Non-Oscillatory Hierarchical Reconstruction for Central and Finite Volume Schemes

This is the continuation of the paper "central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction" by the same authors. The hierarchical reconstruction introduced therein is applied to central schemes on overlapping cells and to nite volume schemes on non-staggered grids. This takes a new nite volume approach for approximating non-smooth solutions. A critical step for high order nite volume schemes is to reconstruct a nonoscillatory high degree polynomial approximation in each cell out of nearby cell averages. In the paper this procedure is accomplished in two steps: first to reconstruct a high degree polynomial in each cell by using e.g., a central reconstruction, which is easy to do despite the fact that the reconstructed polynomial could be oscillatory; then to apply the hierarchical reconstruction to remove the spurious oscillations while maintaining the high resolution. All numerical computations for systems of conservation laws are performed without characteristic decomposition. In particular, we demonstrate that this new approach can generate essentially non-oscillatory solutions even for 5th order schemes without characteristic decomposition.The research of Y. Liu was supported in part by NSF grant DMS-0511815. The research of C.-W. Shu was supported in part by the Chinese Academy of Sciences while this author was visiting the University of Science and Technology of China (grant 2004-1-8) and the Institute of Computational Mathematics and Scienti c/Engineering Computing. Additional support was provided by ARO grant W911NF-04-1-0291 and NSF grant DMS-0510345. The research of E. Tadmor was supported in part by NSF grant 04-07704 and ONR grant N00014-91-J-1076. The research of M. Zhang was supported in part by the Chinese Academy of Sciences grant 2004-1-8

Central Schemes for Porous Media Flows

We are concerned with central differencing schemes for solving scalar hyperbolic conservation laws arising in the simulation of multiphase flows in heterogeneous porous media. We compare the Kurganov-Tadmor, 2000 semi-discrete central scheme with the Nessyahu-Tadmor, 1990 central scheme. The KT scheme uses more precise information about the local speeds of propagation together with integration over nonuniform control volumes, which contain the Riemann fans. These methods can accurately resolve sharp fronts in the fluid saturations without introducing spurious oscillations or excessive numerical diffusion. We first discuss the coupling of these methods with velocity fields approximated by mixed finite elements. Then, numerical simulations are presented for two-phase, two-dimensional flow problems in multi-scale heterogeneous petroleum reservoirs. We find the KT scheme to be considerably less diffusive, particularly in the presence of high permeability flow channels, which lead to strong restrictions on the time step selection; however, the KT scheme may produce incorrect boundary behavior