Improving the convergence rate of the Petrov‐Galerkin techniques for the solution of transonic and supersonic flows

Published in Int. J. Numer. Meth. Engng. Vol. 34 (2), pp. 543-568, 1992
doi: 10.1002/nme.1620340211

Abstract

This paper report progress on a technique to accelerate the convergence to steady solutions when the streamline‐upwind/Petrov‐Galerkin (SUPG) technique is used. Both the description of a SUPG formulation and the documentation of the development of a code for the finite element solution of transonic and supersonic flows are reported. The aim of this work is to present a formulation to be able to treat domains of any configuration and to use the appropriate physical boundary conditions, which are the major stumbling blocks of the finite difference schemes, together with an appropriate convergence rate to the steady solution.

The implemented code has the following features: the Hughes' SUPG‐type formulation with an oscillation‐free shock‐capturing operator, adaptive refinement, explicit integration with local time‐step and hourglassing control. An automatic scheme for dealing with slip boundary conditions and a boundary‐augmented lumped mass matrix for speeding up convergence.

It is shown that the velocities at which the error is absorbed in and ejected from the domain (that is damping and group velocities respectively) are strongly affected by the time step used, and that damping gives an O($N^{2}$) algorithm contrasting with the O(N) one given by absorption at the boundaries. Nonetheless, the absorbing effect is very low when very different eigenvalues are present, such as in the transonic case, because the stability condition imposes a too slow group velocity for the smaller eigenvalues. To overcome this drawback we present a new mass matrix that provides us with a scheme having the highest group velocity attainable in all the components.

In Section 1 we will describe briefly the theoretical background of the SUPG formulation. In Section 2 it is described how the foregoing formulation was used in the finite element code and which are the appropriate boundary conditions to be used. Finally in Section 3 we will show some results obtained with this code.