Finite volume HWENO schemes for nonconvex conservation laws
For researchers in computational fluid dynamics and conservation laws, this work addresses a known bottleneck in applying high-order HWENO schemes to nonconvex problems, but the modifications are incremental.
The paper identifies that high-order finite volume HWENO schemes perform poorly or converge slowly for nonconvex conservation laws, and proposes modifications based on first-order monotone schemes or entropic projection to improve performance. Numerical tests demonstrate the effectiveness of the proposed modifications.
We illustrate that numerical solutions of high order finite volume Hermite weighted essentially non-oscillatory (HWENO) scheme for some nonconvex conservation laws perform poorly or converge to the entropy solution in a slow speed. The modified finite volume HWENO schemes based either on first order monotone schemes or a second order entropic projection following the work of Qiu and Shu [SIAM J. Sci. Comput., 31 (2008), 584-607] are proposed and compared for solving one-dimensional scalar problems. We extend the modified finite volume HWENO based on first order monotone schemes for one-dimensional systems and two-dimensional scalar conservation laws. Numerical tests for several representative examples will be reported.