Numerical Schemes for Evolution Equations
Replace the derivatives by differences and a differential equation becomes a recursion. Consistency and stability together give convergence, and the stability condition is what caps the time step.
What a learner can do afterwards
- Derive a finite difference scheme and state its order of accuracy
- Test a scheme for stability and read off the step-size restriction
- Explain why an explicit scheme for the heat equation needs a small time step
1 · Read
Differential equations describe change, and direction fields draw that change as tiny arrows across the plane. Following the arrows by hand sketches solutions without solving anything. Euler automates the walk: step along the local arrow, re-aim, repeat.
The heat equation says the future is local averaging, and explicit schemes copy it onto a grid: each new value blends its old self with its neighbours, weighted by r. The mesh ratio is r = k dt over dx squared; with k = 1, dt = 0.01, dx = 0.1, dx squared is 0.01, so r is 1. The scheme uses a forward time difference with a centred space difference: first order in time, second in space.
Stability analysis asks whether the blending calms or amplifies grid noise, and it caps the step: stability needs r below or at 1 over 2. Since r ties dt to dx squared, halving dx quarters the allowed dt. That is why an explicit heat scheme forces tiny time steps on fine grids.
Consistency plus stability gives convergence, and neither alone suffices. A consistent scheme without stability returns nonsense, however fine the grid. Taylor logic rates the accuracy, the same way rectangles, trapezoids, and parabolas rate quadrature.
Replace derivatives with differences, keep the mesh ratio inside its stability bound, and consistency will mature into convergence.
2 · Watch
Take it off screen
Where it sits
This opens up
Nothing builds on it yet.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.