Energy-Stable Residual Distribution Methods For System Of Shallow Water Equations - Wei Shyang, Chang (2019)
A state-of-the-art Energy-Stable Residual Distribution (ESRD) method is expanded for a system of Shallow Water Equations (SWE) as an improvement over the finite volume counterpart (ESFV) for inheriting multi-dimensional feature, minimal sensitivity to grid distortions and the ability to achieve higher order accuracy with smaller stencil. ESRD imposes energy control simultaneously with the computation of the main variables through the mapping of primary conservative variables to energy variables. The energy conservation and energy stable conditions are achieved via the design of isotropic signals and artificial signals respectively. To preserve the cost-effectiveness of the scheme, the work is limited to only full explicit approach. The main contribution of this work is the source term discretisation which is designed to achieve numerical well-balanceness property. The effects of grid skewness variations on the order of accuracy and stability of ESRD were examined based on scalar analyses. Different degrees of freedom were manipulated to achieve positivity (first order scheme) and linear preserving (second order scheme) properties. A non-linear limited scheme is also constructed with the blending of the first and second order schemes. Unlike ESFV, ESRD demonstrates its ability to preserve the order of accuracy even on high randomized triangular grids. The well-balancedness of the proposed scheme was validated numerically and the order of accuracy of the well-balanced version of the schemes are still preserved.