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Concept
Temporal discretization
Résumé
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:
The first-order temporal discretization using backward differences is
And the second-order discretization is
where
is a scalar
is the value at the next time,
is the value at the current time,
is the value at the previous time,
The function is evaluated using implicit- and explicit-time integration.
Temporal discretization is done by integrating the general discretized equation over time. First, values at a given control volume at time interval are assumed, and then value at time interval is found. 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:
where is a weight between 0 and 1.
yields the fully explicit scheme.
yields the fully implicit scheme.
yields the Crank-Nicolson scheme.
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 remains to be evaluated. The function is now evaluated using implicit and explicit-time integration.
This methods evaluates the function at a future time.
The evaluation using implicit-time integration is given as:
This is called implicit integration as in a given cell is related to in neighboring cells through :
In case of implicit method, the setup is unconditionally stable and can handle large time step (). But, stability doesn't mean accuracy. Therefore, large affects accuracy and defines time resolution.
Source officielle
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