Workshop on Foundations of Numerical PDEs

July 7 - July 9, 2005

Universidad de Cantabria in Santander, Spain

High Order Finite Volume Schemes for Balance Laws with Stiff Source

Giovanni Russo

Dipartimento di Matematica ed Informatica at Universita di Catania

Abstract:   Finite difference methods are usually preferred over their finite volume counterparts for the development of high order schemes for the numerical solution of hyperbolic systems of balance laws with a stiff source. The unstaggered version os such schemes is usually developed by a method-of-line approach: the derivative of the flux is discretized in a conservative way, using some high order nonlinear reconstruction on the flux, to prevent spurious oscillations, and the system of PDE's is reduced to a large system of ODE's, which contains a non stiff part (the flux term) and a stiff part (the relaxation). At this point, the system of ODE's can be solved by some Implicit-Explicit (IMEX) scheme, in which the source term is treated implicitly, while the flux term is treated explicitly. The advantage of finite difference is that the source term does not couple the cells, and therefore the implicit step can be efficiently solved, treating each cell separately. The same procedure cannot be applied with a finite volume scheme, since the cell average of the source is not equal to the source evaluated at the cell average, and the source term couples the cells. In our finite volume methods, a different approach to time discretization, called Central Runge Kutta (CRK), treats the numerical solution (cell average) with a conservative scheme for the flux, while the stage values at the edge of each cell (pointwise) are treated by a non conservative scheme, in which the source term is decoupled from the other cells, and therefore implicit schemes can be effectively used for the source. During the talk the development of finite difference on staggered grids will be also discussed, and high order finite volume methods for stiff balance laws will be illustrated both in the staggered and unstaggered version.