Explicit and implicit methods

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Explicit and implicit methods are approaches used in numerical analysis for obtaining numerical approximations to the solutions of time-dependent ordinary and partial differential equations, as is required in computer simulations of physical processes. Explicit methods calculate the state of a system at a later time from the state of the system at the current time, while implicit methods find a solution by solving an equation involving both the current state of the system and the later one. Mathematically, if ${\displaystyle Y(t)}$ is the current system state and ${\displaystyle Y(t+\Delta t)}$ is the state at the later time (${\displaystyle \Delta t}$ is a small time step), then, for an explicit method

${\displaystyle Y(t+\Delta t)=F(Y(t))\,}$

while for an implicit method one solves an equation

${\displaystyle G{\Big (}Y(t),Y(t+\Delta t){\Big )}=0\qquad (1)\,}$

to find ${\displaystyle Y(t+\Delta t).}$