Finite Difference Method

Finite differences

The low-down ho-down

Too little time before the exam to watch an 8 minute video???

Here’s a quick summary:

Solving temperature across a plane wall can be difficult, even harder with 2 or 3 dimensions.

To simplify this we set up a set of linear equations which can be solved using a matrix.

1D conduction

To approach the problem we have to split the system into a network of nodes.

Starting with the heat equation we can then develop an equation for Ti based on the surrounding nodes.

This then leaves us with a set of equations that all need to be solved to obtain a solution.

But how??? Matrices are our friend, by taking the inverse of a matrix, an equation can be found.

For an example matrix for a 4 node network we get:

Key Points:

• • Matrix A is n+1 x n+1

  • Matrix T and b are 1 column and n+1 long

  • The diagonal is filled with the factors from the linear equation (1,-2,1)

  • Matrix A is multiplied by the factor k/∆x2

  • The top and bottom rows are determined by boundary conditions

  • The rest is filled with zeros

Boundary Conditions:

The top and bottom rows of the matrices but be defined to fulfill the boundary conditions of the matrices. This may seem complicated but just needs to be looked at methodically.

For example the Dirichlet condition:

T0  = TL   - where the temperature at x = 0  is TL

For this we need the solution to define T0 as TL, this requires the top left value of A to be set to 1 to make it equal the a value and corresponding b value to equal TL.

This process would then be repeated for the other boundary condition.

Video on Finite Difference