the WAF method for non-homogeneous SWE with pollutant

This paper deals with the extension of the WAF method to discretize Shallow Water Equations with pollutants. We consider two different versions of the WAF method, by approximating the intermediate waves using the flux of HLL or the direct approach of HLLC solver. It is seen that both versions can be written under the same form with different definitions for the approximation of the velocity waves. We also propose an extension of the method to non-homogeneous systems. In the case of homogeneous systems it is seen that we can rewrite the third component of the numerical flux in terms of an intermediate wave speed approximation. We conclude that – in order to have the same relation for non-homogeneous systems – the approximation of the intermediate wave speed must be modified. The proposed extension of the WAF method preserves all stationary solutions, up to second order accuracy, and water at rest in an exact way, even with arbitrary pollutant concentration. Finally, we perform several numerical tests, by comparing it with HLLC solver, reference solutions and analytical solutions

Perfectly matched layers with high rate damping for hyperbolic systems

We propose a simple method for constructing non-reflecting boundary conditions via Perfectly Matched Layer approach. The basic idea of the method is to build a layer with high rate damping properties with are provided by adding the stiff relaxation source terms to all equations of the system. No complicated modification of the system to be solved is then required

Conservative numerical methods for advanced model kinetic equations

Model kinetic equations are often used to describe rarefied gas flows in the broad range of flow regimes. Accurate model kinetic equations were proposed by Shakhov [1] for monatomic gases and by Rykov [2] for diatomic gases. The main advantage of these model equations over other models reported in the literature is that the model collision integral contains expressions for heat fluxes and ensures their correct relaxation, e.g. correct Prandtl number for the monatomic gas. Developing conservative numerical methods for kinetic equations is a demanding task due to the nonlinear character and very high dimension of these equations. These methods are defined as methods from which the discrete conservation laws for the mass, momentum and energy follow. Non-conservative methods produce non-physical source terms in the conservation equations and may fail to converge to the proper solution as the Knudsen number is decreased unless the velocity grid is constantly refined. Due to the highly nonlinear character of the kinetic equation, both exact and model ones, the conservation property is very difficult to satisfy. The aim of this paper is to present a direct approach for ensuring conservation in the numerical methods for model kinetic equations, including Shakhov and Rykov models. The approach is based on the approximation of the conditions which are used in deriving these model equations. We construct a nonlinear system of equations defining the macro parameters in collision integrals. This system can be easily solved numerically. The resulting numerical methods do not require prohibitively fine meshes and ensures correct relaxation of heat fluxes in the continuum limits. We apply the proposed method to a plane hypersonic flow over a plate. The obtained results illustrate the robustness and accuracy of the method

ADER schemes for hyperbolic conservation laws with reactive terms

In this paper we generalizew the semi-analytical method16 for solving the Derivative Riemann Problem to hyperbolic systems for which the Riemann problem solution is not available. As an application example we implement the new derivative Riemann solver in the high-order finite-volume ADER advection schemes. We provide numerical examples for two-dimensional hyperbolic systems which illustrate robustness and high ac- curacy of the resulting schemes

Development and application of generalized MUSTA schemes

Numerical methods for solving non-linear systems of hyperbolic conservation laws via finite volume methods or discontinuous Galerkin finite element methods require, as the building block, a monotone numerical flux. The simplest approach for providing a monotone numerical utilizes a symmetric stencil and does not explicitly make use of wave propagation information, giving rise to centred or symmetric schemes. A more refined approach utilizes wave propagation information through the exact or approximate solution of the Riemann problem, giving rise to Godunov methods. Conventional approximate Riemann solvers are usually complex and are not available for many systems of practical interest, such as for models for compressible multi-phase flows. It is thus desirable to construct a numerical flux that emulates the best flux available (upwind) with the simplicity and generality of symmetric schemes. Here we build upon MUSTA approach [2,3], which leads to schemes that have the simplicity and generality of symmetric schemes and the accuracy of upwind schemes. First we present a new flux that is an average of symmetric fluxes and which reproduces the Godunov upwind scheme for the model hyperbolic equation. For non-linear systems it is found that this flux gives superior results to those of the whole family of incomplete Riemann solvers that do not explicitly account for linearly degenerate fields. Then we incorporate this flux into the MUSTA multi-staging approach, as predictor and corrector. It is found that the resulting MUSTA schemes reproduce the Godunov upwind scheme for the model hyperbolic equation for any number of stages, including multiple space dimensions. They are linearly stable in two and three space dimensions and the stability region is identical to that of the Godunov upwind method. For non-linear systems the MUSTA scheme with one or two stages gives results that are indistinguishable from those of Riemann solvers, such as the exact Riemann solver or Roe approximate Riemann solver. Finally, we assess the schemes on carefully chosen test problems for the one-dimensional equations of magneto hydrodynamics and nonlinear elasticity. High-order examples are provided for the multidimensional Euler equations in the framework of finite-volume WENO schemes [1]. The results illustrate the accuracy and efficiency of new methods combined with the ease of coding

The Derivative Riemann Problem: The basis for high order ADER Schemes

The corner stone of arbitrary high order schemes (ADER schemes) is the solution of the derivative Riemann problem at the element interfaces, a generalization of the classical Riemann problem first used by Godunov in 1959 to construct a first-order upwind numerical method for hyperbolic systems. The derivative Riemann problem extends the possible initial conditions to piecewise smooth functions, separated by a discontinuity at the interface. In the finite volume framework, these piecewise smooth functions are obtained from cell averages by a high order non-oscillatory WENO reconstruction, allowing hence the construction of non-oscillatory methods with uniform high order of accuracy in space and time