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Damping characteristics of finite difference methods for one-dimensional reaction-diffusion equations. (English) Zbl 1107.65076
Summary: The exact solution of finite difference methods for the one-dimensional, linear reaction-diffusion equation in infinite spatial domains is obtained analytically and compared with the exact solution of the corresponding partial differential equation by considering a fixed time interval and a limit process. It is shown that the exact solution of the difference equation differs from the exact one only in a temporal damping term which is asymptotically identical to the exact one when the grid spacing is much smaller than the wave length. The analysis presented in the paper also shows and emphasizes that the linear stability analysis of finite difference equations for the one-dimensional linear reaction-diffusion equation can be performed without any recourse whatsoever to the norms and spectrum of the amplification matrix.

65M06Finite difference methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
35K05Heat equation
Full Text: DOI
[1] Ames, W. F.: Numerical methods for partial differential equations. (1977) · Zbl 0577.65077
[2] Lapidus, L.; Pinder, G. F.: Numerical solution of partial differential equations in science and engineering. (1982) · Zbl 0584.65056
[3] Mitchell, A. R.; Griffiths, D. F.: The finite difference method in partial differential equations. (1980) · Zbl 0417.65048
[4] Strikwerda, J. C.: Finite difference schemes and partial differential equations. (1989) · Zbl 0681.65064
[5] Kaplan, W.: Advanced calculus. (1991) · Zbl 0746.00003