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 usefulchoice.
