Lesson 254
Differential Equations<br>& Numerical PDEs
Euler · Runge–Kutta · Finite Differences · Heat Equation
1:00How to simulate differential equations numerically — from forward Euler and RK4 for ODEs to finite-difference schemes and the heat equation for PDEs.
By the end, you can
- Explain what a differential equation is and why the unknown is a function, not a number.
- Apply one or two steps of forward Euler by hand and compute the result.
- State Euler's global error order O(h) and its stability condition |1 + h·k| ≤ 1.
- Explain how RK4 achieves O(h⁴) error and how many slope samples it uses.
- Distinguish ODEs from PDEs by the number of independent variables.
- Approximate a second derivative using the (+1, −2, +1)/Δx² finite-difference stencil.
- Write out the FTCS update rule and identify the diffusion number r.
- Apply the FTCS stability criterion r ≤ 1/2 to compute the maximum allowed time step.
- Explain what stiffness means and why implicit methods are preferred for stiff problems.
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