An Asymptotic Parallel Linear Solver and Its Application to Direct Numerical Simulation for Compressible Turbulence

Abstract

When solving numerically partial differential equations such as the Navier-Stokes equations, higher-order finite difference schemes are occasionally applied for spacial descretization. Compact finite difference schemes are one of the finite difference schemes and can be used to compute the first-order derivative values with smaller number of stencil grid points, however, a linear system of equations with a tridiagonal or pentadiagonal matrix derived from the schemes have to be solved. In this paper, an asymptotic parallel solver for a reduce matrix, that obtained from the Mattor’s method in a computation of the first-order derivatives with an eighth-order compact difference scheme under a periodic boundary condition, is proposed. The asymptotic solver can be applied as long as the number of grid points of each Cartesian coordinate in the parallelized subdomain is 64 or more, and its computational cost is lower than that of the Mattor’s method. A direct numerical simulation code has also been developed using the two solvers for compressible turbulent flows under isothermal conditions, and optimized on the vector supercomputer SX-Aurora TSUBASA. The optimized code is 1.7 times faster than the original one for a DNS with $2048^3$ grid points and the asymptotic solver achieves approximately a 4-fold speedup compared to the Mattor’s solver. The code exhibits excellent weak scalability.