Explicit and implicit methods: Difference between revisions

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In [[applied mathematics]], '''explicit and implicit methods''' are approaches for mathematical [[computer simulation|simulation]] of [[physics|physical]] [[process]]es, or in other words, they are [[numerical analysis|numerical methods]] for solving time-variable [[ordinary differential equation|ordinary]] and [[partial differential equation]]s.

Explicit methods calculate the state of a system atafter theone next instanceinterval of time from the state of the system at the current time, while an implicit method finds it by solving an equation involving both the current system state and the future one. To put it in symbols, if <math>Y(t)</math> is the current system state and <math>Y(t+\Delta t)</math> is the state at the next instance in time (<math>\Delta t</math> is a small time step), then, for an explicit method

: <math>Y(t+\Delta t) = F(Y(t))\,</math>

while for an implicit method one solves an equation