10.2 Section 2 Background .1 Cosmology .11 Math Section 10.4

Book 9 Computational Mathematics Chapter 10 Partial Differential Equations

Section 3 Iteration of difference equations

A direct solution of systems of finite difference equations for many grid points by successive elimination turns out to be too laborious and it becomes advisable to employ iterative methods which take into account their special form ([5],[60]).

Consider one of the simplest methods, Liebmann's averaging process for the systems (2.5) ([13], [35], [39]; [43], the last of which presents a survey of other methods).

This method carries out the computations in the following manner: Select an initial approximation u(0)ij and determine successive approximations for the internal points of a grid by means of the formula

(3.1)

The search for an initial approximation can be completed by two methods:

1. Interpolate for the interior points the known boundary values (cf. Example 1);
2. Form a system of finite-difference equations for a grid with a larger value of h, solve it by elimination and then interpolate the values obtained with respect to the given grid (cf.
Example 2).

It has been shown ( [18], [37], [52]) that, for any spacing h, Liebmann's process converges to the exact solution independently of the choice of the initial values, i.e., that there exists the limit

This iteration will converge much more rapidly, if subsequent arithmetic means involve the values of the previous approximation as well as the newly found values (Seidel's Method; cf. Example 2). As a rule, iterations are continued until the required numbers of decimals in two successive steps coincide. The error of approximations to solutions of Laplace's equation can be estimated by the Runge principle ([2], [35]), according to which the error e h of the approximate solution uh, which is obtained for the spacing h, is given by the approximate formula

(3.2)

where u2h is the approximate solution obtained for the spacing 2h. Note that this iteration demands that standard averaging operations be carried out at each interior point, whence it turns out to be very suitable for computer programming. For execution of the process on calculators, it is useful to prepare a sufficient number of computation schedules ([13], [35], [39])*.

*For other effective methods and complete bibliography see [43].

Example 1. Solve Laplace's equation for a square with the boundary conditions indicated in the next figure.

Figure 9

Solution: The next figure displays an appropriate schedule for the problem:

Figure 10

Each grid point is replaced by a square. The squares corresponding to boundary points contain the given boundary values. Since in the iteration process these values do not change, the remaining schedule, shown in the next figures, are 5*5 squares attached to the basic patterns.

Table 101

Values of successive approximations (Example 1)


Filling in of the patterns:

1. Interpolate the boundary values at the internal points as follows: Start with the upper row and assume that the function u(x,y) decreases linearly from 15.45 to 0, i.e., that for the initial value

i.e.,

Continue with the right column setting u(0)5j = u(0)i5. Then proceed to the next row. Assume that the function u(x,y) decreases linearly from 29.39 to 5.15. Reasoning as before, obtain u(0)i4, i = 2, 3, 4, 5, and hence u(0)4j, j = 2, 3, 4, 5. Continue until the entire table of Pattern 1 is completed.

2. Compute successive approximations by (3.1). Attach Pattern 1 to the basic pattern and compute

step by step using (3.1) for k = 1: etc.

Enter all results into Pattern 2 and find the next approximations u(2)ij until the values of two subsequent differ by less than 0.05. The results of the l6th and l7th iterations satisfy this condition. The patterns corresponding to the successive iterations are shown above. Since the patterns are symmetric, only half of them have been completed.

Example 2: Find the solution of Laplace's Equation for a square and the boundary conditions shown in the next figure, using , h = 1/6,:

Figure 11

Solution:

1. Computing the initial approximation: Construct to start with a grid with spacing h = 1/3, denoting the values of the unknown function at the grid points by a, b, c, d. In view of the symmetry of the boundary conditions,

a = b = c = d.

Therefore one need only set up the single equation

40 + b + 20 + d - 4a = 0.

By (3.3), one finds that

a = 30.

Now compute the initial approximation for h = 1/6. Firstly, construct the pattern

Figure 12

and enter into it the boundary conditions and the values obtained at the four points:

From these quantities find the values u(0)ij at the remaining points of the grid. In detail, the computation of u(0)i1, i=l, 2, 3, 4, 5 are as follows. The values u(0)11and u(0)31 are obtained from (2.6):

The value u(0)21 is found by (2.5):

In view of the symmetry,

The remaining values u(0)ij are computed in the same manner.

2. Computing successive approximations: In view of the symmetry of the problem, only results for a quarter of the square have to be obtained. In order to accelerate the convergence of the iterations, proceed as follows. Find u(0)11 by (3.1):

This value is employed to compute u(1)21, i.e.,

When computing u(1)31, use u(1)21 = u(1)41, etc. The iteration process is continued until the results of two consecutive approximations differ by not more than 0.1. The results of the successive approximations for a quarter of the square are presented in the next table.

Table 102

Values of successive approximations (Example 2)

      

Example 3: The next table presents the approximate solution of Laplace's Equation with h = 0.1 for a unit square and the indicated boundary values . Estimate the error of this answer by the Runge Method.

Table 103

Approximate Solution of a boundary value problem with h = 0.1

Solution: Solve the problem again with 2h = 0.2, using the initial approximation from the Table 103 above. The results of the computations are presented in Table 104 next.

Table 104

Approximate Solution of a boundary
value problem with 2h = 0.2

Table 105

The Differences
uh - u2h

Then find the differences uh = u2h between the values of the required solutions obtained with h = 0.1 and 2h = 0.2 and compute the error e h with(3.2) (cf. the next two tables).

Table 106
Errors
e h

Exercises

1. Apply Liebmann's Method of averaging to obtain an approximate solution of Laplace's Equation with h = 1/8 for a square with the vertices A (0,0), B(0, 1), C(1, 1) and D(1, 0). The boundary conditions are given in Table 107 following:

Table 107

Boundary Conditions for problem 1

Carry out the iterations accurate within 10-2.

2. Find an approximate solution of Laplace's Equation in the square ABCD for the boundary conditions. indicated in the next table with h = 1/6 for the following values of the parameters

Table 108

Boundary Conditions for problem 2

Iterate to within 10-2.

3. Find approximate solutions of Laplace's Equation for the domains and boundary conditions a to c in the Figure 13 next.

Figure 13



with iterations until the difference between subsequent values of the function for all points become less than 0.005.

4. Solve Laplace's Equation for a unit square with h = 1/8. The boundary values on left side of the square are 2.5, 5.0, 7.5, 10.0, 7.5, 5.0, 2.5, while they are zero elsewhere. Iterate to within 10-4. Use as first approximation the solution of Problem 3c above.