Hostname: page-component-7bb8b95d7b-cx56b Total loading time: 0 Render date: 2024-09-11T14:10:26.393Z Has data issue: false hasContentIssue false
  • English
  • Français

A Dimensional Splitting of ETD Schemes for Reaction-Diffusion Systems

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  17 May 2016

E. O. Asante-Asamani  and
Bruce A. Wade
E. O. Asante-Asamani*
Affiliation:
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI, 53201-0413, USA
Bruce A. Wade*
Affiliation:
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI, 53201-0413, USA
*
*Corresponding author. Email addresses:eoa@uwm.edu (E. O. Asante-Asamani), wade@uwm.edu (B. A. Wade)
*Corresponding author. Email addresses:eoa@uwm.edu (E. O. Asante-Asamani), wade@uwm.edu (B. A. Wade)
Get access

Abstract

Novel dimensional splitting techniques are developed for ETD Schemes which are second-order convergent and highly efficient. By using the ETD-Crank-Nicolson scheme we show that the proposed techniques can reduce the computational time for nonlinear reaction-diffusion systems by up to 70%. Numerical tests are performed to empirically validate the superior performance of the splitting methods.

Keywords

Exponential time differencingdimensional splittingreaction diffusion equations

MSC classification

Secondary: 65M12: Stability and convergence of numerical methods 65M15: Error bounds 65M20: Method of lines
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 5 , May 2016 , pp. 1343 - 1356
Copyright
Copyright © Global-Science Press 2016 

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]Cox, S.M. and Matthews, P.C., Exponential time differencing for stiff systems, Journal of Computational Physics, 176 (2002), 430–455.Google Scholar
[2]Du, Q. and Zhu, W., Analysis and applications of the exponential time differencing schemes and their contour integration modifications, BIT Numerical Mathematics, 45 (2005), 307–328.Google Scholar
[3]Hochbruck, M. and Ostermann, A., Explicit exponential Runge-Kutta methods for semilinear parabolic problems, SIAM Journal on Numerical Analysis, 43 (2005), 1069–1090.Google Scholar
[4]Voss, D.A. and Khaliq, A.Q.M., Parallel LOD methods for second order time dependent PDEs, Computers & Mathematics with Applications, 30 (1995), 25–35.Google Scholar
[5]Kleefeld, B., Khaliq, A.Q.M. and Wade, B.A., An ETD Crank-Nicolson method for reaction-diffusion systems, Numer. Methods Partial for Partial Differential Equations, 28 (2012), 1309–1335.Google Scholar
[6]Yousuf, M., Khaliq, A. Q. M. and Kleefeld, B., The numerical approximation of nonlinear Black-Scholes model for exotic path–dependent American options with transaction cost, International Journal of Computer Mathematics, 89 (2012), 1239–1254.CrossRefGoogle Scholar
[7]Bhatt, H.P. and Khaliq, A.Q.M., The locally extrapolated exponential time differencing LOD scheme for multidimensional reaction-diffusion systems, Journal of Computational and Applied Mathematics, 285 (2015), 256–278.Google Scholar
[8]Akrivis, G., Crouzeiz, M. and Makridakis, C., Implicit-explicit multistep methods for quasi-linear parabolic equations, Numerische Mathematik, 82 (1999), 521–541.Google Scholar
[9]Fernandes, R. I. and Fairweather, G., An ADI extrapolated Crank-Nicolson orthogonal spline collocation method for nonlinear reaction-diffusion systems, Journal of Computational Physics, 231 (2012), 6248–6267.CrossRefGoogle Scholar
[10]Gatenby, R.A. and Gawlinski, E.T., A reaction-diffusion model of cancer invasion, Cancer research, 56 (1996), 5745–5753.Google Scholar
[11]Holmes, E.E., Lewis, M.A., Banks, J.E. and Viet, R.R., Partial differential equations in ecology: spatial interactions and population dynamics, Ecology, 75 (1994), 17–29.Google Scholar
[12]Khaliq, A.Q.M., Martin-Vanquero, J., Wade, B.A. and Yousuf, M., Smoothing schemes for reaction-diffusion systems with nonsmooth data, Journal of Computational and Applied Mathematics, 223 (2009), 374–386.CrossRefGoogle Scholar
[13]Kondo, S. and Miura, T., Reaction-diffusion model as a framework for understanding biological pattern formation, Science, 329 (2010), 1616–1620.Google Scholar
[14]Leveque, Randall J., Finite difference methods for ordinary and partial differential equations: Steady-state and time-dependent problems, SIAM, 98(2007).Google Scholar
[15]Martín-Vaquero, J. and Wade, B.A., On efficient numerical methods for an initial-boundary value problem with nonlocal boundary conditions, Applied Mathematical Modelling, 36(2012) 3411–3418.Google Scholar
[16]Sherratt, J.A. and Murray, J.D., Models of epidermal wound healing, Proceedings of the Royal Society of London B: Biological Sciences, 241(1990) 29–36.Google ScholarPubMed
[17]Chen, L.Q. and Shen, J., Applications of semi-implicit Fourier-spectral method to phase field equations, Computer Physics Communications, 108(1998) 147–158.CrossRefGoogle Scholar
[18]Gear, W. and Kevrekidis, I., Projective methods for stiff differential equations: Problems with gaps in their eigenvalue spectrum, SIAM Journal on Scientific Computing, 24(2003) 1091–1106.Google Scholar
[19]Vigo-Aguiar, J. and Wade, B.A., Adapted BDF algorithms applied to parabolic problems, Numerical Methods for Partial Differential Equations, 23(2007) 350–365.CrossRefGoogle Scholar
[20]Khaliq, A.Q.M. and Wade, B.A., On smoothing of the Crank-Nicolson scheme for nonhomogeneous parabolic problems, Journal of Computational Methods in Science and Engineering, 1(2001) 107–123.Google Scholar