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Numerical Methods
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022
A local truncation error (from the Latin truncare, meaning to cut off) measures the accuracy of the method at a specific step, assuming that the method was exact at the previous step: hTn(h)=(u(xn+1)−u(xn))−(yn+1−yn),
Numerical Methods
Published in Suman Kumar Tumuluri, A First Course in Ordinary Differential Equations, 2021
This numerical disaster is due to the round-off error. Thus decreasing the step size, so that certain ‘important’ terms in the computation of the approximate solution are neglected, can lead to increase in errors which in turn causes adverse effects. For a detailed analysis of round-off errors refer to [4]. Though reduction of step size decreases the local truncation error, it can create two types of problems. They are (i) increase of computation cost/time due to increase in the number of arithmetic calculations (indirectly increases the round-off errors), (ii) direct increase in the round-off errors as explained before. In all the examples and exercises that we consider henceforth, we do not take h too small and assume that the round-off errors are negligible. To summarize the discussion, if h is large then the local truncation error is large, and if h is too small then the round-off error is large. In order to minimize both the local truncation error as well as the round-off error we need to use higher order methods, in particular, multi-step methods.
Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2018
A local truncation error (from the Latin truncare, meaning to cut off) measures the accuracy of the method at a specific step, assuming that the method was exact at the previous step: hTn(h)=uxn+1-uxn-yn+1-yn,
Optimal local truncation error method for solution of elasticity problems for heterogeneous materials with irregular interfaces and unfitted Cartesian meshes
Published in Mechanics of Advanced Materials and Structures, 2023
with the following local truncation error for the first stencil (see our article [35] for details): i.e., the order of the local truncation error cannot exceed four for any 9-point uniform stencils independent of the method used for their derivation (the finite element method, the finite volume method, the finite difference method, or any other method). The fourth order of the local truncation error corresponds to the second order of accuracy for the global numerical solution (e.g., see our article [35]) and is the same as that for linear finite elements. These results are different from the application of OLTEM to the Poisson equation for which at the same 9-point stencils the accuracy was improved by two orders for rectangular meshes and by four orders for square meshes compared to linear finite elements; see [42].
Calculus of variations for estimation in ODE–PDE landslide-like models with discrete-time asynchronous measurements
Published in International Journal of Control, 2022
Mohit Mishra, Gildas Besançon, Guillaume Chambon, Laurent Baillet
The Euler method is based on a truncated Taylor series expansion (Ascher & Petzold, 1998), i.e. expansion of y in the neighbourhood of , where with , N being the number of time steps, gives At each time step, higher order terms are neglected which induces local truncation error proportional to the square of the step size , and the global error (error at a given time) is proportional to the step size. The value of variable y at time in Equation (1) is computed as The Euler method is used for numerical integration of system and adjoint ODEs, both forward and backward in time.
Optimal local truncation error method for solution of 2-D elastodynamics problems with irregular interfaces and unfitted Cartesian meshes as well as for post-processing
Published in Mechanics of Advanced Materials and Structures, 2023
For homogeneous materials all aj () coefficients are aj = 1 (see Eq. 12 if we consider material ) as well as all () coefficients are zero. The case of material can be similarly treated. For material the local truncation error, Eq. (24), does not include the terms with symbol i.e., the corresponding terms () in Eq. (24). Then, the local system of equations, Eq. (28), reduces to the following 48 algebraic equations for the 36 stencil coefficients () and 12 Lagrange multipliers (): where similar to Section 2.3.1, equations should be replaced by and for the first stencil with j = 1 as well as should be replaced by and for the second stencil with j = 2; see Remark 3. We should mention that the explicit values of the stencil coefficients for homogeneous materials with Poisson ratio are given in our paper [46]. The stencil coefficients of OLTEM for homogeneous materials provide the fourth order of the local truncation error, Eq. (24); i.e., the order of the local truncation error cannot exceed four for any 9-point uniform stencils independent of the method used for their derivation (the finite element method, the finite volume method, the finite difference method, or any other method). The fourth order of the local truncation error corresponds to the second order of accuracy for the global numerical solution (e.g., see our paper [46]) and is the same as that for linear finite elements. These results are different from the application of OLTEM to the scalar wave equation for which at the same 9-point stencils the accuracy was improved by two orders compared to linear finite elements; see [42].