Uniformly high order accurate essentially non-oscillatory schemes, 111
Journal of Computational Physics
Uniformly high-order accurate nonoscillatory schemes
SIAM Journal on Numerical Analysis
Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Efficient implementation of essentially non-oscillatory shock-capturing schemes,II
Journal of Computational Physics
Efficient implementation of weighted ENO schemes
Journal of Computational Physics
Weighted essentially non-oscillatory schemes on triangular meshes
Journal of Computational Physics
ADER: A High-Order Approach for Linear Hyperbolic Systems in 2D
Journal of Scientific Computing
ADER: Arbitrary High Order Godunov Approach
Journal of Scientific Computing
Fast high order ADER schemes for linear hyperbolic equations
Journal of Computational Physics
Journal of Computational Physics
ADER schemes for three-dimensional non-linear hyperbolic systems
Journal of Computational Physics
Derivative Riemann solvers for systems of conservation laws and ADER methods
Journal of Computational Physics
A direct Eulerian GRP scheme for compressible fluid flows
Journal of Computational Physics
Journal of Computational Physics
Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem
Numerische Mathematik
Journal of Computational Physics
Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms
Journal of Computational Physics
The generalized Riemann problems for compressible fluid flows: Towards high order
Journal of Computational Physics
Hyperbolic reformulation of a 1D viscoelastic blood flow model and ADER finite volume schemes
Journal of Computational Physics
Hi-index | 31.47 |
We study three methods for solving the Cauchy problem for a system of non-linear hyperbolic balance laws with initial condition consisting of two smooth vectors, with a discontinuity at the origin, a high-order Riemann problem. Two of the methods are new; one of the them results from a re-interpretation of the high-order numerical methods proposed by Harten et al. [A. Harten, B. Engquist, S. Osher, S.R. Chakravarthy, Uniformly high order accuracy essentially non-oscillatory schemes III, J. Comput. Phys. 71 (1987) 231-303] and the other is a modification of the solver in [E.F. Toro, V.A. Titarev, Solution of the generalised Riemann problem for advection-reaction equations, Proc. Roy. Soc. London A 458 (2002) 271-281]. A systematic assessment of all three solvers is carried out and their relative merits are discussed. We also implement the solvers, locally, in the context of high-order finite volume numerical methods of the ADER type, on unstructured meshes. Schemes of up to fifth order of accuracy in space and time for the two-dimensional compressible Euler equations and the shallow water equations with source terms are constructed. Empirically obtained convergence rates are studied systematically and, for the tests considered, these correspond to the theoretically expected orders of accuracy. We also address the question of balance between flux gradients and source terms, for steady flow. We find that the ADER schemes may be termed asymptotically well-balanced, in the sense that the well-balanced property is attained as the order of the method increases, and this without introducing any ad-hoc fixes to the schemes or the equations.