Robert P. K. Chan

On explicit two-derivative Runge-Kutta methods

The theory of Runge-Kutta methods for problems of the form y′ = f(y) is extended to include the second derivative y′′ = g(y): = f′(y)f(y). We present an approach to the order conditions based on Butcher’s algebraic theory of trees (Butcher, Math Comp 26:79–106, 1972), and derive methods that take advantage of cheap computations of the second derivatives. Only explicit methods are considered here where attention is given to the construction of methods that involve one evaluation of f and many evaluations of g per step. Methods with stages up to five and of order up to seven including some embedded pairs are presented. The first part of the paper discusses a theoretical formulation used for the derivation of these methods which are also of wider applicability. The second part presents experimental results for non-stiff and mildly stiff problems. The methods include those with the computation of one second derivative (plus many first derivatives) per step, and embedded methods for changing stepsize as well as those involving one first derivative (plus many second derivatives) per step. The experiments have been performed on standard problems and comparisons made with some standard explicit Runge-Kutta methods.

On symmetric Runge-Kutta methods of high order

The usual characterization of symmetry for Runge-Kutta methods is that given by Stetter. In this paper an equivalent characterization of symmetry based on theW-transformation of Hairer and Wanner is proposed. Using this characterization it is simple to show symmetry for some well-known classes of high order Runge-Kutta methods which are based on quadrature formulae. It can also be used to construct a one-parameter family of symmetric and algebraically stable Runge-Kutta methods based on Lobatto quadrature. Methods constructed in this way and presented in this paper extend the known class of implicit Runge-Kutta methods of high order.ZusammenfassungDie übliche Charakterisierung der Symmetrie für Runge-Kutta Methoden ist die von Stetter angegebene. In dieser Arbeit wird eine äquivalente Charakterisierung vorgeschlagen, die auf derW-Transformation von Hairer und Wanner beruht. Mit dieser Charakterisierung kann die Symmetrie für einige Klassen von Runge-Kutta Methoden einfach gezeigt werden. Sie kann auch dazu benützt werden, eine einparametrige Familie von symmetrischen und algebraisch stabilen Runge-Kutta Methoden, die auf der Lobatto-Quadratur beruhen, zu konstruieren. Damit kann die Klasse impliziter Runge-Kutta Methoden höherer Ordnung erweitert werden.

Extrapolation of symplectic methods for Hamiltonian problems

Abstract We consider two modes of the extrapolation of symplectic and symmetric Runge–Kutta and related integrators over long time-intervals applied with constant stepsize. In the passive mode, we compute two solution sequences with stepsizes h and h /2 independently and perform extrapolation whenever output is required. In the active mode, we extrapolate at every step and propagate the extrapolated solution. We study and compare in detail both modes of extrapolation applied to the simple harmonic oscillator. We show that passive extrapolation will improve the accuracy of the numerical solution over the whole integration interval even though it destroys the linear error growth of the basic method and that active extrapolation exhibits linear error growth for the harmonic oscillator and, in general, yields higher accuracy than passive extrapolation. The error growth of integrations over long time-intervals is also studied in a more general setting. We obtain asymptotic error formulas for the periodic case and for integrable Hamiltonian systems where linear error growth has been established in the study of Calvo and Hairer [Appl. Numer. Math. 18 (1995) 95–105].

Two-derivative Runge–Kutta methods for PDEs using a novel discretization approach

We develop a novel and general approach to the discretization of partial differential equations. This approach overcomes the rigid restriction of the traditional method of lines (MOL) and provides flexibility in the treatment of spatial discretization. This method is essential for developing efficient numerical schemes for PDEs based on two-derivative Runge–Kutta (TDRK) methods, where the first and second derivatives must be discretized in an efficient way. This is unlikely to be achieved by using MOL. We then apply the explicit TDRK methods to the advection equations and analyze the numerical stability in the linear advection equation case. We conduct numerical experiments on the Burgers’ equation using the TDRK methods developed. We also apply a two-stage semi-implicit TDRK method of order-4 and stage-order-4 to the heat equation. A very significant improvement in the efficiency of this TDRK method is observed when compared to the popular Crank-Nicolson method. This paper is partially based on the work in Tsai’s PhD thesis (2011) [10].

Generalized symmetric Runge-Kutta methods

In this paper the concept of symmetry for Runge-Kutta methods is generalized to include composite methods. The extrapolations of the usual compositions of a symmetric method ℛ of the form are shown not to beA-stable. However, this limitation can be overcome by considering composite methods of the form where represents a non-symmetric and possiblyL-stable method called a symmetrizer satisfying. While no longer symmetric, these composite methods yet satisfy and thus share with symmetric methods the important property of admitting asymptotic error expansions in even powers of 1/n. Composite methods that are constructed in this way and presented in this paper have implementation costs comparable to that for the symmetric method. They generalize those based on the implicit midpoint and trapezoidal rules used with the standard smoothing formulae and thus extend the class of methods for acceleration techniques of extrapolation and defect correction. A characterization ofL-stable symmetrizers for 2-stage symmetric methods is given and studied for a particular stiff model problem. The analysis suggests that certainL-stable symmetrizers can play an important role in suppressing order defect effects for stiff problems.ZusammenfassungIn dieser Arbeit verallgemeinern wir den Symmetriebegriff für Runge-Kutta-Verfahren auf zusammengesetzte Verfahren. Die übliche Aneinanderreihung symmetrischer Schritte ℛ zu keineA-stabilen Extrapolationsstufen ergibt. Diese Einschränkung läßt sich jedoch mit Hilfe zusammengesetzter Verfahren der Form umgehen, wobei das als “Symmetrizer” bezeichnete nicht-symmetrische und möglicherweiseL-stabile Verfahren die Bedingung erfüllt. Obwohl sie nicht im engeren Sinn symmetrisch sind, erfüllen diese zusammengesetzen Verfahren und haben deshalb wie die symmetrischen Verfahren asymptotische Fehlerentwicklungen in geraden Potenzen von 1/n. Die Implementierung solcher in dieser Arbeit behandelter Verfahren führt zu Kosten, die denen bei symmetrischen Verfahren vergleichbar sind. Diese Verfahren stellen eine Verallgemeinerung der impliziten Mittelpunkt- und Trapezregeln mit Standardglättung dar und erweitern die Methoden, für die Konvergenzbeschleunigung mittels Extrapolation und Defektkorrektur möglich ist. DieL-stabilen Symmetrizer für 2-stufige symmetrische Verfahren werden charakterisiert und an Hand eines speziellen steifen Modellproblems studiert. Die Analyse läßt erwarten, daß gewisseL-stabile Symmetrizer eine wichtige Rolle bei der Unterdrückung von Ordnungsabfalleffekten bei steifen Problemen spielen können.

Post-projected Runge-Kutta methods for index-2 differential-algebriac equations

A new projection technique for Runge-Kutta methods applied to index-2 differential-algebraic equations is presented in which the numerical approximation is projected only as part of the output process. It is shown that for methods that are strictly stable at infinity, the order of convergence is unaffected compared to standard projected methods. Gauss methods, for which this technique is of special interest when some symmetry is to be preserved, are studied in more detail.

On symmetrizers for Gauss methods

SummaryIn this paper the maximum attainable order of a special class of symmetrizers for Gauss methods is studied. In particular, it is shown that a symmetrizer of this type for thes-stage Gauss method can attain order 2s-1 only for 1 ≦s ≦ 3, and that these symmetrizers areL-stable. A classification of the maximum attainable order of symmetrizers for some higher stages is presented. AnL-stable symmetrizer is also shown to exist for each of the methods studied.

Two-derivative Runge-Kutta methods for differential equations

Two-derivative Runge-Kutta (TDRK) methods are a special case of multi-derivative Runge-Kutta methods first studied by Kastlunger and Wanner [1, 2]. These methods incorporate derivatives of order higher than the first in their formulation but we consider only the first and second derivatives. In this paper we first present our study of both explicit [3] and implicit TDRK methods on stiff ODE problems. We then extend the applications of these TDRK methods to various partial differential equations [4]. In particular, we show how a 2-stage implicit TDRK method of order 4 and stage order 4 can be adapted to solve diffusion equations more efficiently than the popular Crank-Nicolson method.

Reversible methods of Runge-Kutta type for Index-2 DAEs

Summary.A new interpretation of Runge-Kutta methods for differential algebraic equations (DAEs) of index 2 is presented, where a step of the method is described in terms of a smooth map (smooth also with respect to the stepsize). This leads to a better understanding of the convergence behavior of Runge-Kutta methods that are not stiffly accurate. In particular, our new framework allows for the unified study of two order-improving techniques for symmetric Runge-Kutta methods (namely post-projection and symmetric projection) specially suited for solving reversible index-2 DAEs.

A-stability of implicit Runge-Kutta extrapolations

Abstract This paper concerns A-stability barriers for polynomial extrapolations of implicit Runge-Kutta methods. It is shown that, under some mild restrictions, a method of even order cannot admit an A-stable first extrapolation, while an odd order method cannot admit first and second extrapolations that are both A-stable. It is also shown that a method which is not A-stable cannot admit an A-stable first extrapolation, and furthermore, that neither the first nor the second extrapolation of a symmetric method can be A-stable. More specifically, the first and second extrapolations of the diagonal and second sub-diagonal of the Pade table for the exponential function are shown not to be A-stable as is the second extrapolation of the first sub-diagonal of the Pade table. The implications of these results for some well known classes of high order methods are discussed. Composite methods for which these barriers do not apply are studied, especially those formed by compositions of a symmetric method with an L-stable method. Those based on well known symmetric methods, in particular the 2-stage Gauss or 3-stage Lobatto IIIA, are shown to admit first and second but no higher L-stable extrapolations. Those based on methods whose stability functions are not Pade approximations are shown not to admit an A-stable first extrapolation. To date no third extrapolation of any method is known to be A-stable.