Boundary modifications of the dissipation operators for the three-dimensional Euler equations

Abstract

Explicit methods for the solution of fluid flow problems are of considerable interest in supercomputing. These methods parallelize well. The treatment of the boundaries is of particular interest with respect to both the numeric behavior of the solution and the computational efficiency. We have solved the three-dimensional Euler equations for a twisted channel using second-order, centered difference operators, and a three-stage Runge-Kutta method for the integration. Three different fourth-order dissipation operators were studied for numeric stabilization: one positive definite, one positive semidefinite, and one indefinite. The operators only differ in the treatment of the boundary. For computational efficiency all dissipation operators were designed with a constant bandwidth in matrix representation, with the bandwidth determined by the operator in the interior. The positive definite dissipation operator results in a significant growth in entropy close to the channel walls. The other operators maintain constant entropy. Several different implementations of the semidefinite operator obtained through factoring of the operator were also studied. We show the difference both in convergence rate and robustness for the different dissipation operators, and the factorizations of the operator due to Eriksson. For the simulations in this study one of the factorizations of the semidefinite operator required 70%–90% of the number of iterations required by the positive definite operator. The indefinite operator was sensitive to perturbations in the inflow boundary conditions. The simulations were performed on a 8,192 processor Connection Machine system CM-2. Full processor utilization was achieved, and a performance of 135 Mflops/sec in single precision was obtained. A performance of 1.1 Gflops/sec for a fully configured system with 65,536 processors was demonstrated.