6.3.1 Poisson's Equation in One Dimension

We begin with a one-dimensional version of Poisson's equation,

where f( x) is a given function and v( x) is the unknown function that we want to compute. v( x) must also satisfy the boundary conditions ^[24] v( 0) = v(1) = 0. We discretize the problem by trying to compute an approximate solution at N + 2 evenly spaced points x _i between 0 and 1: x _i = ih, where and 0 ? i ? N + 1. We abbreviate v _i = v( x _i) and f _i = f( x _i). To convert differential equation (6.1) into a linear equation for the unknowns v ₁, , v _N, we use finite differences to approximate

Subtracting these approximations and dividing by h yield the centered difference approximation

where ? _i, the so-called truncation error, can be shown to be . We may now rewrite equation (6.1) at x = x _i as

where 0 < i < N+1. Since the boundary conditions imply that v ₀ = v _N+1 = 0, we have N equations in N unknowns v ₁, , v _N:

or

To solve this equation, we will ignore ?, since it is small compared...