Temporal discretization
In the fields of applied physics and engineering, transient problems are often solved by conducting simulations using computer-aided engineering (CAE) packages, which require discretizing the governing equations in both space and time. Such problems are unsteady (e.g. flow problems), and therefore require solutions in which position varies as a function of time. Temporal discretization involves the integration of every term in different equations over a time step (Δt). The spatial domain can be discretized to produce a semi-discrete form:
If the discretization is done using backward differences; the first-order temporal discretization is given as:
And the second-order discretization is given as:
where
The function F() is evaluated using implicit- and explicit-time integration.
The temporal discretization is done through integration over time on the general discretized equation. First, values at a given control volume P at time interval t are assumed and then value at time interval t+Δt is found. This method states that the time integral of a given variable is equal to a weighted average between current and future values. The integral form of the equation can be written as:
where ƒ is a weight between 0 and 1.
For any control volume this integration holds true for 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 F() remains to be evaluated. The function is now evaluated using implicit and explicit-time integration.
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