J.M. Sanz-Serna

Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations

SummaryWe address the question of convergence of fully discrete Runge-Kutta approximations. We prove that, under certain conditions, the order in time of the fully discrete scheme equals the conventional order of the Runge-Kutta formula being used. However, these conditions, which are necessary for the result to hold, are not natural. As a result, in many problems the order in time will be strictly smaller than the conventional one, a phenomenon called order reduction. This phenomenon is extensively discussed, both analytically and numerically. As distinct from earlier contributions we here treat explicit Runge-Kutta schemes. Although our results are valid for both parabolic and hyperbolic problems, the examples we present are therefore taken from the hyperbolic field, as it is in this area that explicit discretizations are most appealing.

An easily implementable fourth-order method for the time integration of wave problems

Abstract We are concerned with the time-integration of systems of ordinary differential equations arising from the space discretization of partial differential wave equations with smooth solutions. A method is suggested that, while being as easily implementable as the standard implicit mid-point rule, is fourth-order accurate. The new method is symplectic so that it is very well suited for long-time integrations of problems with a Hamiltonian structure. Numerical experiments are reported that refer to a fourth-order Galerkin space discretization of the Korteweg-de Vries equation and to a pseudospectral space discretization of the same equation.

A simple adaptive technique for nonlinear wave problems

Abstract A method for the numerical integration of the nonlinear Schrodinger equation is derived which uses variable time steps and a moving spatial grid. The benefits of adaptation are clearly demonstrated in the numerical experiments reported. The simple technique employed to move the nodes can be applied with little coding effort to general one-dimensional systems of PDEs.

An adaptive moving grid method for one-dimensional systems of partial differential equations

We describe a fully adaptive, moving grid method for solving initial-boundary value problems for systems of one-space dimensional partial differential equations whose solutions exhibit rapid variations in space and time. The method, based on finite-differences, is of the Lagrangian type and has been derived through a co-ordinate transformation which leads to equidistribution in space of the second derivative. Our technique is “intermediate” between static regridding methods, where nodes remain fixed for intervals of time, and continuously moving grid methods, where the node movement and the PDE integration are fully coupled. In our approach, the computation of the moving grids and the solution on these grids are carried out separately, while the nodes are moved at each time-step. Two error monitors have been implemented, one to govern the time-step selection and the other to eventually adapt the number of moving nodes. The method allows the use of different moving grids for different components in the PDE system. Numerical experiments are presented for a set of five sample problems from the literature, including two problems from combustion.

A Method for the Integration in Time of Certain Partial Differential Equations

Abstract A method for the numerical solution of ordinary differential equations is analyzed that is explicit and yet can conserve the quadratic quantities conserved by the equations. The method can be a useful alternative to the usual leapfrog technique, in that it does not suffer from the occurrence of blowup phenomena. Numerical examples concerning the Korteweg-de Vries equation and the nonlinear Schrodinger equation are given.

Split-step spectral schemes for nonlinear Dirac systems

Abstract The paper considers split-step spectral schemes for the numerical integration of nonlinear Dirac systems in [1 + 1]-dimensions. Proofs of stability and convergence are given along with numerical experiments which clearly show the superiority of the suggested methods over standard and split-step finite-difference algorithms.

The numerical study of blowup with application to a nonlinear Schrodinger equation

Abstract We discuss the use of numerical methods in the study of the solutions of evolution problems which exhibit finite-time unbounded growth. We first examine a naive approach in which the growth rate of the numerical solution is accepted as an approximation of the true growth rate. As we shall demonstrate for a radial nonlinear Schrodinger equation, this approach is inadequate since different discretizations exhibit different growth rates. The spurious behaviour of discretizations in the neighbourhood of the singularity is discussed. A reliable procedure for the estimation of the blowup parameters is considered which eliminates the discrepancies between different numerical methods.

A Hamiltonian explicit algorithm with spectral accuracy for the `good' Boussinesq system

We construct an explicit pseudo-spectral method for the numerical solution of the soliton-producing ‘good’ Boussinesq system wt = uxxx + ux + (u2)x, ut = wx. The new scheme preserves a discrete Poisson structure similar to that of the continuous system. The scheme is shown to converge with spectral spatial accuracy. A numerical illustration is given.

Numerical solution of a hyperbolic system of conservation laws with source term arising in a fluidized bed model

Abstract A model a gas fluidized bed is considered which leads to a hyperbolic system of conservation laws with a source term. The system is solved numerically by a second-order operator splitting technique based on a Roe approximate Riemann solver. Numerical experiments demonstrate the ability of the model to reproduce qualitatively the slugging phenomenon in the case when the bed is subject to a relatively large gas flux.

Studies in numerical nonlinear instability. II. A new look at u t + uu x = 0

Abstract It is shown that, in general leap-frog schemes, any particular unstable solution behaves as an attractor of other solutions. For a leap-frog discretization of u t + uu x = 0 a particular kind of unstable solution is constructed which generically attracts any other solution. Estimates of the overflow time are presented and related to the notions of stability threshold and restricted stability.