1,720,988 research outputs found
ALGEBRAICALLY STABLE TWO-STEP RUNGE-KUTTA METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS
No full textWe investigate algebraic stability of two-step Runge-Kutta methods [2] for ordinary differential equations using the criterion proposed by Hewitt and Hill [1] for general linear methods. This criterion is based on suitable transformations on the coefficient matrices of the methods under consideration, in such a way that the G-matrix of algebraically stable formulae is the identity matrix. This gives a remarkable improvement, since the determination of the G-matrix is, in general, a nontrivial task. Examples of algebraically stable two-step Runge-Kutta methods possessing the above feature are presented. This work is in collaboration with Zdzislaw Jackiewicz (Arizona State University), Beatrice Paternoster (University of Salerno) and Dajana Conte (University of Salerno).
REFERENCES [1] L. L. Hewitt, A. T. Hill. Algebraically stable diagonally implicit general linear methods. Appl. Numer. Math., 60 (6):629–636, 2010.
[2] Z. Jackiewicz. General linear methods for ordinary differential equations. John Wiley & Sons, 2009
Highly stable two step collocation methods for stiff differential systems
No full text12th Seminar "NUMDIFF" on Numerical Solution of Differential and Differential-Algebraic Equation
Explicit Nordsieck methods with quadratic stability
No full textWe describe the construction of explicit Nordsieck methods with s stages of order p = s−1 and stage order q = p with inherent quadratic stability and quadratic stability with large regions of absolute stability. Stability regions of these methods compare favorably with stability regions of corresponding general linear methods of the same order with inherent Runge–Kutta stability
Search for Highly Stable General Linear Methods for Ordinary Differential Equations
No full textIn this talk we describe the construction of highly stable general linear methods (GLMs) for the
numerical solution of ordinary differential equations (ODEs). We describe the construction of
some classes of GLMs which are A-stable and L-stable using the Schur criterion, and algebraically
stable methods using criteria proposed recently by Hill, Nonlinear stability of general linear methods,
Numer. Math. 103(2006), 611–629, and Hewitt and Hill, Algebraically stable general linear
methods and the G-matrix, to appear in BIT. We illustrate the results for the class of two-step
Runge-Kutta methods with inherent Runge-Kutta stability for which one of the coefficient matrices
is assumed to have a one-point spectrum. We also describe our search for algebraically stable
methods in this class without imposing any restrictions on the coefficient matrices
Two-step Runge-Kutta methods with quadratic stability functions
Full text linkWe describe the construction of implicit two-step Runge-Kutta methods with stability properties determined by quadratic stability functions. We will aim for methods which are A-stable and L-stable and such that the coefficients matrix has a one point spectrum. Examples of methods of order up to eight are provided
Construction of strong stability preserving general linear methods
No full textWe describe a tool for investigating the strong stability preserving (SSP) general linear methods (GLMs) with two external stages and s internal stages, and derive example of methods which have larger effective Courant-Friedrichs-Levy coefficients than the class of two-step Runge-Kutta (TSRK) methods introduced by Jackiewicz and Tracogna, whose SSP properties were analyzes recently by Ketcheson, Gottlieb, and MacDonald. Numerical examples illustrate that the class of methods derived in this paper achieve the expected order of accuracy and do not produce spurious oscillations for discretizations of hyperbolic conservation laws, when combined with appropriate discretizations in spatial variables
Construction and implementation of highly stable two-step continuous methods for stiff differential systems
No full textWe describe a class of two-step continuous methods for the numerical integration of initial-value problems based on stiff ordinary differential equations (ODEs). These methods generalize the class of two-step Runge-Kutta methods. We restrict our attention to methods of order p = m, where m is the number of internal stages, and stage order q = p to avoid order reduction phenomenon for stiff equations, and determine some of the parameters to reduce the contribution of high order terms in the local discretization error. Moreover, we enforce the methods to be A-stable and L-stable. The results of some fixed and variable stepsize numerical experiments which indicate the effectiveness of two-step continuous methods and reliability of local error estimation will also be presented. © 2011 IMACS. Published by Elsevier B.V. All rights reserved
Algebraically stable two-step almost collocation methods for ordinary differential equations
No full textWe investigate algebraic stability of the new class of two-step almost collocation methods for ordinary differential equations. These continuous methods are obtained by relaxing some of the interpolation and collocation conditions to achieve uniform order of convergence on the whole interval of integration. We describe the search for algebraically stable methods using the crierion based on Nyquist stability function proposed recently by Hill. This criterion leads to the minimization problem in one variable which was solved using fminsearch routine in MATLAB. Examples of algebraically stable methods in this class obtained in this way are presented
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