Parallel Simulation For The Two-dimensional Euler Equations On Unstructured Grids

Strategy for solving two dimensional Euler equations in parallel is presented in this paper. It combines mesh partitioning techniques with a parallel Godunov-type solver on unstructured meshes. Message Passing Interface(MPI) is used for the communication steps.Recursive Spectral Bisection(RSB) algorithm partitions the original domain into subdomains that are well balanced. However it allows edges or elements to straddle between subdomains. In order to be used in conjunction with element-based Godunov algorithm, the RSB algorithm partitions the dual graph of the computational meshes, e.g. Voronoi diagrams for the Delaunay triangulations, such that all information related to a given element is mapped to the same processor.A rank-1 modification for Laplacian matrix associated with a graph is also given. The RSB algorithm relies on the eigenvector, which corresponds to the second smallest eigenvalue of the Laplacian matrix, to compute the partition. In order to calculate this eigenvector efficiently, Lanczos method with no reorthogonalization is used. With rank-1 modification based on the eigenvector corresponding to the smallest Laplacian eigenvalue, Lanczos procedures can usually obtain good approximation for the second smallest eigenvalue of the original Laplacian matrix before numerical difficulties arise.A slope-limiter suitable for cells in unstructured triangulation is developed. To achieve better than first-order accuracy in Godunov method, a piecewise linear function instead of a piecewise constant function is used along the line between the centroids of two neighbored cells. The slope of the linear function in a cell mainly depends on the local gradient of the variables. Through a limiter function, the value of the slope is also affected by the average values and gradients of this variable in neighbored cells such that nonphysical oscillation near discontinuities is avoided.An overlapping mesh partition is used for parallelizing the Godunov algorithm with the above slope-limiter. Cells with common vertices in different subdomains are defined as overlapping cells. After each iteration, only physical state variables in these cells need to be exchanged between subdomains.