Temporal discretization

This methods evaluates the function ${\displaystyle F(\varphi )}$ at a future time.

The evaluation using implicit-time integration is given as:

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

This is called implicit integration as ${\displaystyle \varphi ^{n+1}}$ in a given cell is related to ${\displaystyle \varphi ^{n}}$ in neighboring cells through ${\displaystyle F(\varphi ^{n+1})}$:

$${\displaystyle \varphi ^{n+1}=\varphi ^{n}+\Delta tF(\varphi ^{n+1}),}$$

In case of implicit method, the setup is unconditionally stable and can handle large time step (${\displaystyle \Delta t}$). But, stability doesn't mean accuracy. Therefore, large ${\displaystyle \Delta t}$ affects accuracy and defines time resolution. But, behavior may involve physical timescale that needs to be resolved.

This methods evaluates the function ${\displaystyle F(\varphi )}$ at a current time.

The evaluation using explicit-time integration is given as:

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

And is referred as explicit integration since ${\displaystyle \varphi ^{n+1}}$ can be expressed explicitly in the existing solution values, ${\displaystyle \varphi ^{n}}$:

$${\displaystyle \varphi ^{n+1}=\varphi ^{n}+\Delta t\,F(\varphi ^{n}),}$$

Here, the time step (${\displaystyle \Delta t}$) is restricted by the stability limit of the solver (i.e., time step is limited by the Courant–Friedrichs–Lewy condition). To be accurate with respect to time the same time step should be used in all the domain, and to be stable the time step must be the minimum of all the local time steps in the domain. This method is also referred to as "global time stepping".

Many schemes use explicit-time integration. Some of these are as follows:

• Lax–Wendroff method
• Runge–Kutta method