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

Abstract

In this work, the Roe, the Steger and Warming, the Van Leer, the Chakravarthy and Osher, the Harten, the MacCormack, the Frink, Parikh and Pirzadeh, the Liou and Steffen Jr. and the Radespiel and Kroll first order schemes are implemented employing an implicit formulation to solve the Euler equations in the two-dimensional space. These schemes are implemented according to a finite volume formulation and using a structured spatial discretization. The Roe, the Chakravarthy and Osher, the Harten and the Frink, Parikh and Pirzadeh schemes are flux difference splitting ones, whereas the others are flux vector splitting schemes. The implicit schemes employ an ADI (“Alternating Direction Implicit”) approximate factorization or Symmetric Line Gauss-Seidel to solve implicitly the Euler equations. Explicit and implicit results are compared, as also the computational costs, trying to emphasize the advantages and disadvantages of each formulation. The schemes are accelerated to the steady state solution using a spatially variable time step, which has demonstrated effective gains in terms of convergence rate according to Maciel. The algorithms are applied to the
solution of the physical problem of the moderate supersonic flow along a compression corner. The results have demonstrated that the most accurate solutions are obtained with the Harten first order scheme, when implemented in its explicit version. The best wall pressure distribution is obtained by the Radespiel and Kroll first order scheme, in both explicit and implicit cases.