Equation Discretization

Can be thought of the steady state, static form of the diffusion equaiton - begin with equaiotn, expect relaxation to steady state (i.e. as t→ inf, d/dt → 0), get the following Poisson equation:

Matrix form, relaxation form

• Discretizatiion leads to a syustenm of eqns. There are 2 ways to solve the equations. Matrix direct methods and relaxtiuon iteration mrethods.

Matrix form (not followed)

I did not write a code that uses a direct matrix method for solving the system of equations, but include this section here for completeness.

Relaxation form (followed)

Error, accuracy, convergence

Laplace Equation, source term = 0

Note that the discretization equation with f=0 describes each point as being the average of its neighbors. This is in fact one of the key properties of the harmonic functions, which are the class of solution to the Laplace Equation. Since this is such an ubiquitious equation in physics, and a subset of the Poisson equation, I felt it was important to review some of the properties behind it.

• Boundary conditions
• Minimizing distance b/w boundaries (in 1d this is a line, in 2d this is soap bubble)
• Physical interprestaion - no charge within the domain, but elsewhere.

References