Propagation of Errors in Euler Methods
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The error bound of a numerical algorithm is very crucial to it’s selection for use in computatio of numerical values of Initial Value Problems. In this work, we investigate and compute the error bounds for the new Euler scheme proposed by Abraham in [1]. We compare and contrast this same parameter for the existing Euler Methods and the new proposed method. AMS MSC 2010 Classification: 65L05, 65L06Keywords:
Euler method
Value (mathematics)
Propagation of uncertainty
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We prove sharp, computable error estimates for the propagation of errors in the numerical solution of ordinary differential equations. The new estimates extend previous estimates of the influence of data errors and discretization errors with a new term accounting for the propagation of numerical round-off errors, showing that the accumulated round-off error is inversely proportional to the square root of the step size. As a consequence, the numeric precision eventually sets the limit for the pointwise computability of accurate solutions of any ODE. The theoretical results are supported by numerically computed solutions and error estimates for the Lorenz system and the van der Pol oscillator.
Pointwise
Truncation error
Theory of computation
Round-off error
Ode
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Basis (linear algebra)
Round-off error
Error Analysis
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Duality (order theory)
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This thesis focuses on the a posteriori error analysis for the linear second-order wave equation discretized by the second order Newmark scheme in time and the finite element method in space. We adopt the particular choice for the parameters in the Newmark scheme, namely β = 1/4, γ = 1/2, since it provides a conservative method with respect to the energy norm. We derive a posteriori error estimates of optimal order in time and space for the fully discrete wave equation. The error is measured in a physically natural norm: H1 in space, Linf in time. Numerical experiments demonstrate that our error estimators are of optimal order in space and time. The resulting estimator in time is referred to as the 3-point estimator since it contains the discrete solution at 3 points in time. The 3-point time error estimator contains the Laplacian of the discrete solution which should be computed via auxiliary finite element problems at each time step. We propose an alternative time error estimator that avoids these additional computations. The resulting estimator is referred to as the 5-point estimator since it contains the fourth order finite differences in time and thus involves the discrete solution at 5 points in time at each time step. We prove that our time estimators are of optimal order at least on sufficiently smooth solutions, quasi-uniform meshes in space and uniform meshes in time. The most interesting finding of this analysis is the crucial importance of the way in which the initial conditions are discretized: a straightforward discretization, such as the nodal interpolation, may ruin the error estimators while providing quite acceptable numerical solution. We also extend the a posteriori error analysis to the general second order Newmark scheme (γ = 1/2) and present numerical comparasion between the general 3-point time error estimator and the staggered grid error estimator proposed by Georgoulis et al. In addition, using obtained a posteriori error bounds, we implement an efficient adaptive algorithm in space and time. We conclude with numerical experiments that show that the manner of interpolation of the numerical solution from one mesh to another plays an important role for optimal behavior of the time error estimator and thus of the whole adaptive algorithm.
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Error Analysis
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We propose a novel methodology for evaluating the accuracy of numerical solutions to dynamic economic models.It consists in constructing a lower bound on the size of approximation errors.A small lower bound on errors is a necessary condition for accuracy: If a lower error bound is unacceptably large, then the actual approximation errors are even larger, and hence, the approximation is inaccurate.Our lower-bound error analysis is complementary to the conventional upper-error (worst-case) bound analysis, which provides a su¢ cient condition for accuracy.As an illustration of our methodology, we assess approximation in the …rst-and second-order perturbation solutions for two stylized models: a neoclassical growth model and a new Keynesian model.The errors are small for the former model but unacceptably large for the latter model under some empirically relevant parameterizations.
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Error Analysis
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