Finite difference methods general description

The first widely known approximation method for partial difierential equations is the finite difference method (Southwell 1946 de G AUen 1955) which approximates the governing equations of a field problem using local expansions for the variables, generally truncated Taylor series. Comprehensive descriptions of the method can be found for example in (Forsythe and Wasow 1960 Collatz 1966 Mitchell and Griffiths 1980). 

Full rate modeling Accurate description of transitions Appropriate for shallow beds, with incomplete wave development General numerical solutions by finite difference or collocation methods Various to few... 

In general, discontinuities constitute a problem for numerical methods. Numerical simulation of a blast flow field by conventional, finite-difference schemes results in a solution that becomes increasingly inaccurate. To overcome such problems and to achieve a proper description of gas dynamic discontinuities, extra computational effort is required. Two approaches to this problem are found in the literature on vapor cloud explosions. These approaches differ mainly in the way in which the extra computational effort is spent. 

We presented fully self-consistent separable random-phase-approximation (SRPA) method for description of linear dynamics of different finite Fermi-systems. The method is very general, physically transparent, convenient for the analysis and treatment of the results. SRPA drastically simplifies the calculations. It allows to get a high numerical accuracy with a minimal computational effort. The method is especially effective for systems with a number of particles 10 — 10, where quantum-shell effects in the spectra and responses are significant. In such systems, the familiar macroscopic methods are too rough while the full-scale microscopic methods are too expensive. SRPA seems to be here the best compromise between quality of the results and the computational effort. As the most involved methods, SRPA describes the Landau damping, one of the most important characteristics of the collective motion. SRPA results can be obtained in terms of both separate RPA states and the strength function (linear response to external fields). 

Traditional methods of simulation in hydrodynamics are based on the description of a fluid field obeying to partial differential equations. Finite difference, finite elements, spectral methods are generally used to approximate the equations and they are represented in the computer by floating point numbers. The implementation of the boundary conditions is the main difficulty of these methods. 

Finally, computational aspects and numerical issues are considered. This includes a short description of the finite difference, the finite volume, and the finite element discretization methods. The chapter ends with some general comments. 

The general description of these methods is provided in the present section. More detailed information on the finite differences and boundary elements will be provided in later sections of the article. It will become clear that in electrochemistry applications (of interest to us), the BEM could be the most efficient numerical technique. 

Moment methods come in many different variations, but the general idea is to increase the number of transported moments (beyond the hydrodynamic variables) in order to improve the description of non-equilibrium behavior. As noted earlier, the moment-transport equations are usually not closed in terms of any finite set of moments. Thus, the first step in any moment method is to apply a closure procedure to the truncated set of moment equations. Broadly speaking, this can be done in one of two ways. 

It is emphasized in Ref. 187 that very flexible basis sets are required to deal with the finite-basis 0-dependence of the complex eigenvalues in CCR methods, and in particular to converge the resonance widths. However, the term very flexible is used in comparison to standard basis sets for valence anions, and in fact good results are obtained for auto-ionizing resonances of He, H , and Be using the aug-cc-pVTZ-l-[3s3p] basis,which includes three even-tempered diffuse s and p shells. This is not much different from the basis sets recommended here for proper description of loosely-bound electrons in general gas-phase calculations. 

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