Numerical Comparison Of The Second-order Godunov Scheme MUSCL And The Fifth-order FD-WENO Scheme For Compressible Euler Equation

A numerical study is undertaken comparing the fifth-order finite difference weighted essentially non-oscillatory scheme (FD-WEN05) to the second-order Godunov scheme MUSCL (monotonic upstream-centered scheme for conservation laws). For quantitative comparison purpose, these methods are tested on a series of problems whose true solutions are known or can be computed with high accuracy, such as linear advection problem, Sod Riemann, "peak" Riemann problem, Woodward and Colella interacting shock wave problem and the two-dimensional periodic vortex problem. For qualitative comparison purpose, these methods are tested on Rayleigh-Taylor instability and Richtmyer-Meshkov instability problems. The numerical results show that: for the Sod and Lax Riemann problems, high accuracy and high resolution results can be easily achieved by both schemes, but MUSCL is performed much faster than WEN05. For the " Peak" Riemann problem and the Woodward and Colella interacting shock wave problem, when both methods are performed on same space mesh, we find that WENO5 is better than MUSCL around the shock and the rarefaction, in more de-tail, WENO5 has not only higher resolution but also smaller errors than MUSCL. For the two-dimensional periodic vortex problem, We find that the numerical results of WENO5 when performed on the uniform mesh of 81 x 81 points are much more accurate than that of MUSCL when performed on the refined uniform mesh of 161 ×161 points, and the former takes CPU time shorter than the latter. For Rayleigh-Taylor instability and Richtmyer-Meshkov instability problems, when both methods are performed on same space mesh, we find that the numerical results of WENO5 are rich of higher resolution and the images are more realistic. In view of all the results mentioned above, we thus conclude that the second-order Godunov scheme MUSCL has advantages for many problems containing only simple shocks with almost linear smooth solutions in between, such as the solutions to most Riemann problems. However, when the solution contains both discontinuities and complex solution structures in the smooth regions, a higher order method, such as WENO5, may be more realistic and more economical in CPU time.