Numerical methods discretization error

A numerical method is said to be convergent if the global discretization error tends to zero when the step size tends to zero. The global discretization error is the difference between the computed solution (neglecting round-off errors) and the theoretical solution. Convergence is a minimal property of a numerical method and there is no use of a divergent method. Most numerical methods are convergent if they are consistent and stable. 

A numerical method is said to be consistent if the local discretization error (on one step) tends to zero when the step size tends to zero. In other words, more intuitively, the numerical algorithm tends to the mathematical equations as h - 0. The local discretization error is the error that would be made in one step if the previous values were exact and if there were no round-off errors. 

The preceding concept of stability is not sufficient when stiff problems are considered and it is necessary to introduce the concept of absolute stability. A numerical method is said to be absolutely stable if the global discretization error remains bounded for a given step size h when the number, N, of steps tends to infinity. 

The discretization error (also called the truncation or formulation error), which is caused by the approximations used in the formulatioa of the numerical method. 

The total error in any result obtained by a numerical method is the sum of the discretization error, which decreases with decreasing step size, and the roundoff error, which increases with decreasing step size, as shown in Fig. 5-58. 

In practice, we do not know the exact solution of the problem, and thus we cannot detennine the magnitude of tlie error involved in the numerical method. Knowing that the global discretization error is proportional to the step size is not much help either since there i.s no easy way of determining the value of the proportionality constant. Besides, the global discretization error alone is meaningless without a true estimate of the round-off error. Therefore, we recommend (he following practical procedures to assess the accuracy of the results obtained by a numerical method. 

A consistent numerical scheme produces a system of algebraic equations which can be shown to be equivalent to the original model equations as the grid spacing tends to zero. The truncation error represents the difference between the discretized equation and the exact one. For low order finite difference methods the error is usually estimated by replacing all the nodal values in the discrete approximation by a Taylor series expansion about a single point. As a result one recovers the original differential equation plus a remainder, which represents the truncation error. 

Methods based on an ASC have a long history in quantum-mechanical (QM) calculations with continuum solvent [60, 61, 77], where they are generally known as polarizable continuum models (PCMs). However, PCMs have seen little use in the area of biomolecular electrostatics, for reasons that are unclear to us. In the QM context, such methods are inherently approximate, even with respect to the model problem defined by Poisson s equation, owing to the volume polarization that results from the tail of the QM electron density that penetrates beyond the cavity and into the continuum [13, 14, 89], The effects of volume polarization can be treated only approximately within the ASC formalism [14, 15, 89], For a classical solute, however, there is no such tail and certain methods in the PCM family do afford a numerically exact solution of Poisson s equation, up to discretization errors that are systematically eliminable. Moreover, ASC methods have been generalized to... 

An illustration of the sampling bias (i.e., due to discretization error) is shown in Eig.7.1. As the stepsize is increased, the error in sampling is increased as well, limiting the effectiveness of numerical methods. This bias can be dramatically different for different numerical methods. As we shall show, with the right choice of numerical method it is often possible to substantially reduce this error, and it is also possible to calculate (under some assumptions) the perturbation introduced by the numerical method, and to correct for its presence. 

The numerical analysis of stochastic differential equations is traditionally based on the concepts of weak and strong accuracy. Let a numerical method be given for solving an SDE in the form of a discrete stochastic process X +i = 0(X , h) for n = 0, l,...,v — 1, where vh = x is, fixed. We denote the stochastic solution of the SDE by X(t). In the case of strong accuracy, our measure of the global error is the quantity [200]... 

Another important factor for the successful performance of a discretization scheme is its stability. In simple terms, a numerical solution method is said to be stable if it does not magnify the errors that appear in the course of a numerical solution process. For unsteady problems, stability guarantees that the numerical method yields a bounded solution, provided that the solution of... 

Adaptive computations of nonlinear systems of reaction-diffusion equations play an increasingly important role in dynamical process simulation. The efficient adaptation of the spatial and temporal discretization is often the only way to get relevant solutions of the underlying mathematical models. The corresponding methods are essentially based on a posteriori estimates of the discretization errors. Once these errors have been computed, we are able to control time and space grids with respect to required tolerances and necessary computational work. Furthermore, the permanent assessment of the solution process allows us to clearly distinguish between numerical and modelling errors - a fact which becomes more and more important. 

The integration of deterministic particle trajectories is not problematic since it is possible to draw from the vast body of known algorithms for ODEs. However, whereas exact deterministic trajectories must always remain within the integration domain (except at open boundaries), the discretized versions obtained by any numerical scheme will be subjected to a finite discretization error. This error has two main consequences with far-reaching computational effects in micro/macro methods. 

The model equations can be discretized and solved using numerical methods such as finite element, finite difference, or finite volume methods. These three methods are based on dividing geometrical domains in the model into volume elements of finite size, i.e., going from a continuous problem to a discrete problem. The time variable is often discretized using finite difference, also in the cases where finite elements or finite volumes are used for the space variables. This discretization generates a new set of equations and a new (numerical) model. The error introduced when going from the mathematical model to the numerical model is called the truncation error. 

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