Composite overlapping meshes for the solution of partial differential equations
Journal of Computational Physics
Numerical solution of the Riemann problem for two-dimensional gas dynamics
SIAM Journal on Scientific Computing
Efficient implementation of weighted ENO schemes
Journal of Computational Physics
Wave propagation algorithms for multidimensional hyperbolic systems
Journal of Computational Physics
Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method
Journal of Computational Physics - Special issue: commenoration of the 30th anniversary
Approximate Riemann solvers, parameter vectors, and difference schemes
Journal of Computational Physics - Special issue: commenoration of the 30th anniversary
The Runge-Kutta discontinuous Galerkin method for conservation laws V multidimensional systems
Journal of Computational Physics
Journal of Computational Physics
Toward the ultimate conservative scheme: following the quest
Journal of Computational Physics
A class of approximate Riemann solvers and their relation to relaxation schemes
Journal of Computational Physics
Journal of Computational Physics
High Order Fluctuation Schemes on Triangular Meshes
Journal of Scientific Computing
Journal of Computational Physics
A High-Resolution Rotated Grid Method for Conservation Laws with Embedded Geometries
SIAM Journal on Scientific Computing
Essentially non-oscillatory Residual Distribution schemes for hyperbolic problems
Journal of Computational Physics
Hi-index | 31.45 |
Residual distributions (RD) schemes are a class of high-resolution finite volume methods for unstructured grids. A key feature of these schemes is that they make use of genuinely multidimensional (approximate) Riemann solvers as opposed to the piecemeal 1D Riemann solvers usually employed by finite volume methods. In 1D, LeVeque and Pelanti [R.J. LeVeque, M. Pelanti, A class of approximate Riemann solvers and their relation to relaxation schemes, J. Comput. Phys. 172 (2001) 572] showed that many of the standard approximate Riemann solver methods (e.g., the Roe solver, HLL, Lax-Friedrichs) can be obtained from applying an exact Riemann solver to relaxation systems of the type introduced by Jin and Xin [S. Jin, Z.P. Xin, Relaxation schemes for systems of conservation-laws in arbitrary space dimensions, Commun. Pure Appl. Math. 48 (1995) 235]. In this work we extend LeVeque and Pelanti's results and obtain a multidimensional relaxation system from which multidimensional approximate Riemann solvers can be obtained. In particular, we show that with one choice of parameters the relaxation system yields the standard N-scheme. With another choice, the relaxation system yields a new Riemann solver, which can be viewed as a genuinely multidimensional extension of the local Lax-Friedrichs scheme. This new Riemann solver does not require the use Roe-Struijs-Deconinck averages, nor does it require the inversion of an mxm matrix in each computational grid cell, where m is the number of conserved variables. Once this new scheme is established, we apply it on a few standard cases for the 2D compressible Euler equations of gas dynamics. We show that through the use of linear-preserving limiters, the new approach produces numerical solutions that are comparable in accuracy to the N-scheme, despite being computationally less expensive.