David M. Sloan

A review of pseudospectral methods for solving partial differential equations

Finite Difference (FD) methods approximate derivatives of a function by local arguments (such as d u ( x ) / d x ≈ ( u ( x + h ) − u ( x − h ))/2 h , where h is a small grid spacing) – these methods are typically designed to be exact for polynomials of low orders. This approach is very reasonable: since the derivative is a local property of a function, it makes little sense (and is costly) to invoke many function values far away from the point of interest.

A simple adaptive grid method in two dimensions

This paper gives an interpretation of the concept of equidistribution in the context of adaptive grid generation for multidimensional problems. It is shown that the equidistribution principle cannot be satisfied throughout the domain of the problem and, based on this recognition, a local equidistribution principle is developed. A discrete formulation is described for grid generation in two space dimensions and a smoothing mechanism is presented for improving mesh quality. The adaptive grid method that is constructed contains three grid-quality parameters. Numerical examples illustrate adaptive grid generation using a prescribed monitor function and grid generation for numerical solution of partial differential equations. Results show that the method produces high quality grids and that it is fairly insensitive to the choice of parameters.

Numerical solution of a singularly perturbed two-point boundary value problem using equidistribution: analysis of convergence

Abstract Adaptive grid methods are becoming established as valuable computational techniques for the numerical solution of differential equations with near-singular solutions. Adaptive methods are equally effective in approximating solutions of problems with boundary layers or interior layers (see, for example, Mulholland et al., SIAM J. Sci. Comput. 19(4) (1998) 1261–1289). Much is now being done in developing error analyes for methods that are based on adaptivity. In this paper, we present a rigorous error analysis for the solution of a singularly perturbed two-point boundary value problem on a grid that is constructed adaptively from a knowledge of the exact solution. The discrete solutions are generated by an upwind finite difference scheme and the grid is formed by equidistributing a monitor function based on arc-length. An error analysis shows that the discrete solutions are uniformly convergent with respect to the perturbation parameter, epsilon. The epsilon-uniform convergence is confirmed by numerical computations.

Analysis of difference approximations to a singularly perturbed two-point boundary value problem on an adaptively generated grid

Over the last few years there has been a significant growth in the use of adaptive grid methods for the numerical solution of differential equations with steep solutions. Little has been done, however, on the error analysis of adaptive methods. In this paper, we present an analysis for an upwind finite difference solution of a singular perturbation problem on a grid that is generated adaptively by equidistributing a monitor function based on the exact solution. It is shown that the discrete solutions converge uniformly with respect to the perturbation parameter, epsilon. This epsilon-uniform convergence is illustrated by numerical computations.

Pseudospectral Solution of Near-Singular Problems using Numerical Coordinate Transformations Based on Adaptivity

The work presented here describes a method of coordinate transformation that enables spectral methods to be applied efficiently to differential problems with steep solutions. The approach makes use of the adaptive finite difference method presented by Huang and Sloan [SIAM J. Sci. Comput., 15 (1994), pp. 776--797]. This method is applied on a coarse grid to obtain a rough approximation of the solution and a suitable adapted mesh. The adaptive finite difference solution permits the construction of a smooth coordinate transformation that relates the computational space to the physical space. The map between the spaces is based on Chebyshev polynomial interpolation. Finally, the standard pseudospectral (PS) method is applied to the transformed differential problem to obtain highly accurate, nonoscillatory numerical solutions. Numerical results are presented for steady problems in one and two space dimensions.

The pseudospectral method for third-order differential equations

Generalized quadrature rules are derived which assist in the selection of collocation points for the pseudospectral solution of differential equations. In particular, it is shown that for an nth-order differential equation in one space dimension with two-point derivative boundary conditions, an ideal choice of interior collocation points is the set of zeros of a Jacobi polynomial. The pseudospectral solution of a third-order initial-boundary value problem is considered and accuracy is assessed by examining how well the discrete eigenproblem approximates the continuous one. Convergence is established for a special choice of collocation points and numerical results are included to demonstrate the viability of the approach.

Some numerical aspects of computing inertial manifolds

Various aspects of the problem of computing inertial manifolds for dissipative partial differential equations are investigated. In particular, methods are proposed which generalize previous ones and improve upon some of their limitations. The basic ideas and techniques are exemplified mainly for the Kuramoto–Sivashinsky equation. This equation is chosen because it is fairly simple but the dynamics are sufficiently complicated, and it is a classical case for which a considerable amount of computational experience has been well documented.

Fourier pseudospectral solution of the regularised long wave equation

Abstract An algorithm is presented for the Fourier pseudospectral solution of the regularised long wave (RLW) equation. The semi-discrete equations satisfy the energy conservation condition of the RLW equation. Numerical results are presented to show that the scheme can resolve the fine structure of wave interaction problems. Evidence is presented to show that the popular second-order leap-frog time discretisation will produce high accuracy only if the time-step is well below the linear stability limit. By considering linear and nonlinear models a case is made for the use of higher-order accuracy in time.

A comparison of Fourier pseudospectral methods for the solution of the Korteweg-de Vries equation

Abstract Various Fourier pseudospectral methods are used to approximate the Korteweg-de Vries equation. The methods differ in their treatment of the time discretisation. The methods are compared for computational efficiency using the 1- and 2-soliton test problems. They are also compared from an accuracy viewpoint using the Zabusky-Kruskal recurrence problem.

On nonlinear instabilities in leap-frog finite difference schemes

Abstract Briggs, Newell and Sarie (J. Comput. Phys.51, 83 (1983)) have discussed a mechanism for the destabilisation of finite difference approximations to nonlinear partial differential equations. Their ideas were developed using the leap-frog approximation to the advection equation. Here the same situation is examined in a manner which compares the basic solution to a periodic wavetrain. An investigation is made into the stability of the basic solution to small disturbances which take the form of side-band Fourier modes. The relation between side-band growth and envelope modulation is discussed.