Designing Adaptive Low Dissipative High Order Schemes for Long-time Integrations

Classical stability and convergence theory are based on linear and local linearized analysis as the time steps and grid spacings approach zero. This theory offers no guarantee for nonlinear stability and convergence to the correct solution of the nonlinear governing equations. The use of numerical dissipation has been the key mechanism in combating numerical instabilities. Aside from acting as a post-processor step, most filters serve as some form of numerical dissipation. Without loss of generality, “numerical-dissipation/filter” is, hereafter, referred to as “numerical dissipation”. Proper control of the numerical dissipation to accurately resolve all relevant multiscales of complex flow problems while still maintaining nonlinear stability and efficiency for long-time numerical integrations poses a great challenge to the design of numerical methods. The required type and amount of numerical dissipation are not only physical problem dependent, but also vary from one flow region to another. This is particularly true for unsteady high-speed shock/shear/boundary-layer/turbulence/acoustics interactions and/or combustion problems since the dynamics of the nonlinear effect of these flows are not well-understood, while longtime integrations of these flows have already stretched the limit of the current available supercomputers and the existing numerical methods. It is of paramount importance to have proper control on the type and amount of numerical dissipation in regions where it is needed but nowhere else. Inappropriate type and/or amount can be detrimental to the integrity of the computed solution, even with extensive grid refinement.