Unit 10 of 13
Numerical Methods and Reliable Computation
- 6 lessons
- Videos 56 to 61
Many differential equations have no solution formula and are solved numerically. This unit builds Euler and Runge-Kutta methods, studies their error and stability and checks the results.
Euler's method follows tangent steps, with local truncation error of order h squared and global error of order h on smooth problems. Heun's method and the midpoint method reach second order and the classical Runge-Kutta method fourth order, and embedded pairs estimate the error to adapt the step. Stiff equations have fast decaying modes that force small explicit steps, and implicit Euler stays stable where explicit Euler does not, though stability alone does not give accuracy. Shooting and finite differences handle boundary problems, and residuals, refinement, events and units test the answer.
Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.
Euler and Runge-Kutta Methods
Euler's Method and Numerical Error
How Euler's method follows tangent steps of a chosen step size, how its local truncation error differs from its global error, what first order of accuracy means, and how step refinement and stability checks judge the numbers.
Video 56
Improved Euler and Second-Order Runge-Kutta
How Heun's method and the midpoint method use two stage slopes per step, why every second-order Runge-Kutta method of this kind has second-order global error, and why the two methods still differ.
Video 57
Fourth-Order Runge-Kutta and Adaptive Steps
How the classical Runge-Kutta method takes a weighted average of slopes from four RK4 stages, why its global error is fourth order, and how embedded error estimates and absolute and relative tolerances adapt the step size.
Video 58
Stiffness and Implicit Methods
Stiffness, Stability, and Implicit Methods
Why fast decaying modes make a problem stiff, how the test equation and amplification factors give regions of absolute stability, and why the A-stable implicit Euler method is stable without always being accurate.
Video 59
Numerical Boundary Problems and Verification
Numerical Boundary-Value Problems
How shooting turns a two-point boundary-value problem into a search for a root of the boundary residual, how finite differences lead to a tridiagonal system, and how conditioning and nonunique solutions limit both.
Video 60
Reliable Differential-Equation Computation
How to test a numerical solution with equation residuals, initial and boundary checks, event detection, refinement and independent checks, and why events, singularities and unit errors each need their own check.
Video 61