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• W99895786 abstract "In this paper we will investigate how the local errors accumulate to the global error in Lie group methods for linear ODEs. The concept of the local and global errors has to be redefined to fit in the framework of Lie groups and algebras. Formulas for tracking the global error are proposed and demonstrated on numerical examples. AMS subject classifications. 65L05 1. Introduction. Among all properties that come with the numerical solution of the initial value problem in ordinary differential equations ( ODEs), small global error is usually the most important. If the global error of the computed solut ion is small enough on the interval of interest, other properties of the solution, such as corre ct asymptotic behavior, conservation of invariants, or retaining the geometric structure, becom e less important. In this paper, we will focus on the case when the global error i s small. The reason for this is at least twofold: first, as the numerical solution is c lose to the exact solution, we expect to observe similar dynamics of both solutions, while the behavior of the global error can be more unpredictable, when the distance between both solutions becomes larger. Second, the global error estimate can be used for the step-size control a nd large global error indicates that the step-size control failed. How large the global error can be before it is too large depends on the problem we are solving. In this paper we will consider the global error too large, if the exact and numerical solutions at a certain value of the indep endent variable cannot be covered with the same coordinate chart of the solution manifold. The global error at a certain point t is usually a consequence of local errors committed at each step from the initial point up to t. How the local errors are accumulating into the global error depends on the differential equation, on the numerica l method, and on the step-size selection. While we have some control over the size of the local errors during the process of solving an ODE, the global errors are usually beyond our reach. The connection between global and local errors and possible methods for controlling the global error directly has inspired a lot of research in recen t years. Highham (10) analyzed the connection between the error tolerance and the global er ror in the case of Runge-Kutta methods. Dormand et al. (6) proposed global embedding Runge-Kutta schemes for step-size control based on the estimation of the global error. Calvo et al. (1) studied methods for the global error estimation in the presence of the step-size selection mechanism for Runge- Kutta methods. Stuart (25) analyzed tolerance proportionality of the global error in Runge- Kutta methods. Viswanath (26) was concerned with situations where the usual exponential growth of the global error can be replaced by a less pessimist ic one. Onumanyi et al. (22) studied global error estimates for the finite difference met hods for initial and boundary value problems. Kulikov and Shindin were concerned with estimates for the local and global errors of linear multi-step methods with constant coefficients and fixed step-size in ( 17), and linear multi-step methods combined with Hermite type interpolation in (18). Cao and Petzold (3)" @default.
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• W99895786 date "2010-01-01" @default.
• W99895786 modified "2023-09-27" @default.
• W99895786 title "ACCUMULATION OF GLOBAL ERROR IN LIE GROUP METHODS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS" @default.
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