Difference equations approximating

The complete posing of a difference problem necessitates specifying the difference analogs of those conditions in addition to the approximation of the governing differential equation. The set of difference equations approximating the differential equation in hand and the supplementary boundary and initial conditions constitute what is called a difference scheme. In order to clarify the essence of the matter, we give below several examples. 

In the preceding sections this trend of research was due to serious developments of the Russian and western scientists. Specifically, the method for solving difference equations approximating an elliptic equation with variable coefficients in complex domains G of arbitrary shape and configuration is available in Section 8 with placing special emphasis on real advantages of MATM in the numerical solution of the difference Dirichlet problem for Poisson s equation in Section 9. 

With the background material of the previous sections an approach for obtaining a set of difference equations approximating a partial differential equation over the spatial finite elements can now be discussed. There are two basic approaches used to obtain such a set of equations. These will first be illustrated with a specific PDE which will be taken as a form of the linear Poisson equation ... 

Discretizating by substituting the various finite-difference type approximations for the terms in the integrated equation representing flow processes, which converts the integral equations into a system of algebraic equations. 

If the process lasts long enough in comparison with the duration of the period 2v, one may consider (approximately) A , Ap, and A

difference equations (6-124) to the stroboscopic differential equations. 

The principle of corresponding states will be used as the basis for comparing the different equations of state. This principle states that, to a reasonable approximation, all gases show the same (p. Fni. T) behavior when compared in terms of the reduced variables. The extent to which this principle is followed is... 

Grids and grid functions. The composition of a difference scheme approximating a differential equation of interest amouts to performing the following operations ... 

The right-hand sides V /j and of problem (38) are called the error of approximation of equation (35) by the difference equation (37) and the error of approximation of condition (36) by the difference condition on... 

In this section a unified interpretation of difference equations as operator equations in an abstract space is carried out and, after this, the corresponding definitions of approximation, stability and convergence are presented. This approach is quite applicable in mathematical physics for stationary problems. 

With these relations established, the second continuity condition (O+O) = w (l — 0) is approximated to O(h ) by the difference equation... 

As a result, a considerable amount of effort has been expended in designing various methods for providing difference approximations of differential equations. The simplest and, in a certain sense, natural method is connected with selecting a, suitable pattern and imposing on this pattern a difference equation with undetermined coefficients which may depend on nodal points and step. Requirements of solvability and approximation of a certain order cause some limitations on a proper choice of coefficients. However, those constraints are rather mild and we get an infinite set (for instance, a multi-parameter family) of schemes. There is some consensus of opinion that this is acceptable if we wish to get more and more properties of schemes such as homogeneity, conservatism, etc., leaving us with narrower classes of admissible schemes. 

What differential equation is approximated by the difference equation... 

In particular, u t) may be a solution of a certain differential equation. In that case we say that the difference scheme approximates the difference equation, provided condition (31) holds, etc. 

We note in passing that one is to understand the statement if a scheme is stable and provides an approximation, then it is convergent given in Chapter 2, Section 2 as follows both the difference equation and the initial value generate an approximation (if we accept — (Pj u(0), then ilyoft, - Wft(O) = 0). 

The intuition suggests that in such a setting the governing difference equation and the boundary condition at the point a = 0 have one and the same order of approximation 0(r + h ). To make sure of it, it suffices only to evaluate the residual... 

Thus, the attainable summarized approximation of the additive scheme (8) owes a debt to the simultaneous usual approximations and a summarized approximation. In accordance with what has been said above, equations (5) are approximated by the chain of the difference equations (6)-(7) in a summarized sense and every scheme (8) with the number a approximates the corresponding equation involved in collection (6) in the usual sense. 

Additive schemes. The general formulations and statements. Considerable effort is devoted to a discussion of additive schemes after introducing the notion of summarized approximation. With this aim, we recall the notion of the n-layer difference scheme as a difference equation with respect to t of order n — 1 with operator coefficients ... 

Also, we consider the total approximation method as a constructive method for creating economical difference schemes for the multidimensional equations of mathematical physics. The notion of additive scheme is introduced as a system of operator difference equations that approximates the original differential equation in the total sense. Two quite general heuristic methods (proposed earlier by the author) for obtaining additive economical schemes are discussed in full details. The additive schemes require a new technique for investigating convergence and a new type of a priori estimates that take into account the definition of the property of approximation. 

With the continuous differential operators replaced by difference expressions, we convert the problem of finding an analytic solution of the governing equations to one of finding an approximation to this solution at each point of the mesh M. We seek the solution U of the nonlinear system of difference equations... 

This then provides a physical derivation of the finite-difference technique and shows how the solution to the differential equations can be propagated forward in time from a knowledge of the concentration profile at a series of mesh points. Algebraic derivations of the finite-difference equations can be found in most textbooks on numerical analysis. There are a variety of finite-difference approximations ranging from the fully explicit method (illustrated above) via Crank-Nicolson and other weighted implicit forward. schemes to the fully implicit backward method, which can be u.sed to solve the equations. The methods tend to increase in stability and accuracy in the order given. The difference scheme for the cylindrical geometry appropriate for a root is... 

Equation 12.15 is a difference equation that, for small changes, can be approximated by a differential equation ... 

When an equation describes a system exactly but the equation cannot be solved, there are two general approaches that are followed. First, if the exact equation cannot be solved exacdy, it may be possible to obtain approximate solutions. Second, the equation that describes the system exactly may be modified to produce a different equation that now describes the system only approximately but which can be solved exactly. These are the approaches to solving the wave equation for the helium atom. 

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