Analytical implementation of Roe solver for two-layer shallow water equations with accurate treatment for loss of hyperbolicity
For researchers simulating stratified flows, this provides a faster alternative to the accurate but computationally expensive Roe solver, though the improvement is incremental.
The paper presents an analytical implementation of the Roe solver for two-layer shallow water equations, using a closed-form quartic solution to avoid numerical eigensolvers. The A-Roe scheme matches the accuracy of the standard Roe scheme while achieving computational speeds comparable to simpler schemes like GFORCE and IFCP.
A new implementation of the Roe scheme for solving two-layer shallow-water equations is presented in this paper. The proposed A-Roe scheme is based on the analytical solution to the characteristic quartic of the flux matrix, which is an efficient alternative to a numerical eigensolver. Additionally, an accurate method for maintaining the hyperbolic character of the governing system is proposed. The efficiency of the quartic closed-form solver is examined and compared to numerical eigensolvers. Furthermore, the accuracy and computational speed of the A-Roe scheme is compared to the Roe, Lax-Friedrichs, GFORCE, PVM, and IFCP schemes. Finally, numerical tests are presented to evaluate the efficiency of the iterative treatment for the hyperbolicity loss. The proposed A-Roe scheme is as accurate as the Roe scheme, but much faster, with computational speeds closer to the GFORCE and IFCP scheme.