10.3 Section | 2 Background .1 Cosmology .11 Math | Section 10.5 |
Section 4 Boundary Value Problems for curvilinear domains
If the boundary G of a domain G is curvilinear, then the values of u_{ij} at the boundary points are obtained by transfer from points near the boundary. The errors involved in such a procedure can be considerably reduced by setting up equations for each boundary point ([2], [13], [35]):
1. For the point A_{h} :
(4.1)
2. For the point C_{h} :
(4.2)
Once one such equation has been obtained for each boundary point and added to the system (2.4) or (2.5), one has arrived at a system of algebraic equations fir the u_{ij}. If this system is solved by Liebmann's Method, the successive approximations of the boundary values are given by
(4.3)
(4.4) |
Example 1: Solve Laplace's Equation
(4.5)
for the circle x^{2 }+ y^{2} = 16 and the boundary condition
(4.6)
Solution: Since the solution is symmetric, only a single quadrant of the circle need be considered.
Filling in the patterns:
(1) Figure 15 is a large mesh net with h =2. The point M(12,2) on the boundary G lies nearest to the grid point A(4,2), hence let
u(A) u(M) = 12?2^{2} = 48.
Analogously, the point M'(2,12) is closest to the A'(2,4), whence u(A') u(M') = 12?2^{2} = 48.
Obviously, at C(4,0) and C'(0,4) we have
u(C) = u(C') = 0.
Denoting the values of u(x,y) at the interior grid points by a, b and c and taking into account the fact that the problem is symmetric yields the system of finite difference equations
hence
Figure 15
2. Construct a grid with h =1 and unspecified boundary values and let
Using the values of the function u(x,y) at points of the grid with h=2 as well as at boundary points and taking into consideration symmetry, form finite difference equations for the points a, b, c, d, e, and f of the preceding figure and set up the equations (6) (see Sec, 10.2). Thus we get
Hence we find approximate values
3. Specify the value of u(x,y) at the boundry points using formula (4). For Point A we have
hence
Carry out similar commutations for point B:
Thus for the boundary value points
4. Compile a table of the initial values (Pattern 1) and specify in turn the values of u(x,y) at the internal points by means of (3.1) until the values obtained in two successive iterations differ by less than unity. The results of the computations are then entered into Patterns 2 and 3.
Table 109 - Values of successive applications of (5) and (6).
,
It is seen that the approximate values of u(x,y) in Patterns 2 and 3 differ by less than unity. For purposes of comparison, Pattern 3a contains the values of the exact solution of the problem at the mesh points.
Exercises
Use the difference method with spacing h to find the solution of Laplace's Equation in G for the given boundary boundary conditions. Solve the system of finite difference equations, which specify the boundary values, by Liebmann's method,
1. Use h = 0.1. The domain G is bounded by the curves 2y = 1 - 4x^{2}, y = 0, x = 0, boundry conditions have the form
2. Use h = 0.2. The domain G is bounded by
The boundary conditions are:
(a) where C is the circle x^{2} + (y +3)^{2} = 16.
(b) where C is the circle x^{2} + (y +3)^{2} = 16.