Multiscale Central High-Resolution Schemes with Different Types of Extended Slope Limiters on Wavelet-Based Adapted Cells

Abstract

To have proper multiscale simulations by central high-resolution schemes, the performance of slope limiters is crucial on adapted cells in terms of stability, accuracy, high-resolution, entropy-satisfying and spectral features. Hence, here, different families of slope limiters are extended or developed over non-uniform centered/non-centered cells, obtained by the wavelet-based adapted grids. The developed limiters on adaptive cells, are: (1) The entropy-satisfying limiter, (2) Second-order non-symmetric limited slopes and corresponding symmetric formulations, (3) Limiters with symmetric three-point stencils, second-order accuracy and the total variation diminishing (TVD) feature. The three-point stencil slope limiters do not preserve the symmetry feature on non-uniform cells. However, slopes obtained by non-symmetric stencils nearly preserve the symmetric property for different directions, as the direction-effect is inherent in their formulations. For non-symmetric limiters, firstly, corresponding TVD and monotonicity-preserving conditions are provided. Then, eight non-symmetric limited slopes are developed with four-, five- and three-point stencils. They are then unified to achieve four symmetric limiters. For the symmetric limiters with three-point stencils, also, the concept of blending of two limiters is updated to achieve compression-adaptive limiters. All limiters are used in the cell-adaptive Kurganov-Tadmor (KT) central scheme. Afterwards, the effects of limited slopes are studied on spectral properties. Finally, several problems are presented.