Two-Step Runge–Kutta Methods

Jackiewicz, Zdzisław; Renaut, Rosemary A.; Feldstein, Alan

doi:10.1137/0728062

Cited by 48 publications

(23 citation statements)

References 8 publications

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“…The investigation of TVD time discretization can also be carried out for the generalized Runge-Kutta methods (which have more than one step) in, e.g., [6] and [7]. We have performed this study but failed to find good (in terms of CFL coefficients and whetherL appears) TVD methods in this class.…”

Section: Introductionmentioning

confidence: 99%

Total variation diminishing Runge-Kutta schemes

1998

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Abstract. In this paper we further explore a class of high order TVD (total variation diminishing) Runge-Kutta time discretization initialized in a paper by Shu and Osher, suitable for solving hyperbolic conservation laws with stable spatial discretizations. We illustrate with numerical examples that non-TVD but linearly stable Runge-Kutta time discretization can generate oscillations even for TVD (total variation diminishing) spatial discretization, verifying the claim that TVD Runge-Kutta methods are important for such applications. We then explore the issue of optimal TVD Runge-Kutta methods for second, third and fourth order, and for low storage Runge-Kutta methods.

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Section: Introductionmentioning

confidence: 99%

Total variation diminishing Runge-Kutta schemes

1998

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“…These methods were introduced by Renaut [106], Jackiewicz, Renaut and Feldstein [85] and Jackiewicz, Renaut and Zennaro [86]. These methods were generalised by Jackiewicz and Tracogna [87] and are given by…”

Section: Two Step Runge-kutta Methodsmentioning

confidence: 99%

General Linear Methods with Inherent Runge‐Kutta Stability

Jackiewicz¹

2009

General Linear Methods for Ordinary Differential Equations

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General linear methods were derived approximately thirty years ago as a unifying approach for the study of consistency, stability and convergence of the Runge-Kutta and the linear multistep methods. Their discovery opened the possibility of obtaining essentially new methods which were neither Runge-Kutta nor linear multistep methods nor slight variations of these methods. It was hoped that general linear methods would exist which are practical and have advantages over the traditional methods. Locating such practical methods has proved difficult though, for several reasons. For example, the complexity of the order conditions becomes very high, making it difficult to even find the required conditions in many cases let alone solve them.Several simplifying assumptions must be made, which limit this large class to situations where practical methods are likely to exist. The first assumption is that the stage order is equal to the overall order of the general linear method. This results in methods which, among other things, are not affected by the order reduction phenomenon. The second assumption is that a Nordsieck vector is passed from step to step. This enables such methods to vary the stepsize in a convenient and practical way. The final assumption is that the stability regions of the general linear methods should be identical to those of corresponding Runge-Kutta methods.The first two assumptions are satisfied by the structure of the U and V matrices of the general linear methods. In order to satisfy the last assumption sufficient conditions are developed. These conditions result in a class of general linear methods with a property known as inherent RungeKutta stability (IRKS). The IRKS conditions relate the coefficient matrices of the general linear method with a doubly companion matrix X to satisfyConstructing general linear methods with the IRKS property in the most general way possible is the main aim of this thesis. To derive these methods a transformation is used; this transformation brings all methods of this class into a particular form, which allows the construction using only linear operations.Several special properties of methods with the IRKS property are introduced. For example, conditions which show that the ESIRK methods are a special case of the IRKS methods are introduced. This then allows the introduction of a new class of ESIRK methods which may have advantages over those already known. Also, methods which have a property known as strong stiff accuracy are developed which make them similar to strictly stable Runge-Kutta methods. Methods with strong stiff accuracy are likely to be considered good, particularly because they are the most suitable amongst the IRKS methods for the solution of differential algebraic equations.The theoretical properties of the general linear methods with IRKS are investigated using various implementations from fixed stepsize and fixed order to variable stepsize and variable order codes. The IRKS methods are experimentally compared with several traditional metho...

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“…and y = (y\_., y2n_x,..., j£_,, yn, y\,..., y™, yn+xf . These definitions of y4(0), Aw , y0, and y are not unique; see [6] for a different definition.…”

Section: Order Conditions and Butcher Seriesmentioning

confidence: 99%

“…Here we use Butcher series and follow the approach of Hairer and Wanner [2] to get a formula from which order conditions up to any order of accuracy can be derived. Jackiewicz, Renaut, and Feldstein [6] also used Butcher series to get the order conditions for implicit two-step methods but without the use of the composition theorem of Hairer and Wanner [2]. To apply the theory of Hairer and Wanner [2], we need to write the method (0.2) in the matrix-vector form y = A(0)y0 + hAWf(y).…”

Section: Order Conditions and Butcher Seriesmentioning

confidence: 99%

Two-Step Runge-Kutta Methods and Hyperbolic Partial Differential Equations

Renaut¹

1990

Mathematics of Computation

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Abstract. The purpose of this study is the design of efficient methods for the solution of an ordinary differential system of equations arising from the semidiscretization of a hyperbolic partial differential equation. Jameson recently introduced the use of one-step Runge-Kutta methods for the numerical solution of the Euler equations. Improvements in efficiency up to 80% may be achieved by using two-step Runge-Kutta methods instead of the classical onestep methods. These two-step Runge-Kutta methods were first introduced by Byrne and Lambert in 1966. They are designed to have the same number of function evaluations as the equivalent one-step schemes, and thus they are potentially more efficient. By solving a nonlinear programming problem, which is specified by stability requirements, optimal two-step schemes are designed. The optimization technique is applicable for stability regions of any shape. INTRODUCTIONIn this paper we consider a class of pseudo-Runge-Kutta methods for the solution of an ordinary differential system of equations (0.1) y' = f(y) which arises from the semidiscretization of a hyperbolic partial differential equation. In 1982, Jameson [7] initiated interest in the use of one-step Runge-Kutta methods for the numerical solution of the Euler equations. He applied the van der Houwen [4] optimal schemes in codes for the solution of the Euler equations by central differences. These schemes are optimal because they have regions of stability enclosing a maximal interval on the imaginary axis, as is required when central differences are used for the semidiscretization. Here we demonstrate that greater efficiency is achieved by using two-rather than onestep Runge-Kutta formulae. These Runge-Kutta methods were first considered by Byrne and Lambert [1] in 1966.We define an explicit two-step w-stage Runge-Kutta method as m (0.2) yn+x = (1 -ß)yn + ßyn_x + h ][>,/(>V.) + V0Í)) » i=i

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Two-Step Runge–Kutta Methods

Cited by 48 publications

References 8 publications

Total variation diminishing Runge-Kutta schemes

Total variation diminishing Runge-Kutta schemes

General Linear Methods with Inherent Runge‐Kutta Stability

Two-Step Runge-Kutta Methods and Hyperbolic Partial Differential Equations

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