Technische Universität Darmstadt |

Technische Universität Darmstadt Fachbereich Mathematik Alf Gerisch |

*
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.

**Download:** available on request.

**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.

**MSC(1991):**- 35M10 PDE of mixed type
- 65M06 Finite difference methods

**Keywords:** *
partial differential equations; reaction-diffusion systems; finite difference
methods; MacCormack's method; TVD, ADI, splitting schemes
*

**Notes:**

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Last Modification: 06.09.2010 17:21
Author: Alf Gerisch |