Dr. Jürgen Fuhrmann
Weierstrass Institut für angewandte Analysis und Stochastik (WIAS Berlin)
Finite volume schemes for nonlinear convection-diffusion problems based on the solution of local Dirichlet problems
Abstract:
Finite volume schemes for nonlinear convection-diffusion problems based on the solution of local Dirichlet problemsWe present a method to compute the numerical flux of a finite volume scheme used for the approximation of the solution of a scalar nonlinear convection diffusion equation in a 1D, 2D or 3D domain. The method is based on the solution, at each interface between two control volumes, of a nonlinear elliptic two point Dirichlet boundary value problem arising from the projection of the main part of the initial equation onto the normal to that interface. The boundary values are given by the values of the discrete approximation in both control volumes.
The existence of a solution to this two point boundary value problem has been proven. The expression for the numerical flux can be yielded without referring to this solution. Furthermore, the so designed finite volume scheme has the expected stability properties. Its solution converges to the weak solution of the continuous problem.
The scheme is a generalization of the classical exponential fitting scheme of Allen/Soutwell, Ilin or Scharfetter/Gummel for linear convection diffusion problems to the nonlinear case.
Numerical results show the increase of accuracy due to the use of this scheme, compared to some other schemes. Furthermore we discuss possibilities for an effective implementation.
Reference:
R. Eymard, J. Fuhrmann, and K. Gärtner. A finite volume scheme for
nonlinear parabolic equations derived from one-dimensional local
Dirichlet problems.
Numerische Mathematik, 102(3):463 - 495, 2006.
| Datum: | 20.10.2006 | Zeit: | 14.15 | Ort: | FU Berlin, Institut für Mathematik II, Arnimallee 6, 14195 Berlin. | Raum: | 032 im Erdgeschoss |