Multilevel Adaptive Computations in Fluid Dynamics

Abstract

HE usual approach to solving partial differential boundary-value problems is first to discretize the problem in some preassigned manner (e.g., finite element or finite difference equations on a fixed grid), and then to submit the resulting discrete system to some numerical solver. In the multilevel adaptive technique (MLAT),1-23 discretization and solution processes are intermixed. A sequence of uniform grids (or levels), with geometrically decreasing mesh sizes, participates in the process. The cooperative solution process on these grids involves relaxation sweeps over each of them, coarse-grid-to-fine-grid interpolations of corrections and fineto-coarse transfers of residuals. This process has several important benefits. First, it acts as a very fast solver of the algebraic system of equations, since relaxation on each level is very efficient in liquidating those error components whose wavelength is comparable to that level's mesh size. General nonlinear boundary-value problems, such as Navier-Stokes equations in a general domain, are solved at computational work comparable to seven or so relaxation sweeps on the finest grid. The computer storage used may be much smaller than the number of discrete unknowns. Sections II-IX of this paper survey these fast solvers, emphasizing recent work8'10'11 on elliptic systems, such as Cauchy-Rieman n, Stokes, and Navier-Stokes equations. A new type of relaxation, called distributive Gauss Seidel (DGS), has been developed for such systems, based on the elliptic decomposition of the symbol of the finite-difference operators. 8 Also mentioned are fast solvers for compressible viscous flows, for transonic problems, and for unstable steady-state flows. Moreover, the fast multilevel solutions can actually represent better approximations than the full solutions of the difference equations. For example, one can combine the stability of upstream differencing with the accuracy of central differencing by using the first in relaxation and the latter in the residual transfers. One can obtain extrapolations to higher-order approximations by a trivial change in a lowerorder program. Or one can solve evolution problems very inexpensively by performing most time steps on coarse levels, in such a way that the finest-level accuracy is still maintained. These and related techniques (for ill-posed problems, bifurcation problems, and for parametric optimization) are surveyed in Sec. X. Finally, the multilevel structure provides, in a natural way, very flexible and adaptive discretization schemes, which can automatically and efficiently treat boundary layers and other singularities7 (Sec. XI). This is a survey paper. Naturally, not all the ideas expressed here can be rigorously supported at this time. They are described in more detail elsewhere. 2~8