Non-Oscillatory Hierarchical Reconstruction for Central and Finite Volume Schemes (Communications in Computational Physics, 2007)
This is the continuation of the paper "Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction" by thesame authors. The hierarchical reconstruction introduced therein is applied to centralschemes on overlapping cells and to finite volume schemes on non-staggered grids.This takes a new finite volume approach for approximating non-smooth solutions. Acritical step for high-order finite volume schemes is to reconstruct a non-oscillatoryhigh degree polynomial approximation in each cell out of nearby cell averages. Inthe 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 dodespite the fact that the reconstructed polynomial could be oscillatory; then to applythe hierarchical reconstruction to remove the spurious oscillations while maintainingthe high resolution. All numerical computations for systems of conservation laws areperformed without characteristic decomposition. In particular, we demonstrate thatthis new approach can generate essentially non-oscillatory solutions even for 5th-orderschemes without characteristic decomposition.