Toward Applying Algebraic Multigrid to Transonic Flow Problem

Abstract

This article presents a new approach to solve the transonic flow problem by the algebraic multigrid (AMG) method. The proposed algorithm is demonstrated to attain excellent convergence rates, independent of the problem size, through the entire range of flow regimes. The mathematical difficulties of the problem are associated with the fact that the governing equation changes its type from elliptic (subsonic flow) to hyperbolic (supersonic flow). A pointwise relaxation method when applied directly to the upwind discrete operator, in the supersonic flow regime, is unstable. Resolving this difficulty is the main achievement of this work. A variety of issues regarding the AMG coarsening and construction of transfer operators is addressed in order to achieve the required efficiency for the problems under consideration. We demonstrate the AMG performance on a variety of model problems involving the quasi-linear full potential equation in two dimensions for subsonic, sonic, and supersonic flow and various flow directions (with respect to the grid). In addition, we present a preliminary result verifying the capabilities of the constructed algorithm to deal with a nonlinear problem—transonic small-disturbance equation.