|Technische Universität Darmstadt|
Technische Universität Darmstadt
Finite Difference Methods for Coupled Nonlinear Hyperbolic and Parabolic Partial Differential Equations in One and Two Dimensions
by A. Gerisch
Master's thesis, University of Dundee, September 1997.
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Abstract: The solution of a system consisting of a hyperbolic and a parabolic partial differential equation arising in mathematical biology is considered. Numerical approximations are obtained by finite difference methods on equidistant grids in one and two dimensions. A modification of MacCormack's method is given for the solution of the hyperbolic equation in one dimension which possesses improved stability properties. The results are comparable with the solutions generated by a total variation diminishing (TVD) scheme. Such a scheme is developed for constant coefficient homogeneous linear hyperbolic equations with negative wave speed. The hyperbolic problem in two dimensions is solved using a splitting approach and hence involves applying the methods developed for one dimension (MacCormack's scheme). Alternating direction implicit (ADI) methods are employed for the solution of the parabolic equation in two dimensions to keep computational costs down. A set of numerical experiments is conducted to test and compare the algorithms and to solve the biological model.
Keywords: partial differential equations; reaction-diffusion systems; finite difference methods; MacCormack's method; TVD, ADI, splitting schemes
Last Modification: 06.09.2010 17:21
Author: Alf Gerisch