Convergence Acceleration for the Linearized Navier--Stokes Equations using Semicirculant Approximations

Abstract

The iterative solution of systems of equations arising from partial differential equations (PDEs) governing boundary layer flow for large Reynolds numbers is studied. The PDEs are discretized using finite volume or finite difference approximations on tensor product grids. We consider a convergence acceleration technique, where a semicirculant approximation of the spatial difference operator is employed as preconditioner. A relevant model problem is derived, and the spectrum of the preconditioned coefficient matrix is analyzed. It is proved that, asymptotically, the time step for the forward Euler method could be chosen as a constant, which is independent of the number of gridpoints and the Reynolds number Re. The same type of result is also derived for finite size grids, where the solution fulfills a given accuracy requirement. By linearizing the Navier-Stokes equations around an approximate solution, we form a system of linear PDEs with variable coefficients. When utilizing the semicirculant preconditioner for this problem, which has properties very similar to the full nonlinear equations, the results show that the favorable convergence properties hold also here. We compare the semicirculant preconditioner to a multigrid scheme. The number of iterations and the arithmetic complexities are considered, and it clear that semicirculant method is much more efficient for problems where the Reynolds number is large. The number of iterations for the multigrid method grows like √ Re, while the convergence rate for the scheme using semicirculant preconditioning is independent of Re. Also, the multigrid scheme is very sensitive to the level of artificial dissipation, while the method using semicirculant preconditioning is not. Introduction.The following strategy for solving the Navier-Stokes equations governing compressible, stationary flow is frequently employed in computer codes used by the computational fluid dynamics (CFD) community:(i) Generate a tensor product grid or a multiblock grid where the partition grids are tensor product grids.(ii) Discretize the equations in space utilizing a finite volume or finite difference scheme.(iii) Integrate the corresponding time-dependent problem in time by using an explicit Runge-Kutta time-marching method.(iv) Improve the convergence properties by employing some convergence acceleration technique.In section 2 of this paper, we review the semicirculant convergence acceleration, or preconditioning, technique, which is applicable to the type of computations described above. Then in section 3, we describe the boundary layer equations, which yield an approximate solution to the compressible Navier-Stokes equations for the 2D flow over a flat plate. We employ the boundary layer equations for formulating a relevant scalar model problem, and for linearizing the Navier-Stokes equations, resulting in a

A semi-circulant preconditioner for the convection-diffusion equation

“…2-3 we draw the conclusion that we can not expect too much from a minimal residual iteration for the original system of equations (1) when Pe d is kept constant and h → 0. Equation (10) tells us that we shall have good clustering of the eigenvalues and preferably also a well conditioned eigenvector matrix to ensure good convergence.…”

“…In this section we will study the eigenvalues and the eigenvectors of the coefficient matrix A defined in (1). For this reason we will study the matrix A d defined in (5).…”

“…In [1] semi-circulant preconditioners are used as convergence accelerators in a Runge-Kutta time-marching method for the time-dependent compressible Navier-Stokes equations for a flat plate flow problem. A similar model problem to the one presented here is analyzed with respect to eigenvalues.…”

Grid independent convergence using multilevel circulant preconditioning: Poisson’s equation

“…For diffusive flow problems, the rate of convergence is significantly improved by a semicirculant preconditioner, but the required number of iterations is not bounded as the grid is refined, see e.g. [8]. This holds in particular for elliptic problems [3].…”