= Solution
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.
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