convectionreaction equation
Godunov's method
Laplace transformation
operatorsplitting method
embedded analytical solutions
Finite Volume method
fluxbased characteristic method
Discretization methods with analytical solutions for a convectionreaction equation with higherorder discretizations.
Juergen Geiser
Geiser
Juergen
Institut für Mathematik, HumboldtUniversität zu Berlin (ISSN 08630976), 25 pp.
Discretization methods with analytical solutions for a convectionreaction equation with higherorder discretizations.
Juergen Geiser
Preprint series:
Institut für Mathematik, HumboldtUniversität zu Berlin (ISSN 08630976), 25 pp.
MSC 2000
 35K15 Initial value problems for secondorder, parabolic equations

35K57 Reactiondiffusion equations
Abstract
We introduce an improved secondorder discretization method for
the convectionreaction equation by combining analytical and
numerical solutions.
The method is derived from Godunov's scheme, see [Godunov 1959] and [Leveque 2002], and uses analytical solutions to solve the onedimensional convectionreaction equation.
We can also generalize the secondorder methods for discontinuous solutions, because of the analytical test functions.
Onedimensional solutions are used in the higherdimensional solution of the numerical method.
The method is based on the fluxbased characteristic
methods and is an attractive alternative to the classical higherorder TVDmethods, see [Harten 1983].
In this article we will focus on the derivation of
analytical solutions embedded into a finite volume method, for general and special solutions of the characteristic methods.
For the analytical solution, we use the Laplace transformation
to reduce the equation to an ordinary differential equation.
With general initial conditions, e.g. spline functions,
the Laplace transformation is accomplished with the help of
numerical methods.
The proposed discretization method skips the classical error
between the convection and reaction equation by using
the operatorsplitting method.At the end of the article, we illustrate the higherorder method for different benchmark problems. Finally, the method is shown to produce realistic results.
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