The maximum principle

The canonical form of a grid equation of common structure. The maximum principle is suitable for the solution of difference elliptic and parabolic equations in the space C and is certainly true for grid equations of common structure which will be investigated in this section. 

Let CO be a finite set of nodes (a grid) in some bounded domain of the n-dimensional Euclidean space and let P G co be a point of the grid u . Consider the equation 

Similar equations do arise in grid approximations of integral equations. In what follows we will suppose that coefficients A(P) and B P,Q) are subject to the conditions 

A point P is called a boundary node of the grid w if at this point the value of the function y P) is known in advance  

We call the nodes, at which equation (1) is valid under conditions (2), inner nodes of the grid uj is the set of all inner nodes and ui = ui + y is the set of all grid nodes. The first boundary-value problem completely posed by conditions (l)-(3) plays a special role in the theory of equations (1). For instance, in the case of boundary conditions of the second or third kinds there are no boundary nodes for elliptic equations, that is, w = w. 

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Variational calculus, Dreyfus (1962), may be employed to obtain a set of differential equations with certain boundary condition properties, known as the Euler-Lagrange equations. The maximum principle of Pontryagin (1962) can also be applied to provide the same boundary conditions by using a Hamiltonian function. 

In fact, assuming that the solution yi of problem (23) becomes nonzero at least at one point i = we come to a contradiction with the maximum principle if j/, >0, then yi attains its maximal positive value at some point 0 < Iq < N, violating with Theorem 1 the case yi < 0 may be viewed on the same footing. 

The maximum principle implies that problem (12)-(15) is uniquely solvable. With this in mind, we start from the scheme for 0 < x = i/i < 1 ... 

The main idea here is connected with the design of a new difference scheme of second-order approximation for which the maximum principle would be in full force for any step h. The meaning of this property is that we should have (see Chapter 1, Section 1)... 

The natural replacement of the central difference derivative u x) by the first derivative Uo leads to a scheme of second-order approximation. Such a scheme is monotone only for sufficiently small grid steps. Moreover, the elimination method can be applied only for sufficiently small h under the restriction h r x) < 2k x). If u is approximated by one-sided difference derivatives (the right one for r > 0 and the left one % for r < 0), we obtain a monotone scheme for which the maximum principle is certainly true for any step h, but it is of first-order approximation. This is unacceptable for us. 

The proof of convergence of scheme (19) reduces to the estimation of a solution of problem (21) in terms of the approximation error. In the sequel we obtain such estimates using the maximum principle for domains of arbitrary shape and dimension. In an attempt to fill that gap, a non-equidistant grid... 

Theorem 1 (the maximum principle) Let y P) const be a grid function defined on a connected grid w and let both conditions (2) and (4) hold. Then the condition Cy P) < 0 (C y P) > 0) on the grid w implies that y(P) cannot attain the maximal positive (minimal negative) value at the inner nodes P E u>. 

In this chapter difference schemes for the simplest time-dependent equations are studied, namely, for the heat conduction equation with one or more spatial variables, the one-dimensional transfer equation and the equation of vibrations of a string. Two-layer and three-layer schemes are designed for the first, second and third boundary-value problems. Stability is investigated by different methods such as the method of separation of variables and the method of energy inequalities as well as by means of the maximum principle. Asymptotic stability of difference schemes is discovered for the heat conduction equation in ascertaining the viability of difference approximations. Finally, stability theory is being used, increasingly, to help us understand a variety of phenomena, so it seems worthwhile to discuss it in full details. 

The maximum principle and, in particular. Theorem 3 in Chapter 1, Section 2 will be quite applicable once we rearrange problem (II) supplied by homogeneous boundary conditions (scheme (16)) with obvious modifications and minor changes. The traditional tool for carrying out this work is connected with... 

Stability of scheme (II) with the third kind boundary conditions can be discovered following established practice either by the method of separation of variables or on account of the maximum principle. 

Scheme (3) is conditionally stable in the space C with respect to the initial data, the right-hand side and the boundary data. The maximum principle for the difference problem (3) may be of help in establishing the indicated properties with further reference to the canonical form... 

As a matter of fact, we obtain for y equation (6) with F = 0. On the strength of the maximum principle (see corollaries to Theorem 2, Chapter 4, Section 2.3) we have... 

The maximum principle can be applied to any such scheme with weights under the constraint t- < Tg, where... 

This estimate can be improved for the forward difference scheme with (7 = 1 by means of the maximum principle and the method of extraction of stationary nonhomogeneities , what amounts to setting... 

In this regard, the. maximum principle with regard to equation (44) yields... 

Remark 2 Uniform convergence with the rate 0 h + r) of the forward difference scheme with cr = 1 can be established by means of the maximum principle and the reader is invited to carry out the necessary manipulations on his/her own. 

In order to construct a monotone scheme for problem (78) for which the maximum principle would be valid for any li and r, we involve in subsequent considerations the equation of the same type, but with the perturbed operator Z ... 

The maximum principle applies equally well to the estimation of the problem (80) solution with zero boundary conditions y = = 0 in tack-... 

For cr = 1 the maximum principle is in full force for any r and h, due to which the resulting scheme is uniformly stable with respect to the initial data and the right-hand side. What is more, the uniform convergence occurs with the rate 0(r - - h ). 

Indeed, in conformity with the maximum principle (for more detail see Chapter 4, Section 2) problem (9) has for f y) < 0 the majorant... 

However, due to the maximum principle a solution of the boundary-value problem (29)-(30) is non-negative ... 

This serves to motivate the validity of the maximum principle with regard to equation (58) supplied by the homogeneous boundary conditions (55). By utilizing this fact it is plain to show that the estimate holds ... 

Stability of lOS. The main goal of stability consideration is to establish that the uniform convergence with the rate 0 t + /ip) follows from a summarized approximation obtained. This can be done using the maximum principle and a priori estimates in the grid norm of the space C for a solution of problem (21)-(23) expressing the stability of the scheme concerned with respect to the initial data, the right-hand side and boundary conditions. 

Recall that in Chapter 4, Section 2 we have proved the maximum principle and derived a priori estimates for a solution to the grid equation of the general form... 

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