An algebraic characterization ofB-convergent Runge-Kutta methods

Summary In the analysis of discretization methods for stiff intial value problems, stability questions have received most part of the attention in the past.B-stability and the equivalent criterion algebraic stability are well known concepts for Runge-Kutta methods applied to dissipative problems. However, for the derivation ofB-convergence results — error bounds which are not affected by stiffness — it is not sufficient in many cases to requireB-stability alone. In this paper, necessary and sufficient conditions forB-convergence are determined.This paper was written while J. Schneid was visiting the Centre for Mathematics and Computer Science with an Erwin-Schrödinger stipend from the Fonds zur Förderung der wissenschaftlichen Forschung     

A note on stability of the Douglas splitting method

In this note some stability results are derived for the Douglas splitting method. The relevance of the theoretical results is tested for an advection-reaction equation.

     

A note on flux limiting for diffusion discretizations

In this note, a limiting technique is presented to enforcemonotonicity for higher-order spatial diffusion discretizations.The aim is to avoid spurious oscillations and to improve thequalitative behaviour on coarse grids. The technique is relatedto known ones for convection equations, using limiters to boundthe numerical fluxes. Applications arise in pattern formationproblems for reaction–diffusion equations.     

On the existence of solutions to the algebraic equations in implicit runge-kutta methods

This paper deals with the systems of algebraic equations arising in the application ofB-stable Runge-Kutta methods. It is shown that under natural assumptions such systems do not always have a solution. In addition, general sufficient conditions are presented under which such systems do have unique solutions.     

Convergence of linear multistep and one-leg methods for stiff nonlinear initial value problems

To prove convergence of numerical methods for stiff initial value problems, stability is needed but also estimates for the local errors which are not affected by stiffness. In this paper global error bounds are derived for one-leg and linear multistep methods applied to classes of arbitrarily stiff, nonlinear initial value problems. It will be shown that under suitable stability assumptions the multistep methods are convergent for stiff problems with the same order of convergence as for nonstiff problems, provided that the stepsize variation is sufficiently regular.     

Convergence properties of the Runge-Kutta-Chebyshev method

Summary The Runge-Kutta-Chebyshev method is ans-stage Runge-Kutta method designed for the explicit integration of stiff systems of ordinary differential equations originating from spatial discretization of parabolic partial differential equations (method of lines). The method possesses an extended real stability interval with a length proportional tos ². The method can be applied withs arbitrarily large, which is an attractive feature due to the proportionality of withs ². The involved stability property here is internal stability. Internal stability has to do with the propagation of errors over the stages within one single integration step. This internal stability property plays an important role in our examination of full convergence properties of a class of 1st and 2nd order schemes. Full convergence means convergence of the fully discrete solution to the solution of the partial differential equation upon simultaneous space-time grid refinement. For a model class of linear problems we prove convergence under the sole condition that the necessary time-step restriction for stability is satisfied. These error bounds are valid for anys and independent of the stiffness of the problem. Numerical examples are given to illustrate the theoretical results.Dedicated to Peter van der Houwen for his numerous contributions in the field of numerical integration of differential equations.Paper presented at the symposium Construction of Stable Numerical Methods for Differential and Integral Equations, held at CWI, March 29, 1989, in honor of Prof. Dr. P.J. van der Houwen to celebrate the twenty-fifth anniversary of his stay at CWI     

Stability of Approximate Factorization with θ-Methods

Approximate factorization seems for certain problems a viable alternative to time splitting. Since a splitting error is avoided, accuracy will in general be favourable compared to time splitting methods. However, it is not clear to what extent stability is affected by factorization. Therefore we study here the effects of factorization on a simple, low order method, namely the -method. For this simple method it is possible to obtain rather precise results, showing limitations of the approximate factorization approach.     

Spontaneous branching of anode-directed streamers between planar electrodes

Nonionized media subject to strong fields can become locally ionized by penetration of finger-shaped streamers. We study negative streamers between planar electrodes in a simple deterministic continuum approximation. We observe that, for sufficiently large fields, the streamer tip can split. This happens close to the limit of "ideal conductivity." Qualitatively, the tip splitting is due to a Laplacian instability quite like that in viscous fingering. For future quantitative analytical progress, our stability analysis of planar fronts identifies the screening length as a regularization mechanism.     

Spatially hybrid computations for streamer discharges with generic features of pulled fronts: I. Planar fronts

Streamers are the first stage of sparks and lightning; they grow due to a strongly enhanced electric field at their tips; this field is created by a thin curved space charge layer. These multiple scales are already challenging when the electrons are approximated by densities. However, electron density fluctuations in the leading edge of the front and non-thermal stretched tails of the electron energy distribution (as a cause of X-ray emissions) require a particle model to follow the electron motion. But present computers cannot deal with all electrons in a fully developed streamer. Therefore, super-particle have to be introduced, which leads to wrong statistics and numerical artifacts.The method of choice is a hybrid computation in space where individual electrons are followed in the region of high electric field and low density while the bulk of the electrons is approximated by densities (or fluids). We here develop the hybrid coupling for planar fronts. First, to obtain a consistent flux at the interface between particle and fluid model in the hybrid computation, the widely used classical fluid model is replaced by an extended fluid model. Then the coupling algorithm and the numerical implementation of the spatially hybrid model are presented in detail, in particular, the position of the model interface and the construction of the buffer region. The method carries generic features of pulled fronts that can be applied to similar problems like large deviations in the leading edge of population fronts, etc.     

Positivity for explicit two-step methods in linear multistep and one-leg form

Positivity results are derived for explicit two-step methods in linear multistep form and in one-leg form. It turns out that, using the forward Euler starting procedure, the latter form allows a slightly larger step size with respect to positivity. AMS subject classification (2000) 65L06