Uniformly high order accurate essentially non-oscillatory schemes, 111
Journal of Computational Physics
Total-variation-diminishing time discretizations
SIAM Journal on Scientific and Statistical Computing
Efficient implementation of essentially non-oscillatory shock-capturing schemes,II
Journal of Computational Physics
Higher order Godunov methods for general systems of hyperbolic conservation laws
Journal of Computational Physics
Multidimensional upwind methods for hyperbolic conservation laws
Journal of Computational Physics
ICOSAHOM '89 Proceedings of the conference on Spectral and high order methods for partial differential equations
Numerical experiments on the accuracy of ENO and modified ENO schemes
Journal of Scientific Computing
An unsplit 3D upwind method for hyperbolic conservation laws
Journal of Computational Physics
Weighted essentially non-oscillatory schemes
Journal of Computational Physics
Accurate upwind methods for the Euler equations
SIAM Journal on Numerical Analysis
Efficient implementation of weighted ENO schemes
Journal of Computational Physics
A high-order Godunov method for multiple condensed phases
Journal of Computational Physics
Accurate monotonicity-preserving schemes with Runge-Kutta time stepping
Journal of Computational Physics
Computational Considerations for the Simulation of Shock-Induced Sound
SIAM Journal on Scientific Computing
Comparison of Several Difference Schemes on 1D and 2D Test Problems for the Euler Equations
SIAM Journal on Scientific Computing
Journal of Computational Physics
Tracking discontinuities in hyperbolic conservation laws with spectral accuracy
Journal of Computational Physics
Accurate monotonicity- and extrema-preserving methods through adaptive nonlinear hybridizations
Journal of Computational Physics
An evaluation of the FCT method for high-speed flows on structured overlapping grids
Journal of Computational Physics
Journal of Computational Physics
A space-time smooth artificial viscosity method for nonlinear conservation laws
Journal of Computational Physics
Hi-index | 31.48 |
A numerical study is undertaken comparing a fifth-order version of the weighted essentially non-oscillatory numerical (WENO5) method to a modern piecewise-linear, second-order, version of Godunov's (PLMDE) method for the compressible Euler equations. A series of one-dimensional test problems are examined beginning with classical linear problems and ending with complex shock interactions. The problems considered are: (1) linear advection of a Gaussian pulse in density, (2) Sod's shock tube problem, (3) the "peak" shock tube problem, (4) a version of the Shu and Osher shock entropy wave interaction and (5) the Woodward and Colella interacting shock wave problem. For each problem and method, run times, density error norms and convergence rates are reported for each method as produced from a common code test-bed. The linear problem exhibits the advertised convergence rate for both methods as well as the expected large disparity in overall error levels; WENO5 has the smaller errors and an enormous advantage in overall efficiency (in accuracy per unit CPU time). For the nonlinear problems with discontinuities, however, we generally see both first-order self-convergence of error as compared to an exact solution, or when an analytic solution is not available, a converged solution generated on an extremely fine grid. The overall comparison of error levels shows some variation flom problem to problem. For Sod's shock tube, PLMDE has nearly half the error, while on the peak problem the errors are nearly the same. For the interacting blast wave problem the two methods again produce a similar level of error with a slight edge for the PLMDE. On the other hand, for the Shu-Osher problem, the errors are similar on the coarser grids, but favors WENO by a factor of nearly 1.5 on the finer grids used. In all cases holding mesh resolution constant though, PLMDE is less costly in terms of CPU time by approximately a factor of 6. If the CPU cost is taken as fixed, that is run times are equal for both numerical methods, then PLMDE uniformly produces lower errors than WENO for the fixed computation cost on the test problems considered here.