Multiscale Polynomial-Based High-Order Central High Resolution Schemes

Polynomial-based high order central high resolution schemes with semi-discrete forms are integrated with multiresolution-based adapted cells/grids. To preserve the positivity condition on non-uniform cells/grids, the corresponding formulations are studied, redesigned or developed. Two general approaches can be used for polynomial-based reconstructions: (a) direct interpolation by a polynomial, (b) proper combination of different polynomials to construct a new polynomial with desired features. Based on these approaches, three polynomial-based reconstructions are considered: (i) parabolic polynomials interpolating average solutions of three successive cells; (ii) piece-wise parabolic methods (PPMs) obtained with two different local stencils; (iii) central-WENO schemes [based on the results of approach (i)]. For the first approach, the corresponding features, stability conditions, formulations and nonlinear limiters are studied and updated. For the second approach, for more localized stencils, new independent variables (e.g., first and second spatial derivatives) are introduced by adding new conservation laws. Two PPM-based central schemes are formulated and a new limiter and a new updating procedure are introduced. For the third approach, the average-interpolating parabolic polynomial [in approach (i)] is used in the framework of the central-WENO formulation. Third and fourth order formulations are provided on non-uniform grids/cells. Finally some numerical examples are presented to verify the results and to assess effectiveness and robustness of the three approaches.