Eli Turkel

Preconditioned methods for solving the incompressible low speed compressible equations

Abstract Acceleration methods are presented for solving the steady state incompressible equations. These systems are preconditioned by introducing artificial time derivatives which allow for a faster convergence to the steady state. We also consider the compressible equations in conservation form with slow flow. Two arbitrary functions α and β are introduced in the general preconditioning. An analysis of this system is presented and an optimal value for β is determined given a constant α. It is further sown that the resultant incompressible equations form a symmetric hyperbolic system and so are well posed. Several generalizations to the compressible equations are presented which extend previous results.

Boundary Conditions for the Numerical Solution of Elliptic Equations in Exterior Regions

Elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the well-posedness of the problem. Examples of practical applications include the Helmholtz equation and Laplace’s equation. Computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface. Direct imposition of the condition at infinity along the finite boundary results in large errors. A sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain. Estimates of the error due to the finite boundary are obtained for several cases. Computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions. The examples are based on a finite element formulation but finite difference methods can also be used.

On central-difference and upwind schemes

A class of numerical dissipation models for central-difference schemes constructed with secondand fourth-difference terms is considered. The notion of matrix dissipation associated with upwind schemes is used to establish improved shock capturing capability for these models. In addition, conditions are given that guarantee that such dissipation models produce a TVD scheme. Appropriate switches for this type of model to ensure satisfaction of the TVD property are presented. Significant improvements in the accuracy of a centraldifference scheme are demonstrated by computing both inviscid and viscous transonic airfoil flows. 1Research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-18605 while the author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665.

Dissipative two-four methods for time-dependent problems

A generalization of the Lax-Wendroff method is presented. This generalization bears the same relationship to the two-step Richtmyer method as the KreissOliger scheme does to the leapfrog method. Variants based on the MacCormack method are considered as well as extensions to parabolic problems. Extensions to two dimensions are analyzed, and a proof is presented for the stability of a Thommentype algorithm. Numerical results show that the phase error is considerably reduced from that of second-order methods and is similar to that of the Kreiss-Oliger method. Furthermore, the (2, 4) dissipative scheme can handle shocks without the necessity for an artificial viscosity.

Implicit schemes and LU decompositions

Implicit methods for hyperbolic equations are analyzed by constructing LU factorizations. It is shown that the solution of the resulting tridiagonal systems in one dimension is well conditioned if and only if the LU factors are diagonally dominant. Stable implicit methods that have diagonally dominant factors are constructed for hyperbolic equations in n space dimesnions. Only two factors are required even in three space dimensions. Acceleration to a steady state is analyzed. When the multidimensional backward Euler method is used with large time steps, it is shown that the scheme approximates a Newton-Raphson iteration procedure.

Absorbing PML boundary layers for wave-like equations

We consider absorbing layers that are extensions of the PML of Berenger (1994). These will be constructed both for time problems and for Helmholtz-like equations. We shall consider applications to electricity and magnetism and acoustics (with a mean flow) in both physical space and in the time Fourier space (Helmholtz equation). Numerical results are presented showing the efficiency of this condition for the time dependent Maxwell equations.

Review of preconditioning methods for fluid dynamics

Abstract We consider the use of preconditioning methods to accelerate the convergence to a steady state for both the incompressible and compressible fluid dynamic equations. Most of the analysis relies on the inviscid equations though some applications for viscous flow are considered. The preconditioning can consist of either a matrix or a differential operator acting on the time derivatives. Hence, in the steady state the original steady solution is obtained. For finite difference methods the preconditioning can change and improve the steady-state solutions. Several preconditioners previously discussed are reviewed and some new approaches are presented.

Far field boundary conditions for compressible flows

Abstract A family of boundary conditions which simulate outgoing radiation are derived. These boundary conditions are applied to the computation of steady state flows and are shown to significantly accelerate the convergence to steady state. Numerical results are presented. Extensions of this theory to problems in duct geometries are indicated.

On accuracy conditions for the numerical computation of waves

Abstract The Helmholtz equation ( Δ + K 2 n 2 ) u = f with a variable index of refraction n and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems. Such problems can be solved numerically by first truncating the given unbounded domain, imposing a suitable outgoing radiation condition on an artificial boundary and then solving the resulting problem on the bounded domain by direct discretization (for example, using a finite element method). In practical applications, the mesh size h and the wave number K are not independent but are constrained by the accuracy of the desired computation. It will be shown that the number of points per wavelength, measured by ( Kh ) −1 , is not sufficient to determine the accuracy of a given discretization. For example, the quantity K 3 h 2 is shown to determine the accuracy in the L 2 norm for a second-order discretization method applied to several propagation models.

An iterative method for the Helmholtz equation

An iterative algorithm for the solution of the Helmholtz equation is developed. The algorithm is based on a preconditioned conjugate gradient iteration for the normal equations. The preconditioning is based on an SSOR sweep for the discrete Laplacian. Numerical results are presented for a wide variety of problems of physical interest and demonstrate the effectiveness of the algorithm.