Numerical Dissipation Control in High-Order Methods for Compressible Turbulence: Recent Development

Abstract

This comprehensive overview presents our continued efforts in high-order finite difference method (FDM) development for adaptive numerical dissipation control in the long-time integration of direct numerical simulation (DNS), large eddy simulation (LES), and implicit LES (ILES) computations of compressible turbulence for gas dynamics and MHD. The focus is on turbulence with shock wave numerical simulations using the adaptive blending of high-order structure-preserving non-dissipative methods (classical central, Padé (compact), and dispersion relation-preserving (DRP)) with high-order shock-capturing methods in such a way that high-order shock-capturing methods are active only in the vicinity of shock/shear waves, and high-gradient and spurious high-frequency oscillation regions guided via flow sensors. Any efficient and high-resolution high-order shock-capturing methods are good candidates for the blending of methods procedure. Typically, the adaptive blending of more than one method falls under two camps: hybrid methods and nonlinear filter methods. They are applicable to unstructured finite volume, finite element, discontinuous Galerkin, and spectral element methods. This work represents the culmination of over 20 years of high-order FDM developments and hands-on experience by the authors and collaborators in adaptive numerical dissipation control using the “high order nonlinear filter approach”. Extensions of these FDM versions to curvilinear nonuniform, freestream-preserving moving grids and time-varying deforming grids were also developed. By examining the construction of these two approaches using the high-order multistage type of temporal discretization, the nonlinear filter approach is made more efficient and less CPU-intensive while obtaining similar accuracy. A representative variety of test cases that compare the various blending of high-order methods with standalone standard methods is illustrated. Due to the fact that our nonlinear filter methods are not well known in compressible turbulence with shock waves, the intent of this comprehensive overview is for general audiences who are not familiar with our nonlinear filter methods. For readers interested in the implementation of our methods into their computer code, it is hoped that the long overview will be helpful.