题  目： Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin and finite volume schemes for conservation laws

报告人： Dr. Xiangxiong Zhang, Department of Mathematics,

         Massachusetts Institute of Technology

时  间：2011年12月21日 15:30

地  点： 管理科研楼1611

摘  要：

    For the initial value problem of scalar conservation laws, the exact solution is always bounded by the maximum and minimum of the initial value, which is an important desired property for numerical solutions. It is very difficult to design a scheme satisfying a strict maximum principle without losing high order accuracy especially for multi-dimensional nonlinear problems. We proposed a general framework (for arbitrary order of accuracy) to 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. We obtained, for the first time, a high order accurate conservative maximum-principle-satisfying scheme for multi-dimensional scalar conservation laws. The main ingredients include a sufficient condition for the DG or FV method with first order Euler forward time discretization to satisfy the maximum principle and a simple limiter to enforce this condition. 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 higher dimensions on either rectangular or triangular meshes. Moreover, the framework also applies to systems of conservation laws. Suitable generalizations result in high order DG or FV schemes satisfying maximum principle for two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field, positivity preserving property of water height for shallow water equations, and positivity preserving property of density and pressure for compressible Euler equations in gas dynamics. Numerical tests demonstrating the good performance of the scheme will be. This is a joint work with Prof. Chi-Wang Shu.

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          国家数学与交叉科学中心合肥分中心
      
 

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