Finite Volume Methods for Systems of Viscous Conservation Laws: Numerical Analysis and Software

Jürgen Fuhrmann

Weierstrass Institute for Applied Analysis and Stochastics Mohrenstr. 39 10117 Berlin.


Abstract

Using the flux function notation developed for the description of time explicite finite volume schemes, we present an exsistence and stability analysis for time implicite finite volume schemes for scalar nonlinear viscous conservation laws. The generalization of this notation to systems leads to an easy to understand application programming interface and an implementation of a solver for this class of problems. It uses various variants of Newton's method as nonlinear solvers and - among other possibilities - agglomeration type algebraic multigrid with point block ILU smoothers as preconditioners for Krylov subspace methods for the linear systems. The corresponding code - sysconlaw - is implemented on top of the pdelib toolbox for the numerical solution of partial differential equations developed at the Weierstrass Institute Berlin. Numerical examples like two phase flow in porous media, thermoconvective flow in porous media, or the Brusselator equations demonstrate the flexibility and usefulness of this approach.