI: Abdus Salam School of Mathematical Sciences, Lahore. Pakistan.

2. JEREMY LEVESLEY: Department of Mathematics, University of Leicester, UK.

3. SALMA ALI: Shaheed Benazir Bhutto Women University, Pakistan

Abstract:

We propose a meshless adaptive solution of the time-dependent partial di erential equations (PDE) using radial basis functions (RBFs). The approximate solution to the PDE is obtained using multiquadrics (MQ). We choose MQ because of its exponential convergence for su ciently smooth functions. The solution of partial di erential equations arising in science and engineering frequently have large variations occurring over small portion of the physical domain. The challenge then is to resolve the solution behaviour there. For the sake of e ciency we require a ner grid in those parts of the physical domain whereas a much coarser grid can be used otherwise. Local scattered data reconstruction is used to compute an error indicator to decide where nodes should be placed. We use polyharmonic spline approximation in this step. The performance of the method is shown for numerical examples of one dimensional Kortwegde-Vries equation, Burger's equation and Allen-Cahn equation.

Page(s): 90-106

DOI: DOI not available

Published: Journal: Journal of Prime Research in Mathematics, Volume: 13, Issue: 1, Year: 2017

Keywords:

Keywords are not available for this article.

References:

[1] Anthonissen M.J.H. .2001 .Local defect correction techniques: analysis and application to combustion. , : .

[2] Behrens J.,Iske A. .2002 .Radial basis functions and partial di erential equations. Comput. Math. Appl., 319(3-5) : 327.

[3] Behrens J.,Iske A.,M. A. .2003 .Kaser. Adaptive meshfree method of backward characteristics for nonlinear transport equations. Notes Comput. Sci. Eng, 21 : 36.

[4] Behrens J.,Iske A. .2003 .Adaptive meshfree method of backward characteristics for nonlinear transport equations. Notes Comput. Sci. Eng, 21 : 36.

[5] Bertozzi A.L.,Brenner M.P.,Dupont T. F.,Kadano L. P. .1994 .Singularities and similarities in interface ows. In Trends and perspectives in applied mathematics, 155 : 208.

[6] Burgers J. M. .1948 .A mathematical model illustrating the theory of turbulence. In Advances in Applied Mechanics, 171 : 199.

[7] Cruza P.,Alvesb M.,Magalhesa F.D.,Mendes A. .2003 .Solution of hyperbolic pdes using a stable adaptive multiresolution method. Chemical Enginerring Science, 58 : 1792.

[8] Davydov O.,Oanh D.T. .2011 .Adaptive meshless centres and RBF stencils for Poisson equation. J. Comput. Phys., 287(2) : 304.

[9] Driscoll T.A.,Heryudono A.R.H. .2007 .Adaptive residual subsampling methods for radial basis function interpolation and collocation problems. Comput. Math. Appl., 927(6) : 939.

[10] Duchon J. .1977 .Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In Constructive theory of functions of several variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 571 : 100.

[11] Hon Y. C.,Schaback R.,Zhou X. .2003 .An adaptive greedy algorithm for solving large RBF collocation problems. Numer. Algorithms, 13(1) : 25.

[12] Hong C.,Li K.,Wen-Jun L. .2010 .Numerical solution of pdes via integrated radial basis function networks with adaptive training algorithm. , 11 : 860.

[13] Iske A. .2007 .Particle ow simulation by using polyharmonic splines. In Algorithms for approximation. , 83 : 102.

[14] Libre N.A.,Emdadi A.,Shekarchi M. Kansa E.J.,Rahimian M. .2008 .A fast adaptive wavelet scheme in RBF collocation for nearly singular potential PDEs. CMES Comput. Model. Eng. Sci., 263(3) : 284.

[15] Madych W. R.,Nelson S. A. .1992 .Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation. J. Approx. Theory, 94(1) : 114.

[16] Meinguet J.,Oberwolfach J. .1979 .Basic mathematical aspects of surface spline interpolation. In Numerische Integration (Tagung, 211 : 220.

[17] Meinguet J.,Reidel J. .1979 .An intrinsic approach to multivariate spline interpolation at arbitrary points. of NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., 163 : 190.

[18] Roussel O.,Schneider K.,Tsigulin A.,Bockhorn H. .2003 .A conservative fully adaptive multiresolution algorithm for parabolic PDEs. J. Comput. Phys., 493(2) : 523.

[19] Sarra S.A. .2005 .Adaptive radial basis function methods for time dependent partial di erential equations. Appl, 79(1) : 94.

[20] Schaback R. .1995 .Error estimates and condition numbers for radial basis function interpolation. Adv. Comput. Math., 251(3) : 264.

[21] Shen Q. .2009 .A meshless method of lines for the numerical solution of KdV equation using radial basis functions. Eng. Anal. Bound, 1171(10) : 1180.

[22] Vrankar L.,Kansa E. J.,Ling L.,Runovc F.,Turk G. .2010 .Moving-boundary problems solved by adaptive radial basis functions. Comput. & Fluids, 1480(9) : 1490.

[23] Wendland H. .2005 .Scattered data approximation. Computational Mathematics, 17 : .

[24] Xue J.,Liao G. .2006 .Least-squares nite element method on adaptive grid for PDEs with shocks. Numer. Methods Partial Di erential Equations, 114(1) : 127.

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