Explicit methods:
Are simple and inexpensive to evaluate per time step.
Parallelize naturally across large systems.
Suffer from conditional numerical stability.
Require prohibitively small step sizes on stiff equations.
Implicit methods:
Are computationally intensive per step (requiring nonlinear root finding or linear system solves).
Do not parallelize as easily.
Offer superior numerical stability (often unconditionally stable).
Can take significantly larger step sizes on stiff problems without diverging.
Key Takeaways Across IVP Families¶
Systems of equations and higher-order ODEs (via reduction of order) are natural, direct vector extensions of single-equation solvers.
Adaptive time stepping is essential in practice and can be achieved with minimal overhead using embedded Runge-Kutta pairs (such as RKF45).
Explicit Runge-Kutta methods efficiently deliver high-order accuracy whenever numerical stability is not the limiting factor.
Implicit Runge-Kutta methods require solving coupled stage equations simultaneously, which increases computational cost with the number of stages.
Explicit linear multistep methods have step-size and stability restrictions that limit their standalone use.
Implicit linear multistep (Backward Differentiation Formulae, BDF) methods combine strong stiff stability with efficient reuse of solution history.
Dr. Mike’s Tips¶
Know your physics. If you do not know whether your system is stiff, start with an adaptive BDF solver (or an automatic stiffness-detecting solver).
Non-stiff smooth dynamics: If you know your system is non-stiff, RK45 is the go-to workhorse and will save substantial compute time.
Discontinuous forcing: If your physics features abrupt changes (e.g., step functions, switching, or sharp pulses), consider Backward Euler (BDF1) across the discontinuity, where higher-order Taylor assumptions break down.