An assessment of compact schemes combined with compact filters for a numerical simulation of compressible flow

Abstract

Characteristics of compact schemes combined with compact filters for compressible flows are numerically investigated on the Sod's problem in this study. Tridiagonal 6th, 8th and pentadiagonal 10th order compact schemes developed by Lele and 4th order compact scheme developed by Kim are used for the spatial derivatives and compact filters proposed by Lele, Gaitonde, Zhanxin, Kim are examined under CFL=1.0 condition. L1 and L2 norm are calculated from errors on velocity field from the end of expansion wave to shock. Results show that the compact filter developed by Gaitonde (free parameter is near 0.5) with Lele's 6th order compact scheme makes large error near the end of expansion wave and it cannot remove numerical oscillations perfectly near boundary. It is also revealed by the tests with Gaitonde's 8th order compact filter that the higher order compact scheme developed by Lele makes smaller L1 and L2 norms and the 4th order compact scheme developed by Kim makes the smallest L1 and L2 norm. In addition, it is found by the tests with Kim's 6th order compact filter that the higher order compact scheme developed by Lele makes smaller L1 and L2 norms and the 4th order compact scheme developed by Kim makes the smallest L1 and L2 norm. In all combinations of compact schemes and compact filters assessed in the present paper, although no combination of them can suppress spurious oscillations near the end of expansion wave and a shock perfectly, Kim's 4th order compact scheme combined with Kim's 6th order compact filter is the most appropriate to capture shock.

Keywords

• compressible flow
• numerical simulation
• compact scheme
• compact filter
• shock capturing
• sod's problem