AIAA 2003–3705 Optimization of Multistage Coefficients for Explicit Multigrid Flow Solvers

Explicit Euler and Navier-Stokes flow solvers based on multigrid schemes combined with modified multistage methods have become very popular due to their efficiency and ease of implementation. An appropriate choice of multistage coefficients allows for high damping and propagative efficiencies in order to accelerate convergence to the steady-state solution. However, most of these coefficients have been optimized either by trial and error or by geometric methods. In this work, we propose to optimize the damping and propagative efficiencies of modified multistage methods by expressing the problem within the framework of constrained non-linear optimization. We study the optimization of different objective functions with a variety of constraints and their effects on the convergence rate of a two-dimensional Euler flow solver. We start with a simple scalar one-dimensional model wave equation to demonstrate that the constrained optimization results yield multistage coefficients that are comparable to those that have been used in the past. We then extend the method to the case of the two-dimensional Euler equations and we distinguish each optimization case not only by choosing a different objective function, but also by specifying the conditions in a generic computational cell in terms of three parameters: the local Mach number, the aspect ratio, and the flow angle. Using this approach, it is possible to find the optimized coefficients for different combinations of these three parameters and use suitable interpolations for each of the cells in the mesh. Finally, we present observations regarding the relative importance of the propagative and damping efficiency of explicit schemes and conclude that, although a certain level of damping is required for stability purposes, the major contribution to convergence appears to be the propagative efficiency of the scheme. Nomenclature i Imaginary unit = √ −1 j, k Cell indices in the two computational coordinate directions m Last (highest) level for multistage scheme l Arbitrary level index for multistage scheme α l , β l Modified Runge-Kutta multistage coefficients w State variable for the scalar wave equation W State vector for conservative variables dW e Differential state vector for entropy variables W j,k Discretized state vector (state at mesh points) Fx, Fy Flux vectors in the x and y directions S Cell surface dS x , dS y Projections of S in the x and y directions u x-component of velocity v y-component of velocity ρ Density p Static pressure E Total energy (internal plus kinetic) H Total enthalpy …