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 (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. Authors supported in part for the first under Taiwan National Science Council grant NSC 99-2115-M-110006-MY3; the second by JSPS KAKENHI (24560187); and the third as part of the Center for Frontiers of Subsurface Energy Security, an Energy Frontier Research Center funded by the U.S. Department of Energy under Award Number DE-SC0001114, and by the U.S. National Science Foundation grant DMS-0835745. Chieh-Sen Huang Department of Applied Mathematics and National Center for Theoretical Sciences, National Sun Yat-sen University, Kaohsiung 804, Taiwan, R.O.C. E-mail: [email protected] Feng Xiao Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku, Yokohama, 226-8502, Japan E-mail: [email protected] Todd Arbogast University of Texas at Austin, Institute for Computational Engineering and Sciences, 201 EAST 24th St., Stop C0200, Austin, Texas 78712-1229 USA E-mail: [email protected] 2 C.-S. Huang, F. Xiao, and T. Arbogast