Convergence and the Order of Accuracy

A typical integrator computes successive steps from the formulas 

Assume that % is a smooth map for all h 0. The exact solution satisfies 

Taking the difference of the numerical and exact solutions, we have 

This assumption is usually verified by expanding the numerical and exact solutions in powers of h, using Taylor series expansions. 

To tackle the question of the growth of local error, we still must make an important assumption on namely that it satisfies a Lipschitz condition of the form 

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The uniform convergence and the order of accuracy of a difference scheme. In the study of convergence and accuracy of scheme (2) we begin by placing the problem for the error... 

If scheme (II) is stable with respect to the right-hand side and approximates problem (I), then it converges and the order of accuracy coincides with the order of approximation. 

With these relations established, we conclude that if the scheme is stable and approximates the original problem, then it is convergent. In other words, convergence follows from approximation and stability and the order of accuracy and the rate of convergence are connected with the order of approximation. 

Everything just said means that in establishing convergence and in determining the order of accuracy of a scheme it is necessary to evaluate the error of approximation, discover stability and then derive estimates of the form (22) known as a priori estimates. 

The method of test functions is quite applicable in verifying convergence and determining the order of accuracy and is stipulated by a proper choice of the function I7(x). Such a function is free to be chosen in any convenient way so as to provide the validity of the continuity conditions at every discontinuity point of coefficients. By inserting it in equation (1) of Section 1 we are led to the right-hand side / = kU ) — qU and the boundary values jj, — U(0) and = U 1). The solution of such a problem relies on scheme (4) of Section 1 and then the difference solution will be compared with a known function U x) on various grids. 

Theorem 1 If scheme (21) is correct and generates an approximation on an element u BA then it converges. More precisely, a solution yh of problem (21) converges to this element u B as /i — 0 and, in addition, the order of accuracy of scheme (21) coincides with the order... 

It is worth noting here that on the square grid (h = h. = h) this condition is automatically fulfilled. A proper choice of (p guarantees the sixth order of accuracy of scheme (9) on any such grid. Convergence of scheme (9) with the fourth order in the space C can be established without concern of condition (11). An alternative way of covering this is to construct an a priori estimate for A z p and then apply the embedding theorem (see Section 4). 

Similarly, improvement in the accuracy of the nuclear dynamics would be fruitful. While in this review we have shown that, in the absence of any approximations beyond the use of a finite basis set, the multiple spawning treatment of the nuclear dynamics can border on numerically exact for model systems with up to 24 degrees of freedom, we certainly do not claim this for the ab initio applications presented here. In principle, we can carry out sequences of calculations with larger and larger nuclear basis sets in order to demonstrate that experimentally observable quantities have converged. In the context of AIMS, the cost of the electronic structure calculations precludes systematic studies of this convergence behavior for molecules with more than a few atoms. A similar situation obtains in time-independent quantum chemistry—the only reliable way to determine the accuracy of a particular calculation is to perform a sequence of... 

Quasi-Newton methods may seem crude, but they work well in practice. The order of convergence is (1 + /5)/2 1.6 for a single variable. Their convergence is slightly slower than a properly chosen finite difference Newton method, but they are usually more efficient in terms of total function evaluations to achieve a specified accuracy (see Dennis and Schnabel, 1983, Chapter 2). 

In order to find a reasonable configuration for our calculation, we take test calculation to optimize the bulk structure of pyrite with GGA and LDA exchange-correlation functional. In the calculation, the plane wave cutoff energy set is 280 eV and the key point set is 4 x 4 x 4, the convergence tolerances set is 10 eV/atom. The optimized cell parameter of the two methods is 0.5415 nm and 0.5425 nm respectively, which is in good agreement with the experiment data (0.5417 nm) reported. It indicates that this configuration is sufficient to satisfy the request of accuracy. 

MBPT starts with the partition of the Hamiltonian into H = H0 + V. The basic idea is to use the known eigenstates of H0 as the starting point to find the eigenstates of H. The most advanced solutions to this problem, such as the coupled-cluster method, are iterative well-defined classes of contributions are iterated until convergence, meaning that the perturbation is treated to all orders. Iterative MBPT methods have many advantages. First, they are economical and still capable of high accuracy. Only a few selected states are treated and the size of a calculation scales thus modestly with the basis set used to carry out the perturbation expansion. Radial basis sets that are complete in some discretized space can be used [112, 120, 121], and the basis... 

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