On composite mesh difference methods for hyperbolic differential equations

Summary Difference schemes on more than one mesh, called composite mesh difference methods (CMDM), are considered for hyperbolic equations. A stability proof for a one-dimensional CMDM is presented and a numerical experiment by a CMDM for the inviscid shallow-water equations is described.Research supported by the Swedish Natural Science Research Council (Nfr. 2711-18)     

On projected Runge-Kutta methods for differential-algebraic equations

Ascher and Petzold recently introducedprojected Runge-Kutta methods for the numerical solution of semi-explicit differential-algebraic systems of index 2. Here it is shown that such a method can be regarded as the limiting case of a standard application of a Runge-Kutta method with a very small implicit Euler step added to it. This interpretation allows a direct derivation of the order conditions and of superconvergence results for the projected methods from known results for standard Runge-Kutta methods for index-2 differential-algebraic systems, and an extension to linearly implicit differential-algebraic systems.     

The method of fractional steps for conservation laws

Summary The stability, accuracy, and convergence of the basic fractional step algorithms are analyzed when these algorithms are used to compute discontinuous solutions of scalar conservation laws. In particular, it is proved that both first order splitting and Strang splitting algorithms always converge to the unique weak solution satisfying the entropy condition. Examples of discontinuous solutions are presented where both Strang-type splitting algorithms are only first order accurate but one of the standard first order algorithms is infinite order accurate. Various aspects of the accuracy, convergence, and correct entropy production are also studied when each split step is discretized via monotone schemes, Lax-Wendroff schemes, and the Glimm scheme.Partially supported by an Alfred Sloan Foundation fellowship and N.S.F. grant MCS-76-10227Sponsored by US Army under contract No. DAA 629-75-0-0024     

Contractivity of Runge-Kutta methods

In this paper we present necessary and sufficient conditions for Runge-Kutta methods to be contractive. We consider not only unconditional contractivity for arbitrary dissipative initial value problems, but also conditional contractivity for initial value problems where the right hand side function satisfies a circle condition. Our results are relevant for arbitrary norms, in particular for the maximum norm.For contractive methods, we also focus on the question whether there exists a unique solution to the algebraic equations in each step. Further we show that contractive methods have a limited order of accuracy. Various optimal methods are presented, mainly of explicit type. We provide a numerical illustration to our theoretical results by applying the method of lines to a parabolic and a hyperbolic partial differential equation.Research supported by the Netherlands Organization for Scientific Research (N.W.O.) and the Royal Netherlands Academy of Arts and Sciences (K.N.A.W.)     

On improving the absolute stability of local extrapolation

Summary A widely used technique for improving the accuracy of solutions of initial value problems in ordinary differential equations is local extrapolation. It is well known, however, that when using methods appropriate for solving stiff systems of ODES, the stability of the method can be seriously degraded if local extrapolation is employed. This is due to the fact that performing local extrapolation on a low order method is equivalent to using a higher order formula and this high order formula may not be suitable for solving stiff systems. In the present paper a general approach is proposed whereby the correction term added on in the process of local extrapolation is in a sense a rational, rather than a polynomial, function. This approach allows high order formulae with bounded growth functions to be developed. As an example we derive anA-stable rational correction algorithm based on the trapezoidal rule. This new algorithm is found to be efficient when low accuracy is requested (say a relative accuracy of about 1%) and its performance is compared with that of the more familiar Richardson extrapolation method on a large set of stiff test problems.     

On the local error and the local truncation error of linear multistep methods

Two different measures of the local accuracy of a linear multistep method — the local error and the local trunction error — appear in the literature. It is shown that the principal parts of these errors are not identical for general linear multistep methods, but that they are so for a sub-class which contains all methods of Adams type. It is sometimes argued that local error is the more natural measure; this view is challenged.     

On the stability properties of Brown's multistep multiderivative methods

Summary Brown introducedk-step methods usingl derivatives. We investigate for whichk andl the methods are stable or unstable. It is seen that to anyl the method becomes unstable fork large enough. All methods withk2(l+1) are stable. Fork=1,2,..., 18 there exists a _k such that the methods are stable for anyl _k and unstable for anyl < _k . The _k are given.     

On the relation between algebraic stability andB-convergence for Runge-Kutta methods

Summary This paper is concerned with the numerical solution of stiff initial value problems for systems of ordinary differential equations using Runge-Kutta methods. For these and other methods Frank, Schneid and Ueberhuber [7] introduced the important concept ofB-convergence, i.e. convergence with error bounds only depending on the stepsizes, the smoothness of the exact solution and the so-called one-sided Lipschitz constant . Spijker [19] proved for the case <0 thatB-convergence follows from algebraic stability, the well-known criterion for contractivity (cf. [1, 2]). We show that the order ofB-convergence in this case is generally equal to the stage-order, improving by one half the order obtained in [19]. Further it is proved that algebraic stability is not only sufficient but also necessary forB-convergence.This study was completed while this author was visiting the Oxford University Computing Laboratory with a stipend from the Netherlands Organization for Scientific Research (N.W.O.)     

On the order conditions of Runge-Kutta methods with higher derivatives

Summary Runge-Kutta methods have been generalized to procedures with higher derivatives of the right side ofy=f(t,y) e.g. by Fehlberg 1964 and Kastlunger and Wanner 1972. In the present work some sufficient conditions for the order of consistence are derived for these methods using partially the degree of the corresponding numerical integration formulas. In particular, methods of Gauß, Radau, and Lobatto type are generalized to methods with higher derivatives and their maximum order property is proved. The applied technique was developed by Crouzeix 1975 for classical Runge-Kutta methods. Examples of simple explicit and semi-implicit methods are given up to order 7 and 6 respectively.     

Partitioned adaptive Runge-Kutta methods and their stability

Summary This paper deals with the solution of partitioned systems of nonlinear stiff differential equations. Given a differential system, the user may specify some equations to be stiff and others to be nonstiff. For the numerical solution of such a system partitioned adaptive Runge-Kutta methods are studied. Nonstiff equations are integrated by an explicit Runge-Kutta method while an adaptive Runge-Kutta method is used for the stiff part of the system.The paper discusses numerical stability and contractivity as well as the implementation and usage of such compound methods. Test results for three partitioned stiff initial value problems for different tolerances are presented.     

On the D-suitability of implicit Runge-Kutta methods

The concept of suitability means that the nonlinear equations to be solved in an implicit Runga-Kutta method have a unique solution. In this paper, we introduce the concept of D-suitability and show that previous results become special cases of ours. In addition, we also give some examples to illustrate the D-suitability of a matrixA.     

On integrating vertex singularities using extrapolation

A new approach to the integration of vertex singularities is described. This approach is based on a non-uniform subdivision of the region of integration and the technique fits well to the subdivision strategy used in many adaptive algorithms. A nice feature with this approach is that it can be used in any dimension and on any region of integration which can be subdivided into subregions of the same form. The strategy can be applied both to vertex singularities and internal point singularities. In the latter case this can be done without an initial subdivision of the region in order to put the singular point in a vertex. It turns out that the technique has excellent numerical stability properties.Dedicated to Carl-Erik Fröberg on the occasion of his 75th birthday.This work was supported by The Norwegian Research Council for Science and the Humanities.     

A-priori error estimates of Galerkin backward differentiation methods in time-inhomogeneous parabolic problems

Summary Backward differentiation methods up to orderk=5 are applied to solve linear ordinary and partial (parabolic) differential equations where in the second case the space variables are discretized by Galerkin procedures. Using a mean square norm over all considered time levels a-priori error estimates are derived. The emphasis of the results lies on the fact that the obtained error bounds do not depend on a Lipschitz constant and the dimension of the basic system of ordinary differential equations even though this system is allowed to have time-varying coefficients. It is therefore possible to use the bounds to estimate the error of systems with arbitrary varying dimension as they arise in the finite element regression of parabolic problems.     

Stability and accuracy of time discretizations for initial value problems

Summary This paper continues earlier work by the same authors concerning the shape and size of the stability regions of general linear discretization methods for initial value problems. Here the treatment is extended to cover also implicit schemes, and by placing the accuracy of the schemes into a more central position in the discussion general method-free statements are again obtained. More specialized results are additionally given for linear multistep methods and for the Taylor series method.This research has been supported by Swiss National Foundation, Grant No. 82-524.077This research has been supported by the Heinrich-Hertz-Stiftung, B 32 No. 203/79     

Order conditions for Rosenbrock type methods

Summary Based on the theory of Butcher series this paper developes the order conditions for Rosenbrock methods and its extensions to Runge-Kutta methods with exact Jacobian dependent coefficients. As an application a third order modified Rosenbrock method with local error estimate is constructed and tested on some examples.     

On a numerical approach to Stefan-like problems

Summary This paper is devoted to the numerical analysis of a bidimensional two-phase Stefan problem. We approximate the enthalpy formulation byC ⁰ piecewise linear finite elements in space combined with a semi-implicit scheme in time. Under some restrictions related to the finite element mesh and to the timestep, we prove positivity, stability and convergence results. Various numerial tests are presented and discussed in order to show the accuracy of our scheme.This work is supported by the Fonds National Suisse pour la Recherche Scientifique.     

Accuracy and stability of multistage multistep formulas

Summary We continue our research concerning relations between accuracy and stability of time discretizations. Here we concentrate, rather than on the size and shape of the stability region, on the obtainable accuracy when the number of parameters in the formulas is large.     

Superstable implicit Runge-Kutta methods for second order initial value problems

Using the well known properties of thes-stage implicit Runge-Kutta methods for first order differential equations, single step methods of arbitrary order can be obtained for the direct integration of the general second order initial value problemsy=f(x, y, y),y(x _o)=y _o,y(x _o)=y _o. These methods when applied to the test equationy+2y+ ² y=0, ,0, +>0, are superstable with the exception of a finite number of isolated values ofh. These methods can be successfully used for solving singular perturbation problems for which f/y and/or f/y are negative and large. Numerical results demonstrate the efficiency of these methods.     

A class of pseudo Runge-Kutta methods

This paper develops a general theory for a class of Runge-Kutta methods which are based, in addition to the stages of the current step, also on the stages of the previous step. Such methods have been introduced previously for the case of one and two stages. We show that for any numbers of stages methods of orderp withs+1 p 2s can be constructed. The paper terminates with a study of step size change and stability.     

Improved estimates of statistical regularization parameters in fourier differentiation and smoothing

Summary We investigate the statistical methods of cross-validation (CV) and maximum-likelihood (ML) for estimating optimal regularization parameters in the numerical differentiation and smoothing of non-exact data. Various criteria for optimality are examined, and the (asymptotic) notions of strong optimality, weak optimality and suboptimality are introduced relative to these criteria. By restricting attention to certainN-dimensional Hilbert spaces of smooth and stochastic functions, whereN is the number of data, we give regularity conditions on the data under which _CV, the regularization parameter predicted by CV, is strongly optimal with respect to the predictive mean-square signal error. We show that _ML is at best weakly optimal with respect to this criterion but is strongly optimal with respect to the innovation variance of the data. For numerical differentiation, _CV and _ML are both shown to be suboptimal with respect to the predictive mean-square derivative error.