1,720,982 research outputs found
Composition of Linear Multistep Methods
No full textWe use the theoretical framework of General Linear Methods (GLMs) to analyze and generalize the class of Cash’s Modified Extended Backward Differentiation Formulae (MEBDF). Keeping the structure of MEBDF and their computational cost we propose a more general class of methods that can be viewed as a composition of modified linear multistep methods. These new methods are characterized by smaller error constants and possibly larger angles of A(α)-stability. Numerical experiments which confirm the good performance of these methods on a set of stiff problems are also reported
Continuous two-step Runge-Kutta methods for ordinary differential equations
No full textNew classes of continuous two-step Runge-Kutta methods for the numerical solution of ordinary differential equations are derived. These methods are developed imposing some interpolation and collocation conditions, in order to obtain desirable stability properties such as A-stability and L-stability. Particular structures of the stability polynomial are also investigated. © 2009 Springer Science+Business Media, LLC
Strong Stability Preserving Multistage Integration Methods
Full text linkIn this paper we systematically investigate explicit strong stability preserving (SSP) multistage integration methods, a subclass of general linear methods (GLMs), of order p and stage order q ≤ p. Characterization of this class of SSP GLMs is given and examples of SSP methods of order p ≤ 4 and stage order q = 1, 2, . . . , p are provided. Numerical tests are reported which confirm that the constructed methods achieve the expected order of accuracy and preserve monotonicity
Strong Stability Preserving General Linear Methods
No full textWe describe the construction of strong stability preserving (SSP) general linear methods (GLMs) for ordinary differential equations. This construction is based on the monotonicity criterion for SSP methods. This criterion can be formulated as a minimization problem, where the objective function depends on the Courant-Friedrichs-Levy (CFL) coefficient of the method, and the nonlinear constrains depend on the unknown remaining parameters of the methods. The solution to this constrained minimization problem leads to new SSP GLMs of high order and stage order
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
Highly Stable General Linear Methods for Differential Systems
No full textWe describe search for A‐stable and algebraically stable general linear methods of order p and stage order q=p or q=p−1. The search for A‐stable methods is based on the Schur criterion applied for specific methods with stability polynomial of reduced degree. The search for algebraically stable methods is based on the sufficient conditions proposed recently by Hill
Implicit-explicit general linear methods for ordinary differential equations
No full textMany practical problems in science and engineering are modeled by large systems of ordinary differential equations (ODEs) which arise from discretization in space of partial differential equations (PDEs) by finite difference methods, finite elements or finite volume methods, or pseudospectral methods. For such systems there are often natural splittings of the right hand sides of the differential systems into two parts, one of which is non-stiff or mildly stiff, and suitable for explicit time integration, and the other part is stiff, and suitable for implicit time integration. The efficient solution can be provided by implicit-explicit (IMEX) schemes. In present research we consider the class of general linear methods (GLMs) for ordinary differential equations. We construct IMEX GLMs of order p = 1, ..., 4 with desired stability properties. We assume A-stability of implicit part of IMEX scheme and we search for methods with large regions of absolute stability. Next, we apply constructed methods to a series of test problems
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