An
explicit numerical method normally
has a simple, inexpensive step and needs no simultaneous solve, but stability can
force very small steps, especially for
stiff differential equations. An
implicit numerical method requires
a linear or nonlinear solve, storage and suitable solver tolerances; its larger stability region can permit larger steps when stability is the main restriction. Accuracy, phase fidelity and preservation of physical structure still constrain the step. For this
oscillator, backward
Euler method is stable but dissipative, while forward
Euler method is cheap but unstable without
damping. Neither statement makes all implicit methods superior to all explicit ones; method order, problem structure and total cost determine the
useful choice.
