Computer-aided analysis of the convergence to steady state of discrete approximations to the euler equations

Abstract

The behaviour of a centered finite volume scheme for the isoenergetic Euler equations in two space dimensions is studied by numerical differentiation and approximate eigensystem analysis. The entire semidiscrete approximation including boundary conditions is formulated as a large system of ODES, which are linearized by numerically approximating the Frechet derivative. An approximate eigensystem procedure that only needs the Frechet derivative is used to extract the least damped eigenmodes. The overall method has been applied to the case of transonic flow past an airfoil and has revealed that the most persistent transient modes are highly structured and are associated with eigenvalues of small modulus. Furthermore, they appear to be centered around the shock region, the stagnation region and the trailing edge/wake region of the airfoil. The beneficial effect of local time-step scaling and artificial dissipation is also demonstrated by the method.