On Source Terms and Boundary Conditions Using Arbitrary High Order Discontinuous Galerkin Schemes

Abstract

This article is devoted to the discretization of source terms and boundary conditions using discontinuous Galerkin schemes with an arbitrary high order of accuracy in space and time for the solution of hyperbolic conservation laws on unstructured triangular meshes. The building block of the method is a particular numerical flux function at the element interfaces based on the solution of Generalized Riemann Problems (GRPs) with piecewise polynomial initial data. The solution of the generalized Riemann problem, originally introduced by Toro and Titarev in a finite volume context, provides simultaneously a numerical flux function as well as a time integration method. The resulting scheme is extremely local since it integrates the PDE from one time step to the successive one in a single step using only information from the direct side neighbors. Since source terms are directly incorporated into the numerical flux via the solution of the GRP, our very high order accurate method is also able to maintain very well smooth steady-state solutions of PDEs with source terms, similar to the so-called well-balanced schemes which are usually specially designed for this purpose. Boundary conditions are imposed solving inverse generalized Riemann problems. Furthermore, we show numerical evidence proving that by using very high order schemes together with high order polynomial representations of curved boundaries, high quality solutions can be obtained on very coarse meshes.Keywords: discontinuous Galerkin schemes, ADER approach, source terms, boundary conditions, unstructured meshes Discretization of Source Terms with Arbitrary High Order DG SchemesWe consider the non-homogeneous system of conservation lawswith the fluxS is a source term that depends on the state vector u and that may also explicitly depend on space x = (x, y) and time t. For (1), proper initial conditions at t = 0 have to be provided. On the boundary ∂Ω of the domain, appropriate boundary conditions must be imposed on u or on the fluxes.The computational domain Ω is divided into conforming triangles T (m) addressed by a unique index (m). The numerical solution u h is sought in the space V of piecewise polynomials up to degree N. It is hence approximated inside each triangle T (m) by a linear combination of some time independent polynomial basis functions Φ l ( ξ) of degree N defined on T (m) and with time dependent degrees of freedomû (m) (t):We note that the coordinates ξ refer to a reference element T E , see Fig. 2. The number of degrees of freedom per ele-. For the discretization of the inhomogeneous equation (1), we multiply it by test functions Φ k from the same space V of piecewise polynomials of order N . After integration by parts, the semi-discrete DG scheme for the numerical solution u h is

Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems

“…It was originally developed by Dyson [22] for the Euler equations and was then modified by Dumbser and Munz [21,20] and Taube et al [46] in their work on ADER-DG schemes for the Euler and MHD equations. The two-dimensional case is obtained by setting o oz ¼ 0 and w = 0.…”

NURBS‐enhanced finite element method (NEFEM)

“…A comparison between isoparametric curved FEs and subparametric FEs with linear approximation for the geometry is often presented in order to illustrate the necessity of using curved elements. That is, the polygonal approximation is maintained but the degree of the approximation is increased, see [9,34,57] to name a few. The main problem of considering subparametric formulations with only a linear approximation of the geometry is that the solution of a different problem is considered, that is the flow around a polygon rather than the flow around a circle.…”

“…More recently, in [55] a new methodology is presented for the computation of the fluxes across curved boundaries but, unfortunately, the proposed method is not conservative. In [34], the advantages of using high-order isoparametric elements for the numerical solution of inviscid compressible flow problems are also illustrated. Using ultra-coarse meshes and high-order approximations the authors show the benefits of curved elements.…”

“…The importance of the geometrical model in the numerical solution of Euler equations is crucial, see for instance [8,9,23,27,34,37,55,93,98,99] to name a few. In this section NEFEM is presented as a powerful method for solving the Euler equations of gas dynamics in the presence of curved boundaries.…”

A sixth-order finite volume method for the 1D biharmonic operator: Application to intramedullary nail simulation

“…The finite volume method is a popular technique usually employed to carry out numerical approximations for conservation laws (see, for example, Kroner, 1997;Leveque, 2002;Audusse and Bristeau, 2007;Dumbser and Munz, 2007;Trangenstein, 2009). …”