Accuracy and Convergence Studies of the Numerical Solution of Compressible Flow Problems

Abstract: The numerical solution of compressible flow problems governed by the Navier-Stokes equations is considered. A finite volume method is used for the discretization in space. Different techniques to accelerate the convergence to a steady state are suggested, and the accuracy of the spatial difference operator is analyzed.By treating one spatial direction implicitly, it is possible to modify an explicit Runge-Kutta time-marching method, leading to a semi-implicit scheme. A thorough investigation of the stability and convergence properties is presented. Moreover, the scheme is used as a smoother in a multigrid method, and is reformulated as a preconditioner for a number of Newton-Krylov methods. The semi-implicit approach is shown to be very effective for meshes with high aspect ratios. For the flow over a flat plate with a thin boundary layer, the number of iterations to reach convergence is independent of the Reynolds number (Re).An alternative approach for accelerating the convergence is to apply an optimal semicirculant approximation of the spatial operator as a preconditioner. Also here, significant speedups are demonstrated for high Re flows.Two problems appearing for solvers used in computational fluid dynamics are examined. Methods for updating the ghost cells in a multigrid multiblock algorithm are studied, and the accuracy of the finite volume method applied to a polar mesh is analyzed. Although polar mesh singularities lead to a reduction of the order of the truncation error, the global error is shown to be of practically the same order as for a uniform mesh.