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MathidiaMathematics Academy
Scientific computing
Lesson 59 of 75
Challenge

Computational PDEs

Objective. Discretise derivatives and test numerical solutions.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Computational PDEs discretise derivatives on a grid, replacing a continuous equation by a system of algebraic equations that a computer can solve, and then test the numerical solution for stability and accuracy. The finite difference method approximates ∂u/∂t and ∂²u/∂x² using neighbouring grid values, and for explicit schemes applied to the heat equation the Courant–Friedrichs–Lewy (CFL) condition requires the time step Δt to satisfy Δt ≤ Δx²/(2α) for diffusivity α, or the scheme blows up. For example, with Δx = 0.1 and α = 1, stability requires Δt ≤ 0.005. Checking convergence by halving the grid spacing and confirming the error roughly halves (or quarters) as predicted validates the discretisation.

Worked example

For a computational PDE solving the heat equation with diffusivity α = 2 using an explicit finite difference scheme on a grid with spacing Δx = 0.2, find the maximum stable time step Δt using Δt ≤ Δx²/(2α).

2 … 0.2
Illustrated card showing 2 … 0.2
  1. 1Compute Δx² = 0.2² = 0.04.
  2. 2Compute the denominator 2α = 2 × 2 = 4.
  3. 3Divide: Δt ≤ 0.04/4.
  4. 4Simplify 0.04/4 = 0.01, the maximum stable time step.

Answer: 0.01

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Common mistake. A common mistake is choosing a time step for an explicit finite difference scheme without checking the CFL stability condition, which causes the numerical solution to oscillate and diverge even though the underlying PDE is well-behaved.

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