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 (
The spatial domain can be discretized to produce a semi-discrete form:[1]
The first-order temporal discretization using backward differences is [2]
is evaluated using implicit- and explicit-time integration.
[3] Temporal discretization is done by integrating the general discretized equation over time.
First, values at a given control volume
are assumed, and then value at time interval
This method states that the time integral of a given variable is a weighted average between current and future values.
The integral form of the equation can be written as:
This integration holds for any control volume and any discretized variable.
The following equation is obtained when applied to the governing equation, including full discretized diffusion, convection, and source terms.
After discretizing the time derivative, function
The function is now evaluated using implicit and explicit-time integration.
[5] This methods evaluates the function
The evaluation using implicit-time integration is given as:
This is called implicit integration as
In case of implicit method, the setup is unconditionally stable and can handle large time step (
But, stability doesn't mean accuracy.
affects accuracy and defines time resolution.
But, behavior may involve physical timescale that needs to be resolved.
This methods evaluates the function
The evaluation using explicit-time integration is given as:
And is referred as explicit integration since
can be expressed explicitly in the existing solution values,
) 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.