Low-Mach consistency of a class of linearly implicit schemes for the compressible Euler equations

summary:In this note, we give an overview of the authors' paper [6] which deals with asymptotic consistency of a class of linearly implicit schemes for the compressible Euler equations. This class is based on a linearization of the nonlinear fluxes at a reference state and includes the scheme of Feistauer and Ku\v{c}era [3] as well as the class of RS-IMEX schemes [8,5,1] as special cases. We prove that the linearization gives an asymptotically consistent solution in the low-Mach limit under the assumption of a discrete Hilbert expansion. The existence of the Hilbert expansion is shown under simplifying assumptions

Numerical modeling of shallow flows including bottom topography and friction effects

The aim of the paper is numerical modeling of the shallow water equation with source terms by genuinely multdimensional finite volume evolution Galerkin schemes. The shallow water system, or its one-dimensional analogy the Saint-Venant equation, is used extensively for numerical simulation of natural rivers. Mathematically the shallow water system belongs to the class of balance laws. A special treatment of the source terms describing the bottom topography as well as frictions effects is necessary in order to reflect their balance with the gradients of fluxes. We present behaviour of our new well-balance FVEG scheme for several benchmark test problems and compare our results with those obtained by the finite element scheme of Teschke et al. used for practical river simulations

Finite volume schemes for multidimensional hyperbolic systems based on the use of bicharacteristics

In this survey paper we present an overview on recent results for the bicharacteristics based finite volume schemes, the so-called finite volume evolution Galerkin (FVEG) schemes. These methods were proposed to solve multidimensional hyperbolic conservation laws. They combine the usually conflicting design objectives of using the conservation form and following the characteritics, or bicharacteritics. This is realized by combining the finite volume formulation with approximate evolution operators, which use bicharacteristics of multidimensional hyperbolic system. In this way all of the infinitely many directions of wave propagation are taken into account. The main goal of this paper is to study long-time behaviour of the FVEG schemes. We present several numerical experiments which con¯rm the fact that the FVEG methods are well-suited for long-time simulation

Combined finite element -- finite volume method (convergence analysis)

summary:We present an efficient numerical method for solving viscous compressible fluid flows. The basic idea is to combine finite volume and finite element methods in an appropriate way. Thus nonlinear convective terms are discretized by the finite volume method over a finite volume mesh dual to a triangular grid. Diffusion terms are discretized by the conforming piecewise linear finite element method. In the paper we study theoretical properties of this scheme for the scalar nonlinear convection-diffusion equation. We prove the convergence of the numerical solution to the exact solution

Finite volume schemes for multi-dimensional hyperbolic systems based on the use of bicharacteristics

In this paper we present recent results for the bicharacteristic based finite volume schemes, the so-called finite volume evolution Galerkin (FVEG) schemes. These methods were proposed to solve multi-dimensional hyperbolic conservation laws. They combine the usually conflicting design objectives of using the conservation form and following the characteristics, or bicharacteristics. This is realized by combining the finite volume formulation with approximate evolution operators, which use bicharacteristics of multi-dimensional hyperbolic system. In this way all of the infinitely many directions of wave propagation are taken into account. The main goal of this paper is to present a self contained overview on the recent results. We study the L1-stability of the finite volume schemes obtained by different approximations of the flux integrals. Several numerical experiments presented in the last section confirm robustness and correct multi-dimensional behaviour of the FVEG methods

Comparison study of some finite volume and finite element methods for the shallow water equations with bottom topography and friction terms

We present a comparison of two discretization methods for the shallow water equations, namely the finite volume method and the finite element scheme. A reliable model for practical interests includes terms modelling the bottom topography as well as the friction effects. The resulting equations belong to the class of systems of hyperbolic partial differential equations of first order with zero order source terms, the so-called balance laws. In order to approximate correctly steady equilibrium states we need to derive a well-balanced approximation of the source term in the finite volume framework. As a result our finite volume method, a genuinely multidimensional finite volume evolution Galerkin (FVEG) scheme, approximates correctly steady states as well as their small perturbations (quasi-steady states). The second discretization scheme, which has been used for practical river flow simulations, is the finite element method (FEM). In contrary to the FVEG scheme, which is a time explicit scheme, the FEM uses an implicite time discretization and the Newton-Raphson iterative scheme for inner iterations. We compare the accuracy and performance of both scheme through several numerical experiments, which demonstrate the reliability of both discretization techniques and correct approximation of quasisteady states with bottom topography and friction

Well-balanced finite volume evolution Galerkin methods for the shallow water equations with source terms

The goal of this paper is to present a new well-balanced genuinely multidimensional high-resolution finite volume evolution Galerkin method for systems of balance laws. The derivation of the method will be illustrated for the shallow water equation with geometrical source term modelling the bottom topography. The results can be generalized to more complex systems of balance law