A review on TVD schemes and a refined flux-limiter for steady-state calculations. (English) Zbl 1349.76558
Summary: This paper presents an extensive review of most of the existing TVD schemes found in literature that are based on the One-step Time-space-coupled Unsteady TVD criterion (OTU-TVD), the Multi-step Time-space-separated Unsteady TVD criterion (MTU-TVD) and the Semi-discrete Steady-state TVD criterion (SS-TVD). The design principles of these schemes are examined in detail. It is found that the selection of appropriate flux-limiters is a key design element in developing these schemes. Different flux-limiter forms (CFL-dependent or CFL-independent, and various limiting criteria) are shown to lead to different performances in accuracy and convergence. Furthermore, a refined SS-TVD flux-limiter, referred to henceforth as TCDF (Third-order Continuously Differentiable Function), is proposed for steady-state calculations based on the review. To evaluate the performance of the newly proposed scheme, many existing classical SS-TVD limiters are compared with the TCDF in eight two-dimensional test cases. The numerical results clearly show that the TCDF results in an improved overall performance.
MSC:
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
total variation diminishing; high resolution scheme; normalized variable diagram; convection boundedness criterion; convection discretization; flux limiterSoftware:
SHASTAReferences:
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