# Temporal discretization

## Mathematical technique / From Wikipedia, the free encyclopedia

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

This article may be too technical for most readers to understand. (May 2014) |

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:^{[1]}

The first-order temporal discretization using backward differences is ^{[2]}

And the second-order discretization is

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.^{[3]}