Two-step Runge-Kutta methods and hyperbolic partial differential equations

Renaut, Rosemary A.

doi:10.1090/s0025-5718-1990-1035943-3

Cited by 19 publications

(17 citation statements)

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“…Explicit TSRK methods have found applications in the numerical solution of systems of ODE's arising from semidiscretizations of parabolic and hyperbolic partial differential equations and have been studied by Byrne and Lambert [3], Renaut [16,17], and Verwer [18][19][20]. Related results for explicit A;-step mstage RK methods are given in van der Houwen and Sommeijer [7,8] and van der Houwen [6].…”

Section: Introductionmentioning

confidence: 99%

Variable-Stepsize Explicit Two-Step Runge-Kutta Methods

Jackiewicz¹,

Zennaro²

1992

Mathematics of Computation

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Abstract. Variable-step explicit two-step Runge-Kutta methods for the numerical solution of ordinary differential equations are studied. Order conditions are derived and the results about the minimal number of stages required to attain a given order are established up to order five. The existence of embedded pairs of continuous Runge-Kutta methods and two-step Runge-Kutta methods of order p -1 and p is proved. This makes it possible to estimate local discretization error of continuous Runge-Kutta methods without any extra evaluations of the right-hand side of the differential equation. An algorithm to construct such embedded pairs is described, and examples of (3, 4) and (4, 5) pairs are presented. Numerical experiments illustrate that local error estimation of continuous Runge-Kutta methods based on two-step Runge-Kutta methods appears to be almost as reliable as error estimation by Richardson extrapolation, at the same time being much more efficient.

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Section: Introductionmentioning

confidence: 99%

Variable-Stepsize Explicit Two-Step Runge-Kutta Methods

Jackiewicz¹,

Zennaro²

1992

Mathematics of Computation

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“…The special case of these methods corresponding to 0 = 0 was first studied by Byrne and Lambert [2]. They considered explicit two-step two-stage and two-step three-stage methods of order 3 [10], [11] found methods of the form (1.2) appropriate for the numerical solution of systems of ODEs arising from the semidiscretization of hyperbolic partial differential equations. Verwer [12]- [14] considered two-step and three-step explicit Runge-Kutta methods for the numerical integration of systems resulting from parabolic partial differential equations by applying the method of lines.…”

mentioning

confidence: 99%

“…It is known that the method (1.2) is convergent if and only if it is consistent and zero-stable (see [15] These definitions of Y, Y0, A(? ), and A(l) are not unique (see [11] for a slightly different representation). We also define the (2m + 1)-dimensional vector a in such a way that the vth component of Y0 is an approximation to the solution y(xi + a,h) for v= 1,2, , 2m, and the (2m + l)st component of Y is computed in order to fit y(xi + a2m+jh) (cf.…”

mentioning

confidence: 99%

“…The order conditions up to order 4 computed by using (2.2)-(2.4) are listed in Table 1, where u denotes the vector [1, **I, 1]T of appropriate dimension. These order conditions (in slightly different notation) were obtained before by Renaut [11] using the composition theorem of Hairer and Wanner [4]. In [10] the order conditions were obtained in a more elementary way using the Taylor series expansion.…”

mentioning

confidence: 99%

“…Thus A-stable methods may be useful in this context. Furthermore, the semidiscretization of hyperbolic partial differential equations leads to a requirement of stability along the imaginary axis or in a region in the left half plane (see [10], [11]), and A-stable methods may again be useful. Two-step two-stage methods of order 4 are also studied.…”

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confidence: 99%

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Two-Step Runge–Kutta Methods

Jackiewicz

Renaut

Feldstein

1991

SIAM J. Numer. Anal.

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Abstract. Implicit two-step Runge-Kutta methods are studied. It will be shown that these methods require fewer stages to achieve the same order as one-step Runge-Kutta methods, which means the two-step methods are potentially more efficient than one-step methods. Order conditions are derived and examples of two-step one-stage methods of order 2 and two-step two-stage methods of order 4 are presented. Stability properties of these methods with respect to y'= ay are studied and A-stable two-step methods of order 2 are characterized. Two-step two-stage methods of order 4 which are A-stable are found by an extensive computer search. Semi-implicit two-stage methods of order 4 were also constructed. This is in contrast to the situation encountered in the Runge-Kutta theory where the unique two-stage method of order 4 is not semi-implicit. Society for Industrial and

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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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Two-step Runge-Kutta methods and hyperbolic partial differential equations

Cited by 19 publications

References 11 publications

Variable-Stepsize Explicit Two-Step Runge-Kutta Methods

Variable-Stepsize Explicit Two-Step Runge-Kutta Methods

Two-Step Runge–Kutta Methods

General Linear Methods with Inherent Runge‐Kutta Stability

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