A. Bayliss

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.

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.

Two routes to chaos in condensed phase combustion

The equations governing two models of gasless combustion which exhibit pulsating solutions are numerically solved. The models differ in that one allows for melting of the solid fuel, while the other does not. While both models undergo a Hopf bifurcation from a solution propagating with a constant velocity to one propagating with a pulsating (T-periodic) velocity when parameters related to the activation energy exceed a critical value, the subsequent behavior differs markedly. Numerically both models exhibit a period doubling transition to a

Fronts, relaxation oscillations, and period doubling in solid fuel combustion

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The spectral overlay on finite elements for problems with high gradients

solution when the bifurcation parameter for each model is further increased. For the model without melting, a sequence of additional period doublings occurs, after which apparently chaotic solutions are found. For the model with melting, it is found that the

An adaptive pseudo-spectral method for reaction diffusion problems

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Mappings and accuracy for Chebyshev pseudo-spectral approximations☆

solution returns to the T-periodic solution branch. Then two additional windows of

On the interaction of a sound pulse with the shear layer of an axisymmetric jet

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