Explicit and Implicit Solutions of First Order Algorithms Applied to the Euler Equations in Three-Dimensions

In the present work, the Roe, the Steger and Warming, the Van Leer, the Chakravarthy and Osher, the Harten, the Frink, Parikh and Pirzadeh, the Liou and Steffen Jr. and the Radespiel and Kroll schemes are implemented, on a finite volume context and using an upwind structured spatial discretization, to solve the Euler equations in the three-dimensional space. The Roe, the Harten, the Chakravarthy and Osher and the Frink, Parikh and Pirzadeh schemes are flux difference splitting ones, whereas the others schemes are flux vector splitting ones. All eight schemes are first order accurate in space and their explicit and implicit versions are implemented in three-dimensions. The explicit time integration uses a Runge-Kutta, a time splitting or an Euler method. The former is second order accurate in time, whereas the others are first order accurate in time. In the implicit case, an ADI approximate factorization is employed, which is first order accurate in time. The physical problems of the supersonic flow along a ramp, in the implicit case, and the “cold gas” hypersonic flows around a blunt body and along an air inlet, in the explicit case, are solved. The results have demonstrated that the Liou and Steffen Jr. scheme is the most conservative algorithm among the studied ones, whereas the Van Leer scheme is the most accurate. Key-Words: Flux difference splitting algorithms, Flux vector splitting algorithms, Structured schemes, Euler equations, Three-Dimensions, Supersonic and hypersonic flows.