Adaptive Finite Element Solution Algorithm for the Euler Equations

Abstract

A new finite element algorithm for solving the steady Euler equations describing the flow of an inviscid, compressible, ideal gas is presented. This algorithm uses a finite element spatial discretization coupled with a Runge-Kutta time integration to relax to steady state. It is shown that other algorithms, such as finite difference and finite volume methods, can be derived using finite element principles. A higher-order biquadratic approximation is introduced. Several test problems are computed to verify the algorithms. Adaptive gridding in two and three dimensions using quadrilateral and hexahedral elements is developed and verified. Adaptation is shown to provide CPU savings of a factor of 2-16, and biquadratic elements are shown to provide potential savings of a factor of 2-6. An analysis of the dispersive properties of various discretization methods for the Euler equations is presented, and results allowing the prediction of dispersive errors are obtained. The adaptive algorithm is applied to the solution of several flows in scramjet inlets in two and three dimensions, demonstrating some of the varied physics associated with these flows. Some issues in the design and implementation of adaptive finite element algorithms on vector and parallel computers are discussed. Trust in the LORD with all thine heart; and lean not unto thine own understanding. In all thy ways acknowledge him, and he will direct thy paths.-Proverbs 3:5,6