Maximum-principle-satisfying and positivity-preserving<br/><br/> high order discontinuous Galerkin and finite volume<br/><br/> schemes for conservation laws

Monday, November 1, 2010 - 10:45am - 11:30am
Keller 3-180
Chi-Wang Shu (Brown University)
We construct uniformly high order accurate discontinuous Galerkin
(DG) and weighted essentially non-oscillatory (WENO) finite volume
(FV) schemes satisfying a strict maximum principle for scalar
conservation laws and passive convection in incompressible flows, and
positivity preserving for density and pressure for compressible Euler
equations. A general framework (for arbitrary order of accuracy) is
established to construct a limiter for the DG or FV method with first order
Euler forward time discretization solving one dimensional scalar
conservation laws. Strong stability preserving (SSP) high order time
discretizations will keep the maximum principle and make the scheme
uniformly high order in space and time. One remarkable property of
this approach is that it is straightforward to extend the method to
two and higher dimensions. The same limiter can be shown to preserve
the maximum principle for the DG or FV scheme solving two-dimensional
incompressible Euler equations in the vorticity stream-function
formulation, or any passive convection equation with an incompressible
velocity field. A suitable generalization results in a high order DG
or FV scheme satisfying positivity preserving property for density and
pressure for compressible Euler equations. Numerical tests
demonstrating the good performance of the scheme will be reported.
This is a joint work with Xiangxiong Zhang.
MSC Code: