A new preconditioning algorithm for computable compressible flow

Abstract

Low velocity flows often exhibit incompressible properties, and one of the most prohibitive aspects of these problems is a large number of computer resources required, including both CPU time and memory. Various numerical schemes used to calculate incompressible flow are constantly updated to accelerate convergence and reduce resource occupation, but incompressible flow is an ideal model for studying theoretical problems after all. In addition, it is a common phenomenon that high-speed and low-speed flow regions exist in the same system, and the influence of heat and volume force cannot be ignored in some cases. The artificial compressibility method is based on the idea that the numerical algorithms for compressible flows are used to solve incompressible flow. The system of compressible flow governing equations at very low Mach numbers is stiff due to the large disparity in acoustic wave speed, <i>u</i> + <i>c</i>, and the waves convecting at fluid speed, <i>u</i>. The preconditioning algorithm is effective to change the eigenvalues of the compressible flow equations system so as to remove the large disparity in wave speed, and the essence is to multiply the time derivatives with a suitable matrix. A function in low growth rate with Mach number as a variable is used to construct another new preconditioning matrix. Compared with other matrices of Dailey, Weiss, Choi and Pletcher, the new matrix can well improve the stiffness of the governing equations and the smoothness of eigenvalues in all-speed domain. A one-dimensional numerical example shows that the preconditioning matrix has ability to improve the efficiency of solving low-speed flow problems. These preconditioning matrices are extended to two-dimensional problems to simulate inviscid flow passing through a pipe with bulge and viscous flows passing through a flat and cavity. The results indicate that the new matrix has not only better accuracy but also higher efficiency than Weiss’s and Pletcher’s.