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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:[email protected] (E. O. Asante-Asamani), [email protected] (B. A. Wade)
*Corresponding author. Email addresses:[email protected] (E. O. Asante-Asamani), [email protected] (B. A. Wade)
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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 

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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