A QUANTITATIVE, SECOND-LAW BASED MEASURE OF ACCURACY OF NUMERICAL SCHEMES

摘要

All state-of-the-art schemes used in CFD are at least second-order accurate in continuous regions. Unfortunately, the accuracy of these schemes drops to first order in the vicinity of discontinuities and many schemes also lose accuracy near extrema. A variety of modifications recently have been investigated to combat this drop in accuracy, with different strategies producing different solutions. Still, conventional order-estimates become meaningless near discontinuities, and aesthetics (i.e., the lack of high frequency oscillations, smearing, or other qualitative indicators) are often relied on to assess the quality of a scheme. This approach can be dependable when used to evaluate one-dimensional, low Mach number flow fields. However, without prior knowledge of the actual solution, this approach is unreliable for high Mach number flows, multi-dimensional flows, or flow fields with naturally occurring dissipative phenomena. To remedy this, we use the second law of thermodynamics to assess the accuracy of numerical schemes. This approach supplements traditional error estimates and may be particularly important when comparing schemes with the same formal accuracy. One first-order and four second-order schemes are analyzed, a derivative of the Lax-Wendroff scheme, a MacCormack-based TVD scheme, a TVD scheme based on Roe's approximate Riemann solver, and an UNO scheme. The one-dimensional Riemann problem is used as a test case for comparison.