Modified Riemann Solver Research Articles

An improved HLLC-type solver for incompressible two-phase fluid flows

In this paper, an HLLC-type contact preserving Riemann solver for incompressible two-phase flows using the artificial compressibility formulation is presented. This formulation is an improvement over the previous HLLC-VOF formulation (Bhat and Mandal, 2019). In this work, unlike the previous formulation, the variation of the volume fraction is taken into account when calculating the eigenvalues and eigenvectors. Hence, the equations for the intermediate states and the intermediate wave speed are closely-coupled with density variation during pseudo-time evolution of the solution. Additionally, an interface compression algorithm is used in tandem to ensure sharp interfaces. This modified Riemann solver (called HLLC-VOF-M) is found to be more robust and accurate compared to the older HLLC-VOF solver and the non-contact preserving HLL solver. Several test problems in two- and three-dimensions are solved to demonstrate and evaluate the efficacy of the solver on structured and unstructured meshes.

A weakly compressible SPH method based on a low-dissipation Riemann solver

We present a low-dissipation weakly-compressible SPH method for modeling free-surface flows exhibiting violent events such as impact and breaking. The key idea is to modify a Riemann solver which determines the interaction between particles by a simple limiter to decrease the intrinsic numerical dissipation. The modified Riemann solver is also extended for imposing wall boundary conditions. Numerical tests show that the method resolves free-surface flows accurately and produces smooth, accurate pressure fields. The method is compatible with the hydrostatic solution and exhibits considerably less numerical damping of the mechanical energy than previous methods.

A robust and well‐balanced scheme for the 2D Saint‐Venant system on unstructured meshes with friction source term

SummaryIn the following lines, we propose a numerical scheme for the shallow‐water system supplemented by topography and friction source terms, in a 2D unstructured context. This work proposes an improved version of the well‐balanced and robust numerical model recently introduced by Duran et al. (J. Comp. Phys., 235, 565–586, 2013) for the pre‐balanced shallow‐water equations, accounting for varying topography. The present work aims at relaxing the robustness condition and includes a friction term. To this purpose, the scheme is modified using a recent method, entirely based on a modified Riemann solver. This approach preserves the robustness and well‐balanced properties of the original scheme and prevents unstable computations in the presence of low water depths. A series of numerical experiments are devoted to highlighting the performances of the resulting scheme. Simulations involving dry areas, complex geometry and topography are proposed to validate the stability of the numerical model in the neighbourhood of wet/dry transitions. Copyright © 2015 John Wiley & Sons, Ltd.

Asymptotic preserving scheme for the shallow water equations with source terms on unstructured meshes

The following work is devoted to the construction and validation of a numerical scheme for the 2D shallow water system on unstructured meshes, supplemented by topography and friction source terms. Approximate solutions of frictionless flows are obtained considering a suitable formulation of the conservation laws, involving the water free surface and some fractions of water, accounting for the topography variations. The discretization of the friction source terms relies on the use of a modified Riemann solver for the flux computation. The resulting scheme is then corrected in order to achieve an asymptotic regime preservation. A MUSCL reconstruction is also performed to increase the space order of accuracy. The overall numerical approach is shown to be consistent, well-balanced and to satisfy a domain invariant principle. These results are assessed through several benchmark tests, involving complex geometry and varying bathymetry. In the presence of dry areas, special interest is given to the wave front speed computation, highlighting the stability of the method, even when implementing the asymptotic preserving correction.

A wave propagation algorithm for hyperbolic systems on curved manifolds

An extension of the wave propagation algorithm first introduced by LeVeque [J. Comput. Phys. 131 (1997) 327] is developed for hyperbolic systems on a general curved manifold. This extension is important in a variety of applications, including the propagation of sound waves on a curved surface, shallow water flow on the surface of the Earth, shallow water magnetohydrodynamics in the solar tachocline, and relativistic hydrodynamics in the presence of compact objects such as neutron stars and black holes. As is the case for the Cartesian wave propagation algorithm, this new approach is second order accurate for smooth flows and high-resolution shock-capturing. The algorithm is formulated such that scalar variables are numerically conserved and vector variables have a geometric source term that is naturally incorporated into a modified Riemann solver. Furthermore, all necessary one-dimensional Riemann problems are solved in a locally valid orthonormal basis. This orthonormalization allows one to solve Cartesian Riemann problems that are devoid of geometric terms. The new method is tested via application to the linear wave equation on a curved manifold as well as the shallow water equations on part of a sphere. The proposed algorithm has been implemented in the software package clawpack and is freely available on the web.

A comparison of two different types of shoreline boundary conditions

Two different types of shoreline boundary conditions which can be used in either wave-resolving or wave-averaging models of waves and currents propagation in the nearshore are compared here. The two techniques are essentially different: in the first case the velocity of the shoreline is obtained by the momentum equation and the shoreline position is tracked by changing the grid position, while in the other case the velocity of the shoreline is obtained by a modified Riemann solver and the shoreline is defined as an interface between dry and wet fixed grid points. A number of test cases are described to compare the performance of the two techniques.