Numerical ODEs Summary

Numerical Analysis · Axiom Academy

SUMMARY Ordinary Differential Equations From initial value problems to adaptive methods Initial Value Problems (IVPs): Given y' = f(t, y) with y(t_0) = y_0, find y(t) for t > t_0. Single-Step vs Multistep: Single-step methods (Euler, RK4) use only the current point; multistep methods (Adams) use previous points for efficiency. Explicit vs Implicit: Explicit methods compute y_ n+1 directly; implicit methods require solving equations but handle stiff problems better. Order of Accuracy: Local error O(h^ p+1 ) gives global error O(h^p) for order-p method. Definition: Problem has widely varying time scales - some components decay rapidly while others change slowly. Challenge: Explicit methods require tiny steps for stability, not accuracy. Solution: Use implicit methods (Backward Euler, BDF) which remain stable with larger steps. Detection: Look for large negative eigenvalues in the Jacobian of f(t, y). Non-stiff problems, moderate accuracy: Use RK4 - the workhorse of ODE solvers. Excellent balance of accuracy and efficiency. Non-stiff, adaptive precision: Use RK45 (Dormand-Prince) with automatic step size control. Built into most software (ode45, scipy.integrate). Stiff problems: Use BDF methods (ode15s, LSODA). Implicit methods pay the cost of solving equations but gain stability. High accuracy, long integrations: Use adaptive multistep methods - fewer function evaluations per step after startup.

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