Accuracy of Numerical Methods

As a rule, equations of gas dynamics are discontinuous. From a physical point of view it is fairly common to distinguish weak discontinuities relating to cutting waves and strong discontinuities relating to shock waves . For these reasons successive grid refinement can be made with caution when the accurate account of accuracy of numerical methods is performed. 

As mentioned earlier, overall accuracy of finite element computations is directly detennined by the accuracy of the method employed to obtain the numerical solution of the global system of algebraic equations. In practical simulations, therefore, computational errors which are liable to affect the solution of global stiffness equations should be carefully analysed. 

The function fni(r) given in the standard tables is usually rapidly varying and is therefore difficult to differentiate numerically. The function Fnl[r) is varying much more slowly, and Eq. 11.78 is hence more convenient as the starting point for the numerical work. The accuracy of this method for evaluating the HF energy is now being tested for the atomic case. 

In-day/out-of-day variation Does the precision and accuracy of the method change when conducted numerous times on the same day and repeated on a subsequent day ... 

Our AIMD simulations are all-electron and self-consistent at each 0.4 femtoseconds (fs) time step. Variational fitting ensures accurate forces for any finite orbital or fitting basis sets and any finite numerical grid. These forces are used to propagate the nuclear motion according to the velocity Verlet algorithm [22]. The accuracy of these methods is indicated by the fact that during the 500 time-step simulations of methyl iodide dissociation described below, the center of mass moved by less than 10-6 A. 

The accuracy of this method is dependent entirely on the skill of the person sketching the curvilinear squares. Even a crude sketch, however, can frequently help to give fairly good estimates of the temperatures that will occur in a body and these estimates may then be refined with numerical techniques discussed in Sec. 3-5. An electrical analogy may be employed to sketch the curvilinear squares, as discussed in Sec. 3-9. 

The principal considerations in% choosing a finite-difference method for (7) are accuracy, stability, computation time, and computer storage requirements. Accuracy of a method refers to the degree to which the numerically computed temporal and spatial derivatives approximate the true derivatives. Stability considerations place restrictions on the maxi-... 

Numerical solutions of the Schrodinger equation can be obtained within several degrees of approximation, for almost any system, using its exact Hamiltonian. Density functional theory has proven to be one of the most effective techniques, because it provides significantly greater accuracy than Hartree-Fock theory with just a modest increase in computational cost.io> 3-45 The accuracy of DFT method is comparable, and even greater than other much more expensive theoretical methods that also include electron correlation such as second and higher order perturbation theory. 

As a test for the accuracy of our methods we consider the numerical integration of the Schrodinger equation (1) with l = 0 in the well-known case where the potential V(r) is the Woods-Saxon one. 

Besides examining these properties of numerical methods, specific efforts need to be made to assess the accuracy of numerical solutions of flow processes. Various types of errors and possible ways of estimating and controlling these errors are discussed... 

The great advantage of the methods described in this section over those described earlier is, of course, rapidity in computation. This gain in computational simplicity is, however, at the expense of theoretical rigor. It is, therefore, important to establish the accuracy of the methods described above using the exact method of Section 8.3 as a basis for comparison. The extensive numerical computations made by Smith and Taylor (1983) showed that the explicit method of Taylor and Smith ranked second overall among seven approximate methods tested (the linearized method of Section 8.4 was best). For some determinacy... 

There is still no reports published evaluating the behavior of this procedure for fluid particle flows. In practical applications, the stability problems are commonly adjusted by numerous tricks that reduce the accuracy of the method. 

The performance of numerical methods for chemical continuity equations is generally characterized in terms of accuracy, stability, degree of mass conservation, and computational efficiency. The simplest of such methods is provided by the forward Euler or fully explicit scheme, by which the solution y" 1 at time tn y is given by... 

Analog computers are essentially electronic amplifying circuits capable of solving differential equations without the use of numerical methods required for digital work. The effect of a complex set of variables on a process can be studied without an expensive card programming. The resultant electronic output data as the answer can be automatically plotted for a permanent record. The inherent accuracy of this type of... 

The electronic wave functions are obtained by solving a time-independent Schrddinger equation by methods that include the use of monoelectronic functions (orbitals) as basis sets for the approximation of solutions indeed the kind of functions employed as electronic orbitals strongly affects the convergence rate and accuracy of numerical procedures. 

In the Ref [14], the method of reflections was applied to calculations of three-particle and four-particle interactions. It was shown that, as compared to pair interactions, three- and four-particle interactions introduce corrections of the order 0(l/r" ) and 0(l/r ) to the corresponding velocity perturbations, where r is the characteristic distance between particles. A generalization for the N-particle case was made in [15]. The velocity perturbation is found to be of the order 0(l/r + ). In the same work, expressions for the mobility functions are derived up to the terms of order 0(l/r ). It should be kept in mind that the corresponding expressions are power series in 1/r, so to calculate the velocities at small clearances between particles (it is this case has presents the greatest interest), one has to take into account many terms in the series, or to repeat the procedure of reflection many times. In addition to analytical solutions, numerical solutions of a similar problem are available, for example, in [16]. At small clearances between particles, the application of numerical methods is complicated by the need to increase the number of elements into which particle surfaces are divided in order to achieve acceptable accuracy of the solution. 

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