Accuracy, robustness, and efficiency comparison in iterative computation of convection diff
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Accuracy, Robustness, and E ciency Comparison in Iterative Computation of Convection Di usion Equation with Boundary Layers
Lixin Gey and Jun Zhangz Department of Computer Science, University of Kentucky, 773 Anderson Hall, Lexington, KY 40506-0046, USA August 9, 1999
convection di usion equation, boundary layer, grid stretching, multilevel preconditioner, multigrid method
Key words:
1 Introduction
We consider the two dimensional convection di usion equation with the Dirichlet boundary conditions written in the form uxx + uyy + p(x; y)ux + q(x; y)uy = f (x; y); (x; y) 2 ; (1) u(x; y) = g(x; y); (x; y) 2 @ : The convection coe cients p(x; y) and q(x; y) are functions of the independent variables x and y, and are assumed to be su ciently smooth. Here is a convex domain consisting of a union of rectangles, and @ is the boundary of . For convenience, we refer to the magnitude of p(x; y)
1
and q(x; y) as the Reynolds number (Re). Re determines the strength of the convection relative to that of the di usion. Numerical solution of Equation (1) based on iterative solution methods may be di cult when the Reynolds number is large 19]. If both convection coe cients vanish at some point in , this point is called a stagnation point. Convection di usion equations with stagnation points in their domains are usually used to model recirculating ow problems. For large Re, this type of problems are very hard to solve with classical multigrid method 21]. There are various ways to discretize Equation (1). Traditional nite di erence schemes are the central di erence scheme (CDS) and the upwind di erence scheme (UPS). In the case of CDS, classical iterative methods for solving the resulting linear system do not converge, when the convective terms dominate. Conventional upwind di erence approximation is computationally stable, but is only rst order accurate; and the resulting solution exhibits the e ect of arti cial viscosity 11, 15]. There is considerable interest in developing improved nite di erence discretization schemes for the convection di usion equations that o er both stability and highly accurate approximate solutions 5, 8, 9, 10, 16, 22]. Gupta et al. 8] proposed a nine point fourth order compact nite di erence scheme (FCS) for solving the convection di usion equation on uniform grids. Recently, this scheme has been extended to nonuniform grids 9, 17]. We used it with a multigrid method in 6] and obtained good results for some test problems with large Re. The three di erence schemes, CDS, UPS, and FCS, are easy to implement. The resulting linear systems are frequently used as test examples for iterative methods 4, 14]. In a recent study, the relative advantages and disadvantages of these three schemes on uniform grids are compared, in terms of computational e ciency, solution accuracy, and the algebraic properties of the discretized linear systems 19]. A preconditioned Krylov subspace method was used in 19] to solve the resulting linear systems. The test results in 19] indicate that FCS is computationally more e cient, and represents a compromise between CDS and UPS in terms of stability and solution accuracy. However, none of these schemes with a uniform discretization is desirable, if the computational domain contains some boundary layers, where the solution changes rapidly. This paper is to compare the relative advantages and disadvantages of these three schemes, when a nonuniform discretization mesh is used to resolve the boundary layers. We also compare the relative robustness and e ciency of a classical multigrid method and a multilevel preconditioning technique used to solve the resulting linear systems. This paper is intended to serve as an evaluation of these three di erence schemes for accurate and e cient computations of the convection di usion equations with boundary layers. A set of strategies (discretization, grid stretching, iterative solutions) is proposed for high accuracy, highly e cient, and robust computation of boundary layer problems. Section 2 of this paper discusses the three nite di erence schemes, and, in particular, FCS for the transformed convection di usion equation. Section 3.1 brie y introduces the multigrid method and multilevel preconditioning technique used for solving the resulting linear systems. Numerical results are presented and interpreted in Section 4. Finally, concluding remarks are summarized in Section 5.
Lixin Gey and Jun Zhangz Department of Computer Science, University of Kentucky, 773 Anderson Hall, Lexington, KY 40506-0046, USA August 9, 1999
convection di usion equation, boundary layer, grid stretching, multilevel preconditioner, multigrid method
Key words:
1 Introduction
We consider the two dimensional convection di usion equation with the Dirichlet boundary conditions written in the form uxx + uyy + p(x; y)ux + q(x; y)uy = f (x; y); (x; y) 2 ; (1) u(x; y) = g(x; y); (x; y) 2 @ : The convection coe cients p(x; y) and q(x; y) are functions of the independent variables x and y, and are assumed to be su ciently smooth. Here is a convex domain consisting of a union of rectangles, and @ is the boundary of . For convenience, we refer to the magnitude of p(x; y)
1
and q(x; y) as the Reynolds number (Re). Re determines the strength of the convection relative to that of the di usion. Numerical solution of Equation (1) based on iterative solution methods may be di cult when the Reynolds number is large 19]. If both convection coe cients vanish at some point in , this point is called a stagnation point. Convection di usion equations with stagnation points in their domains are usually used to model recirculating ow problems. For large Re, this type of problems are very hard to solve with classical multigrid method 21]. There are various ways to discretize Equation (1). Traditional nite di erence schemes are the central di erence scheme (CDS) and the upwind di erence scheme (UPS). In the case of CDS, classical iterative methods for solving the resulting linear system do not converge, when the convective terms dominate. Conventional upwind di erence approximation is computationally stable, but is only rst order accurate; and the resulting solution exhibits the e ect of arti cial viscosity 11, 15]. There is considerable interest in developing improved nite di erence discretization schemes for the convection di usion equations that o er both stability and highly accurate approximate solutions 5, 8, 9, 10, 16, 22]. Gupta et al. 8] proposed a nine point fourth order compact nite di erence scheme (FCS) for solving the convection di usion equation on uniform grids. Recently, this scheme has been extended to nonuniform grids 9, 17]. We used it with a multigrid method in 6] and obtained good results for some test problems with large Re. The three di erence schemes, CDS, UPS, and FCS, are easy to implement. The resulting linear systems are frequently used as test examples for iterative methods 4, 14]. In a recent study, the relative advantages and disadvantages of these three schemes on uniform grids are compared, in terms of computational e ciency, solution accuracy, and the algebraic properties of the discretized linear systems 19]. A preconditioned Krylov subspace method was used in 19] to solve the resulting linear systems. The test results in 19] indicate that FCS is computationally more e cient, and represents a compromise between CDS and UPS in terms of stability and solution accuracy. However, none of these schemes with a uniform discretization is desirable, if the computational domain contains some boundary layers, where the solution changes rapidly. This paper is to compare the relative advantages and disadvantages of these three schemes, when a nonuniform discretization mesh is used to resolve the boundary layers. We also compare the relative robustness and e ciency of a classical multigrid method and a multilevel preconditioning technique used to solve the resulting linear systems. This paper is intended to serve as an evaluation of these three di erence schemes for accurate and e cient computations of the convection di usion equations with boundary layers. A set of strategies (discretization, grid stretching, iterative solutions) is proposed for high accuracy, highly e cient, and robust computation of boundary layer problems. Section 2 of this paper discusses the three nite di erence schemes, and, in particular, FCS for the transformed convection di usion equation. Section 3.1 brie y introduces the multigrid method and multilevel preconditioning technique used for solving the resulting linear systems. Numerical results are presented and interpreted in Section 4. Finally, concluding remarks are summarized in Section 5.
