AdvancedTopic 96 of 104Engineering Mathematics
Finite difference method
Replaces derivatives by differences on a grid to solve PDEs.
In depth
Replace derivatives on a grid with differences to turn PDEs into algebraic equations. Explicit schemes march forward in time but need stability limits (e.g. αΔt/Δx²≤½ for the heat equation); implicit schemes are unconditionally stable and need a linear solve.
Key points
- CFL condition for waves
- Truncation error order
- Boundary conditions enter at edge nodes
ExampleCooling of a plate computed on a 50×50 grid of temperatures.
What is Finite difference method?
Replaces derivatives by differences on a grid to solve PDEs.
Why is it important to understand finite difference method?
Finite differences turn PDEs into algebra on a grid, which is the simplest route to numerical solutions of heat and wave problems.
What is a stencil?
The pattern of grid points used to approximate a derivative at one node. Its shape determines the accuracy and the structure of the resulting matrix.
When do explicit schemes fail?
When the time step is too large compared with the grid spacing. The stability condition limits the time step, so check it before running long simulations.
How is Finite difference method used in practice?
Typical applications include solving the steady temperature of a heated plate on a grid, simulating the vibration of a string with fixed ends and modelling groundwater flow through a porous layer.
What should I learn before Finite difference method?
Study "Solving ODEs numerically" first (topic 95 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Finite difference method?
CFL condition for waves. Truncation error order. Boundary conditions enter at edge nodes.
Can you give an example of Finite difference method?
Cooling of a plate computed on a 50×50 grid of temperatures.
What comes after Finite difference method?
The next topic is "Finite element basics", which builds directly on this one.
How difficult is Finite difference method?
It is classed as Advanced in Engineering Mathematics (topic 96 of 104). It is an advanced topic. It assumes you are comfortable with the intermediate topics in Engineering Mathematics.