Fifth Order Multi-moment WENO Schemes for Hyperbolic Conservation Laws

A general approach is given to extend WENO reconstructions to a class of numerical schemes that use different types of moments (i.e., multi-moments) simultaneously as the computational variables, such as point values and grid cell averages. The key is to re-map the multi-moment values to single moment values (e.g., cell average or point values), which can then be used to invoke known, standard reconstruction coefficients and smoothness indicators for single moment WENO reconstructions. The WENO reconstructions in turn provide the numerical approximations for the flux functions and other required quantities. One major advantage of using multi-moments for WENO reconstructions is its compactness. We present two new multi-moment WENO (MM-WENO) schemes of fifth order that use reconstructions supported over only three grid cells, as opposed to the usual five. This is similar to the Hermite WENO schemes of Qiu and Shu (J Comput Phys 193:115–135, 2003), which can also be derived using our general approach. Numerical tests demonstrate that the new schemes achieve their designed fifth order accuracy and eliminate spurious oscillations effectively. The numerical solutions to all benchmark tests are of good quality and comparable to the classic, single moment WENO scheme of the same order of accuracy. The basic idea presented in this paper is universal, which makes the WENO reconstruction an easy-to-follow method for developing a wide variety of additional multi-moment numerical schemes.