Highly-Accurate Filter-Based Artificial-Dissipation Schemes for Stiff Unsteady Fluid Systems

It is well known that the unmodified application of central difference schemes to the Euler equations produces numerically unstable results because such schemes do not naturally damp the high-frequency modes involved in odd-even decoupling. This shortcoming is usually overcome by adding artificial-dissipation terms, thereby producing stable schemes at the price of potential loss of solution accuracy. For unsteady fluid dynamics, solution filtering schemes have been proposed as a more accurate alternative to artificial dissipation especially when explicit physical-time integration is utilized. However, to solve computationally stiff problems efficiently, it is necessary to use a dual-time stepping approach, to which the application of solution filtering is not straightforward. Restricting the solution filtering only to the physical-time level does not guarantee numerical stability, as errors can accumulate in pseudo-time, causing divergence, while including it at the pseudo-time level introduces inconsistencies that lead to convergence problems. In the present work, Shapiroand Purser-type explicit solution filters are used to derive a new class of filter-equivalent artificial-dissipation operators that can be applied in pseudo time to produce stable, convergent, low-dissipation solutions. These novel artificial-dissipation operators are shown to be indistinguishable from their corresponding filtering procedure for explicit, single-time schemes and are just as effective within a dual-time framework. In addition, the filter-based artificial-dissipation schemes are formulated for use with local preconditioning methods and applied to stiff problems such as low-Mach unsteady flows.