A kinetic flux vector splitting scheme for shallow water equations incorporating variable bottom topography and horizontal temperature gradients

Abstract

This paper is concerned with the derivation of a well-balanced kinetic scheme to approximate a shallow flow model incorporating non-flat bottom topography and horizontal temperature gradients. The considered model equations, also called as Ripa system, are the non-homogeneous shallow water equations considering temperature gradients and non-uniform bottom topography. Due to the presence of temperature gradient terms, the steady state at rest is of primary interest from the physical point of view. However, capturing of this steady state is a challenging task for the applied numerical methods. The proposed well-balanced kinetic flux vector splitting (KFVS) scheme is non-oscillatory and second order accurate. The second order accuracy of the scheme is obtained by considering a MUSCL-type initial reconstruction and Runge-Kutta time stepping method. The scheme is applied to solve the model equations in one and two space dimensions. Several numerical case studies are carried out to validate the proposed numerical algorithm. The numerical results obtained are compared with those of staggered central NT scheme. The results obtained are also in good agreement with the recently published results in the literature, verifying the potential, efficiency, accuracy and robustness of the suggested numerical scheme.

A well-balanced moving mesh discontinuous Galerkin method for the Ripa model on triangular meshes

“…The computational domain is (−1, 1). Similar examples have been used in [8,26,27]. The initial data and the bottom topography for this example are given by b…”

“…In recent years studies have been made on the development of well-balanced numerical schemes for the Ripa model. The first work seems to be [8] [31] (a second-order positivity preserving finite volume scheme on rectangular meshes), Sánchez-Linares et al [27] (a second-order positivity preserving HLLC scheme based in path-conservative approximate Riemann solvers, for the onedimensional Ripa model), Han and Li [13] (a high-order finite difference weighted essentially non-oscillatory (WENO) scheme), Saleem et al [26] (a kinetic flux vector splitting scheme on rectangular meshes), Thanh et al [30] (a high-order scheme of van Leer's type for the one-dimensional SWEs with temperature gradient), Rehman et al [24] (a fifth-order finite volume multi-resolution WENO scheme on rectangular meshes), Britton and Xing [6] (a DG scheme for the one-dimensional Ripa model), Qian et al [23] (a DG method based on a source term approximation technique), and Li et al [20] (a DG method based on hydrostatic reconstruction on rectangular meshes). Fixed meshes are employed in the above mentioned works.…”