Mr. IntegralTHE WHOLE UNIT · ONE REFERENCE
Numerical Methods
Cheat sheet.
The key rules, formulas and reminders from all 3 topics, gathered into reference cards.
Key formulas, conditions and traps · Read down each column.
Euler’s Method and Local Slopes
Core rule
Watch for
Compare methods at the same final time. A smoother-looking polyline does not prove accuracy. Step refinement, exact checks when available and stability analysis provide stronger evidence.
Midpoint, Heun and Runge–Kutta Methods
Core rule
Watch for
Higher order does not remove stability limits or repair discontinuous forcing automatically. Split a step at known discontinuities. Adaptive methods estimate local error and adjust steps, but the requested tolerance is not an unconditional bound on every global error.
Numerical Stability, Error and Stiffness
Core rule
Watch for
At , Euler’s factor is , so magnitude is preserved instead of decaying. This boundary is not asymptotic decay. Step refinement should inspect solution values and qualitative behavior, not only whether the program runs.
Euler’s Method and Local Slopes
2 reference blocks
Core rule
Watch for
Compare methods at the same final time. A smoother-looking polyline does not prove accuracy. Step refinement, exact checks when available and stability analysis provide stronger evidence.
Midpoint, Heun and Runge–Kutta Methods
2 reference blocks
Core rule
Watch for
Higher order does not remove stability limits or repair discontinuous forcing automatically. Split a step at known discontinuities. Adaptive methods estimate local error and adjust steps, but the requested tolerance is not an unconditional bound on every global error.
Numerical Stability, Error and Stiffness
2 reference blocks
Core rule
Watch for
At , Euler’s factor is , so magnitude is preserved instead of decaying. This boundary is not asymptotic decay. Step refinement should inspect solution values and qualitative behavior, not only whether the program runs.