Temporal discretization

In applied physics and engineering, temporal discretization is a mathematical technique for solving transient problems, such as flow problems.

Transient problems are often solved using computer-aided engineering (CAE) simulations, which require discretizing the governing equations in both space and time. Temporal discretization involves the integration of every term in various equations over a time step ( $\Delta t$ ).

The spatial domain can be discretized to produce a semi-discrete form:

{\frac {\partial \varphi }{\partial t}}(x,t)=F(\varphi ).~

The first-order temporal discretization using backward differences is

{\frac {\varphi ^{n+1}-\varphi ^{n}}{\Delta t}}=F(\varphi ),

And the second-order discretization is

{\frac {3\varphi ^{n+1}-4\varphi ^{n}+\varphi ^{n-1}}{2\Delta t}}=F(\varphi ),

where

$\varphi$ is a scalar
$n+1$ is the value at the next time, $t+\Delta t$
$n$ is the value at the current time, $t$
$n-1$ is the value at the previous time, $t-\Delta t$

The function $F(\varphi )$ is evaluated using implicit- and explicit-time integration.

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