Temporal discretization

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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 (${\displaystyle \Delta t}$).

The spatial domain can be discretized to produce a semi-discrete form:^[1] $${\displaystyle {\frac {\partial \varphi }{\partial t}}(x,t)=F(\varphi ).~}$$

The first-order temporal discretization using backward differences is ^[2] $${\displaystyle {\frac {\varphi ^{n+1}-\varphi ^{n}}{\Delta t}}=F(\varphi ),}$$

And the second-order discretization is $${\displaystyle {\frac {3\varphi ^{n+1}-4\varphi ^{n}+\varphi ^{n-1}}{2\Delta t}}=F(\varphi ),}$$ where

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

The function ${\displaystyle F(\varphi )}$ is evaluated using implicit- and explicit-time integration.^[3]