A. R. Mitchell

Numerical Solutions of the Good Boussinesq Equation

The “good” Boussinesq equation is studied numerically using Galerkin methods and soliton solutions are shown to exist. An analytic formula for the two-soliton interaction is presented and verified by numerical experiment.

Numerical experience with the nonlinear Schrödinger equation

Abstract Increasing the magnitude of the parameter multiplying the nonlinear term of the nonlinear Schrodinger equation without changing the initial condition u ( x , 0)=sech x , leads to bound states of an increasing number of solitons. This results in very steep gradients in space and time and so provides a more severe test of numerical methods than before. In particular we find that methods which satisfy various conservation laws theoretically may now fail to do so in practice. Various analytical and numerical results relevant to this situation are discussed and illustrated by numerical examples.

Corner singularities in elliptic problems by finite element methods

Abstract Finite element methods, using bilinear basis functions supplemented by singular functions, are described for solving elliptic boundary value problems with corner singularities: The procedure of mesh refinement in the finite element method in the neighborhood of a singularity is illustrated with respect to the harmonic mixed boundary value problem of the slit. Numerical results obtained right up to the tip of the slit are compared at selected points in the field with values obtained by dual series and finite difference methods.

On the solution ofy′=f(x,y) by a class of high accuracy difference formulae of low order

ZusammenfassungDifferenzengleichungenk-ter Ordnung (k=1, 2, 3, 4), in diey und dessen Ableitungen bis zurl-ten Ordnung (l=1, 2, 3) einbezogen sind, werden zur numerischen Integration der Differentialgleichungy′=f(x,y),y0=0 benützt. Die Reduktion des Abbrechfehlers erreicht man am besten, indem man eherl alsk vergrössert.

Some high accuracy difference schemes with a splitting operator for equations of parabolic and elliptic type

SummaryHigh accuracy alternating direction implicit difference schemes for the heat equation, LAPLACEs equation and the biharmonic equation are considered. In addition to surveying the existing methods, several new methods are introduced. Sequences of iteration parameters are obtained for the elliptic problems and a numerical example is given.

Equidistributing principles in moving finite element methods

Abstract Recently Miller and his co-workers proposed a moving finite element method based on a least squares principle. This was followed by a similar method by the present authors using a Petrov—Galerkin approach. In this paper the two methods are compared. In particular, it is shown that both methods move their nodes according to an approximate equidistributing principle. This observation leads to a criterion for the placement of the nodes. It is also shown that the penalty function designed by Miller may also be used with the Petrov—Galerkin method. Finally, numerical examples are given, illustrating the performance of the two methods.

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.

Finite difference solution of the third boundary problem in elliptic and parabolic equations

SummaryFinite difference methods (including the Peaceman-Rachford method) are considered for the solution of the third boundary value problem for parabolic and elliptic equations. Conditions on the coefficients involved in the boundary conditions are obtained from the stability requirements of the difference methods and shown to coincide with those necessary for asymptotic stability of the differential system.

Analysis of a non-linear difference scheme in reaction-diffusion

SummaryA nonlinear partial difference equation resulting from discretising in space and time the parabolic reaction diffusion equation, which models the spruce budworm problem, is analysed and accuracy estimates obtained for solutions over afinite time range and ast→∞. Although the analysis is restricted to the logistic model in one space dimension, the techniques and comparison principles developed in the paper should prove useful in assessing the merits of numerical solutions of other nonlinear parabolic difference equations.

A numerical study of the Belousov-Zhabotinskii reaction using Galerkin finite element methods

The Belousov-Zhabotinskii reaction has been modelled by Field and Noyes [5] as a pair of nonlinear parabolic equations. Previous studies of these, both theoretical and numerical, have assumed wave solutions travelling with constant velocity leading to a simplification of the mathematical model in the form of a system of ordinary differential equations. In the present study a finite element Galerkin method is used directly on the original parabolic system for a range of parameter values.