Prof. Charles Swanson, NASA Langley Research Center and Old Dominion University, USA
Design of RK/Implicit Schemes for Solving the Navier-Stokes Equations
The design and construction of accurate, robust, and efficient solution algorithms for solving the Navier-Stokes equations requires a complete understanding of the role and effectiveness of each component of the algorithm. One class of solution algorithms allowing substantial flexibility in the design process is the RK/Implicit schemes that have been developed in the last several years. Such numerical schemes are constructed with an explicit Runge-Kutta (RK) framework. The usual limitations of explicit schemes with respect to stability and solving stiff systems of equations are removed with an implicit preconditioner. The resulting contractive algorithm has good h-ellipticity and is used as a preconditioner for a full approximation scheme (FAS) multigrid. To ensure that the algorithm has certain properties various analysis techniques are employed. For example, Fourier analysis, and in particular eigensystem analysis, are applied to examine the stabiity and smoothing properties of such schemes. In this presentation the design of the various components and the analysis of these components will be discussed. The distinction between weak and strong solution algorithms will also be included in the discussion, as well as the consequences of numerial difficulties that arise with weak algorithms and the possibility of incorrect physical solutions.