Direct and iterative algorithms for the three-dimensional Euler equations

Direct and iterative algorithms have been developed for solving a finite volume discretization of the three-dimensional Euler equations in curvilinear coordinates. The Euler equations are discretized using numerical derivatives of the numerical flux vector for the Jacobians. Two direct solvers are formulated, one of which has a diagonal plane matrix structure with significantly lower memory requirements. The direct solvers are used as a benchmark in measuring the convergence rate and robustness of more computationally efficient solvers which include two factored approaches, a Newton-relaxation algorithm and a discretized Newton-relaxation algorithm, which uses numerical Jacobians. A diagonal plane formulation for the Newton-relaxation algorithms has also been developed that may have the potential for massive parallelization. It is demonstrated that the Newton-relaxation approach can give convergence rates and robustness equal to that of a direct solver for three-dimensional problems. As a demonstration of the robustness of both the Newton-relaxation algorithm and numerical Jacobians, quadratic convergence to machine zero is demonstrated