We will construct a discrete model whose solution gives an accurate ap

# We will construct a discrete model whose solution

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We will construct a discrete model whose solution gives an accurate ap- proximation to the exact solution at a finite collection of selected points (called nodes) in [0 , 1]. We take the nodes to be equally spaced and to in- clude the interval end points (boundary). So we choose a positive integer N and define the nodes or grid points x 0 = 0 , x 1 = h, x 2 = 2 h, . . . , x N = Nh, x N +1 = 1 , (10.64) where h = 1 / ( N + 1) is the grid size or node spacing. The nodes x 1 , . . . , x N are called interior nodes, because they lie inside the interval [0 , 1], and the nodes x 0 and x N +1 are called boundary nodes. We now construct a discrete approximation to the ordinary differential equation by replacing the second derivative with a second order finite differ- ence approximation. As we know, u 00 ( x j ) = u ( x j +1 - 2 u ( x j ) + u ( x j - 1 h 2 + O ( h 2 ) . (10.65) Neglecting the O ( h 2 ) error and denoting the approximation of u ( x j ) by v j (i.e. v j u ( x j )) f j = f ( x j ) and c j = c ( x j ), for j = 1 , . . . , N , then at each interior node - v j - 1 + 2 v j + v j +1 h 2 + c j v j = f j , j = 1 , 2 , . . . , N (10.66)
170 CHAPTER 10. LINEAR SYSTEMS OF EQUATIONS I and at the boundary nodes, applying (10.63), we have v 0 = v N +1 = 0 . (10.67) Thus, (10.66) is a linear system of N equations in N unknowns v 1 , . . . , v N , which we can write in matrix form as 1 h 2 2 + c 1 h 2 - 1 0 · · · · · · 0 - 1 2 + c 2 h 2 - 1 . . . . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . . . . . . . - 1 0 · · · 0 - 1 2 + c N h 2 v 1 v 2 . . . . . . . . . . . . v N = f 1 f 2 . . . . . . . . . . . . f N . (10.68) The matrix, let us call it A , of this system is tridiagonal and symmetric. A direct computation shows that for an arbitrary, nonzero, column vector v = [ v 1 , . . . , v N ] T v T A v = N X j =1 " v j + c j v j h 2 + c j v 2 j # > 0 , v 6 = 0 (10.69) and therefore, since c j 0 for all j , A is positive definite. Thus, there is a unique solution to (10.68) and can be efficiently found with our tridiagonal solver, Algorithm 5. Since the expected numerical error is O ( h 2 ) = O (1 / ( N + 1) 2 ), even a modest accuracy of O (10 - 4 ) requires N 100. 10.6 A 2D BVP: Dirichlet Problem for the Poisson’s Equation We now look at a simple 2D BVP for an equation that is central to many applications, namely Poisson’s equation. For concreteness here, we can think of the equation as a model for small deformations u of a stretched, square membrane fixed to a wire at its boundary and subject to a force density
10.6. A 2D BVP: DIRICHLET PROBLEM FOR THE POISSON’S EQUATION 171 f . Denoting by Ω, and Ω, the unit square [0 , 1] × [0 , 1] and its boundary, respectively, the BVP is to find u such that - Δ u ( x, y ) = f ( x, y ) , for ( x, y ) Ω (10.70) and u ( x, y ) = 0 . for ( x, y ) Ω (10.71) In (10.70), Δ u is the Laplacian of u, also denoted as 2 u , and is given by Δ u = 2 u = u xx + u yy = 2 u ∂x 2 + 2 u ∂y 2 . (10.72) Equation (10.70) is Poisson’s equation (in 2D) and together with (10.71) specify a (homogeneous) Dirichlet problem because the value of u is given at the boundary.