Solutions of the Euler Equations Using Implicit TVD High Resolution Algorithms in Three-Dimensions

Abstract

In the present work, the Steger and Warming and the Van Leer schemes are implemented, on a finite volume context and using a structured spatial discretization, to solve the Euler equations in the three-dimensional space. The Steger and Warming and the Van Leer schemes are flux vector splitting ones and in their original implementation are first order accurate. A MUSCL approach is implemented in these schemes aiming to obtain second order spatial accuracy. The Minmod non-linear limiter is employed to guarantee such accuracy and TVD high resolution properties. Both schemes are implemented following an implicit formulation. The flux vector splitting schemes employ approximate factorizations in ADI form. Both schemes are first order accurate in time. The algorithms are accelerated to the steady state solution using a spatially variable time step procedure, which has demonstrated effective gains in terms of convergence rate, as shown in Maciel. Both schemes are applied to the solution of the physical problems of the supersonic flow along a ramp and the “cold gas” hypersonic flow along a diffuser. The results have demonstrated that the most accurate results are obtained with the Steger and Warming TVD high resolution scheme.