Abstract
The HWENO (Hermite weighted essentially non-oscillatory) schemes are
high order, high-resolution methods suitable for conservation law and
convection dominated simulations with possible discontinuous or sharp
gradient solutions. In most of the literature, although there are many
other numerical fluxes available, the Lax-Friedrichs numerical flux is
used frequently due to its simplicity. In this paper, we will study an
alternative finite difference HWENO method. The core of this method is
that its numerical flux framework breaks the limitations of the
traditional mathematical form of numerical flux and is suitable for many
different forms of numerical flux. And we systematically investigate the
performance of the HWENO method and present quantitative comparisons for
hyperbolic conservation laws based on different numerical fluxes. The
spatial terms are discretized by using finite difference HWENO scheme
and the time terms are performed by using TVD Runge-Kutta method. The
HWENO method is proposed based on the original WENO methodology for
solving hyperbolic conservation laws. Therefore, the HWENO scheme is
similar to the classic WENO scheme achieved by using numerical flux as a
building block, and their performances are closely related to the
properties of the numerical fluxes. Hence, we study the performance of
HWENO method based on different numerical fluxes, including the
first-order monotone fluxes and second-order TVD fluxes, with the
objective of obtaining better performance for the conservation laws by
choosing suitable numerical fluxes. The detailed discussion focuses on
the one-dimensional system case, including the issues of CPU cost,
accuracy, non-oscillatory property.