Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-27T00:08:11.569Z Has data issue: false hasContentIssue false
  • English
  • Français

A Sylvester-Based IMEX Method via Differentiation Matrices for Solving Nonlinear Parabolic Equations

Published online by Cambridge University Press:  03 June 2015

Francisco de la Hoz  and
Fernando Vadillo
Francisco de la Hoz*
Affiliation:
Department of Applied Mathematics, Statistics and Operations Research, Faculty of Science and Technology, University of the Basque Country UPV/EHU, Barrio Sarriena S/N, 48940 Leioa, Spain
Fernando Vadillo*
Affiliation:
Department of Applied Mathematics, Statistics and Operations Research, Faculty of Science and Technology, University of the Basque Country UPV/EHU, Barrio Sarriena S/N, 48940 Leioa, Spain
*
Corresponding author.Email:francisco.delahoz@ehu.es
Email:fernando.vadillo@ehu.es
Get access

Abstract

In this paper we describe a new pseudo-spectral method to solve numerically two and three-dimensional nonlinear diffusion equations over unbounded domains, taking Hermite functions, sinc functions, and rational Chebyshev polynomials as basis functions. The idea is to discretize the equations by means of differentiation matrices and to relate them to Sylvester-type equations by means of a fourth-order implicit-explicit scheme, being of particular interest the treatment of three-dimensional Sylvester equations that we make. The resulting method is easy to understand and express, and can be implemented in a transparent way by means of a few lines of code. We test numerically the three choices of basis functions, showing the convenience of this new approach, especially when rational Chebyshev polynomials are considered.

Keywords

65M2065M7002.70.Hm02.30.JrSemi-linear diffusion equationspseudo-spectral methodsdifferentiation matricesHermite functionssinc functionsrational Chebyshev polynomialsIMEX methodsSylvester equationsblow-up
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 4 , October 2013 , pp. 1001 - 1026
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ascher, U.M., Ruuth, S.J., and Wetton, B.T.R., Implicit-Explicit Methods for Time-Dependent Partial Differential Equations, SIAM Journal on Numerical Analysis 32 (1995), no. 3, 797–823.Google Scholar
[2]Bartels, R.H. and Stewart, G.W., Solution of the Matrix Equation AX+XB=C, Communications of the ACM 15 (1972), no. 9, 820–826.Google Scholar
[3]Boyd, J.P., The Optimization of Convergence for Chebyshev Polynomial Methods in an Unbounded Domain, J. Comput. Phys. 45 (1982), no. 1, 43–79.Google Scholar
[4]Boyd, J.P., Spectral Methods Using Rational Basic Functions on an Infinite Interval, J. Comput. Phys. 69 (1987), no. 1, 112–142.Google Scholar
[5]Boyd, J.P., Chebyshev and Fourier Spectral Methods, second (revised) ed., Dover, 2001.Google Scholar
[6]Canuto, C., Hussain, M.Y., Quarteroni, A., and Zang, T.A., Spectral Methods in Fluid Dynamics, Springer, 1988.CrossRefGoogle Scholar
[7]Canuto, C., Hussaini, M.Y., Quarteroni, A., and Zang, T.A., Spectral Methods: Fundamentals in Single Domains, Springer, 2007.Google Scholar
[8]de la Hoz, F., A Schur-based algorithm for solving N-dimensional matrix equations, preprint.Google Scholar
[9]de la Hoz, F. and Vadillo, F., A Numerical Simulation for the Blow-up of Semi-linear Diffusion Equations, Int. J. Comput. Math. 86 (2009), no. 3, 493–502.Google Scholar
[10]de la Hoz, F. and Vadillo, F., Numerical simulation of the N-dimensional sine-Gordon equation via operational matrices, Computer Physics Communications 183 (2012), no. 4, 864–879.CrossRefGoogle Scholar
[11]Fornberg, B., A Practical Guide to Pseudospectral Methods, Cambridge University Press, 1998.Google Scholar
[12]Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P, and Zimmermann, P., Mpfr: A multiple-precision binary floating-point library with correct rounding, ACM Transactions on Mathematical Software (TOMS) 33 (2007), no. 2, Article no. 13.CrossRefGoogle Scholar
[13]Fujita, H., On the blowing up of solutions of the Cauchy problem for ut =∆u+u 1+α, J Fac Sci Univ Tokyo Sect I 13 (1966), 109–124.Google Scholar
[14]Galaktionov, V.A. and Vázquez, J.L., The problem of blow-up in nonlinear parabolic equations, Discrete and Continuous Dynamical Systems 8 (2002), no. 2, 339–433.CrossRefGoogle Scholar
[15]Gottlieb, D. and Orszag, S.A., Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, 1977.Google Scholar
[16]Grosch, C.E. and Orszag, S.T., Numerical Solution of Problems in Unbounded Regions: Coordinate Transforms, J. Comput. Phys. 25 (1977), no. 3, 273–295.CrossRefGoogle Scholar
[17]Haidvogel, D.B. and Zang, T., The Accurate Solution of Poisson’s Equation by Expansion in Chebyshev Polynomials, J. Comput. Phys. 30 (1979), no. 2, 167–180.Google Scholar
[18]Herrero, M.A. and Velázquez, J.J.L., Blow-up behavior of one-dimensional semilinear parabolic equations, Ann. Inst. Henri Poincare 10 (1993), no. 2, 131–189.Google Scholar
[19]Higham, N.J., Accuracy and Stability of Numerical Algorithms, SIAM, 1996.Google Scholar
[20]Higham, N.J., Functions of Matrices. Theory and Computation, SIAM, 2008.Google Scholar
[21]Hu, D.Y. and Reichel, L., Krylov-Subspace Methods for the Sylvester Equation, Linear Algebra and its Applications 172 (1992), no. 15, 283–313.Google Scholar
[22]Kassam, A.-K. and Trefethen, L.N., Fourth-Order Time-Stepping for Stiff PDEs, SIAM J. Sci. Comput 26 (2005), no. 4, 1214–1233.Google Scholar
[23]Levine, H.A., The Role of Critical Exponents in Blowup Theorems, SIAM Review 32 (1990), no. 2, 262–288.Google Scholar
[24]Li, Ben-Wen, Tian, Shuai, Sun, Ya-Song, and Hu, Zhang-Mao, Schur-decomposition for 3D matrix equations and its application in solving radiative discrete ordinates equations discretized by Chebyshev collocation spectral method, Journal of Computational Physics 229 (2010), no. 4, 1198–1212.Google Scholar
[25]Ma, Heping, Sun, Weiwei, and Tang, Tao, Hermite Spectral Methods with a Time-Dependent Scaling for Parabolic Equations in Unbounded Domains, SIAM Journal on Numerical Analysis 43 (2006), no. 1, 58–75.Google Scholar
[26]Martin, C.D.M. and Van Loan, C.F., Solving real lineal systems with the complex Schur decomposition, SIAM J. of Matrix Analy. and Appl. 29 (2006), no. 1, 177–183.Google Scholar
[27]Peaceman, D.W. and Rachford, H.H. Jr., The Numerical Solution of Parabolic and Elliptic Differential Equations, Journal of the Society for Industrial and Applied Mathematics 3 (1955), no. 1, 28–41.Google Scholar
[28]Richardson, L.F., The deferred approach to the limit, I-single lattice, Trans. of the Roy. Soc. of London, Series A 226 (1927), 299–349.Google Scholar
[29]Shen, Jie, Tang, Tao, and Wang, Li-Lian, Spectral Methods. Algortims, Analysis and Applications, Springer, 2011.Google Scholar
[30]Stenger, F., Numerical Methods Based on Sinc and Analytic Functions, Springer, 1993.CrossRefGoogle Scholar
[31]Stenger, F., Handbook of Sinc Numerical Methods, Chapman & Hall/CRC, 2011.Google Scholar
[32]Trefethen, L.N., Spectral Methods in MATLAB, SIAM, 2000.Google Scholar
[33]Velázquez, J.J.L., Classification of singularities for blowing up solutions in higher dimensions, Trans. Amer. Math. Soc. 338 (1993), no. 1, 441–464.CrossRefGoogle Scholar
[34]Weideman, J.A.C., The eigenvalues of Hermite and rational spectral differentiation matrices, Numer. Math. 61 (1992).Google Scholar
[35]Weideman, J.A.C. and Reddy, S.C., A MATLAB Differentiation Matrix Suite, ACM Transtions On Mathemtical Software 26 (2000), no. 4, 465–519.Google Scholar