A well-balanced scheme for the numerical processing of source terms in hyperbolic equations
SIAM Journal on Numerical Analysis
High-order accurate discontinuous finite element solution of the 2D Euler equations
Journal of Computational Physics
The Runge-Kutta discontinuous Galerkin method for conservation laws V multidimensional systems
Journal of Computational Physics
Journal of Computational Physics
Asymptotic single and multiple scale expansions in the low Mach number limit
SIAM Journal on Applied Mathematics
ADER: Arbitrary High Order Godunov Approach
Journal of Scientific Computing
An Efficient Inverse Aerodynamic Design Method for Subsonic Flows
An Efficient Inverse Aerodynamic Design Method for Subsonic Flows
Well balanced finite volume methods for nearly hydrostatic flows
Journal of Computational Physics
ADER schemes for three-dimensional non-linear hyperbolic systems
Journal of Computational Physics
Building Blocks for Arbitrary High Order Discontinuous Galerkin Schemes
Journal of Scientific Computing
Journal of Computational Physics
Discontinuous Galerkin Methods: Theory, Computation and Applications
Discontinuous Galerkin Methods: Theory, Computation and Applications
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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.