Ayaboe Edoh

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.

Comparison of Artificial Dissipation and Filtering Schemes for Time-Accurate Simulations

This study makes a comparison between artificial dissipation and filtering schemes as stabilization techniques for time-accurate fluid dynamics simulations. Specifically, we use von Neumann stability analysis to assess these alternatives in terms of their concurrent ability to maintain solution quality through proper preservation of low frequency content, while selectively damping high frequency errors. Our studies reveal the limitations of traditional artificial dissipation schemes as well as of explicit filtering procedures, while also providing a common framework to understand the relationship between these approaches. Importantly, we develop appropriate definitions of Pade-type (implicit) filtering procedures that have favorable properties for time-accurate computations over a wide range of time scales and Mach numbers.

Optimal Runge-Kutta Schemes for High-order Spatial and Temporal Discretizations

Abstract : Numerical discretization for unsteady flow simulations can be broken down into spatial and temporal parts which interplay in complex and sometimes unexpected ways. This paper attempts to address how the properties of the spatial discretization help drive the choice of temporal discretization. In addition, it examines methods for higher than second-order accurate time integration using L-stable singly-diagonally-implicit (ESDIRK) Runge-Kutta methods. Von Neumann analysis is used to examine the theoretical effects of different spatial/temporal discretization combinations. The predictive nature of the von Neumann analysis is then validated through the exploration of the convection of acoustic waves in one dimension and an isentropic vortex in three dimensions. Is is shown that the computational results follow the expected trends taking the von Neumann analysis of the schemes into account. This work highlights that, for unsteady problems, both dissipation and dispersion errors must be accounted for when selecting optimal Runge-Kutta time integrators.