Guoxian Chen, Associate Professor, Wuhan University

Haijun Yu, Professor

A Type of Robust Bound-preserving MUSCL-Hancock Schemes

14:00-15:00 January 19 (Friday) , Tencent Meeting: 909-290-375

The MUSCL-Hancock upwind scheme, as a variation of the MUSCL scheme, only do the initial data reconstruction once in every cell and solve the Riemann problem once at every cell interface in one time step and is widely used to solve hyperbolic conservation laws. The MMP property of the scheme with CFL number 1/2 for scalar equation is proved in [A. Suresh. SIAM Journal on Scientific Computing, 22(4):1184-1198, 2000.].  For general problems the most often cited sufficient stability conditions are obtained in [C. Berthon. Numerische Mathematik, 104(1):27-46, 2006.]: the CFL number is 1/4 of the first order scheme; the initial data reconstruction admits only the basic most dissipative minmod limiter. These contradict the faster speed and higher resolution claims of the scheme. In practice people often use robust but risk settings such as the UNO/super-bee etc. slope limiters and the CFL number up to 1/2.

In this talk we introduce new stability conditions in the sense of bound-preserving for the scheme: a) The CFL number is $(\sqrt{3}-1)/2$ which admits almost 0.73 times the time step of MUSCL scheme and then gives faster simulation; b) The preliminary reconstruction is corrected by a bound-preserving slope limiter. The slope corrector gives the bound-preserving approximation if global bound is considered, and provides a non-oscillation simulation if local bound is considered. The corrector is omitted if  the well-konwn generalized minmod limiter is used for the preliminary reconstruction on scalar problem. Numerical examples verify the sharpness of these two settings and demonstrate the robustness of the schemes for advection problem with spacial variable and general nonlinear problem, we also apply the method on general nonlinear system.