多维离散Pissn方程的矩阵―数组形式及其可解性的判定.docVIP

 A Matrix-array Form for the Multidimensional Discrete Poisson Equation and its Solvability Criterion# WANG Tong, GE Yaojun, CAO Shuyang* 5 10 15 20 25 (State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, ShangHai 200092) Abstract: The multidimensional discrete Poisson equation (MDPE) frequently encountered in science and engineering can be expressed, in many cases, as a brief matrix-array equation firstly defined in this paper. This new-style equation consists of a series of small matrices and can be transformed using the Kronecker sum into a familiar system of linear algebraic equations, AX=b. Then it is proved that the eigenvalues and corresponding eigenvectors of A can be obtained directly from those of these small matrices consisting in that matrix-array equation. Based on this connection, a solvability criterion for the matrix-array equation is proposed. Finally, an application of this criterion is carried out, and an inspiration from the above connection are presented and analyzed. Keywords: Matrix-array equation; Multidimensional discrete Poisson equation; Solvability criterion; Kronecker sum; Eigenvalue; Eigenvector 0 Introduction The motivation of this paper comes from an attempt of modeling the two-dimensional (2D) lid-driven cavity flow by applying the SIMPLE-GDQ method proposed by Shu et al.[1-4] to solve the viscous incompressible Navier-Stokes equations in primitive variable form. The SIMPLE-GDQ method is a combination of the SIMPLE algorithm and the GDQ method. So it is still an iterative method comprising several steps one out of which is the computation of pressure correction. The pressure correction is essentially governed by a Poisson equation. In the SIMPLE-GDQ method, the pressure correction equation is discretized on a non-staggered grid using the GDQ method, i.e. N M k??1 k??1  ?  (0.1) (2) ∑ N??1 k1? 2 N??1 ? paper of Shu et al. [4]. p is the pressure correction to be calculat