A non‐linearly stable implicit finite element algorithm for hypersonic aerodynamics

Abstract

AbstractA generalized curvilinear co‐ordinate Taylor weak statement implicit finite element algorithm is developed for the two‐dimensional and axisymmetric compressible Navier‐Stokes equations for ideal and reacting gases. For accurate hypersonic simulation, air is modelled as a mixture of five perfect gases, i.e. molecular and atomic oxygen and nirogen as well as nitric oxide. The associated pressure is then determined via Newton solution of the classical chemical equilibrium equation system. The directional semi‐discretization is achieved using an optimal metric data Galerkin finite element weak statement, on a developed ‘companion conservation law system’, permitting classical test and trial space definitions. Utilizing an implicit Runge‐Kutta scheme, the terminal algorithm is then non‐linearly stable, and second‐order accurate in space and time on arbitrary curvilinear co‐ordinates. Subsequently, a matrix tensor product factorization procedure permits an efficient numerical linear algebra handling for large Courant numbers. For ideal‐ and real‐gas hypersonic flows, the algorithm generates essentially non‐oscillatory numerical solutions in the presence of strong detached shocks and boundary layer inviscid flow interactions.