Temporal discretization is a mathematical technique applied to
transient problems that occur in the fields of applied physics and engineering.
Transient problems are often solved by conducting simulations using
computer-aided engineering
Computer-aided engineering (CAE) is the broad usage of computer software to aid in engineering analysis tasks. It includes , , , durability and optimization. It is included with computer-aided design (CAD) and computer-aided manufacturing (CAM) ...
(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
Integration may refer to:
Biology
*Multisensory integration
*Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome into a host genome
*DNA integration, by means of site-specific recombinase technology, ...
of every term in different equations over a time step (
).
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:
[Selection of Spatial and Temporal discretization](_blank)
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And the second-order discretization
In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical ...
is given as:
where
* is a scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
quantity.
* is the value at the next time level, .
* is the value at the current time level, .
* is the value at the previous time level, .
The function is evaluated using implicit- and explicit-time integration.
Description
The temporal discretization is done through integration
Integration may refer to:
Biology
*Multisensory integration
*Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome into a host genome
*DNA integration, by means of site-specific recombinase technology, ...
over time on the general discretized equation. 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 equal to a weighted average between current and future values. The integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
form of the equation can be written as:
where is a weight between 0 and 1.
* results in the fully explicit scheme.
* results in the fully implicit scheme.
* results in the Crank-Nicolson scheme.
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
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
, convection
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convec ...
, and source
Source may refer to:
Research
* Historical document
* Historical source
* Source (intelligence) or sub source, typically a confidential provider of non open-source intelligence
* Source (journalism), a person, publication, publishing institute o ...
terms.
Methods for evaluating function ''F''(''φ'')
After discretizing the time derivative, function remains to be evaluated. The function is now evaluated using implicit and explicit-time integration.[Jirka Simunek](_blank)
/ref>
Implicit-time integration
This methods evaluates the function at a future time.
Formulation
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. But, behavior may involve physical timescale that needs to be resolved.
Explicit-time integration
This methods evaluates the function at a current time.
Formulation
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, :
Here, the time step () is restricted by the stability limit of the solver (i.e., time step is limited by the Courant–Friedrichs–Lewy condition In mathematics, the convergence condition by Courant–Friedrichs–Lewy is a necessary condition for convergence while solving certain partial differential equations (usually hyperbolic PDEs) numerically. It arises in the numerical analysis of ex ...
). 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".
Examples
Many schemes use explicit-time integration. Some of these are as follows:
* Lax–Wendroff method The Lax–Wendroff method, named after Peter Lax and Burton Wendroff, is a numerical method for the solution of hyperbolic partial differential equations, based on finite difference
A finite difference is a mathematical expression of the form . ...
* Runge–Kutta method
See also
* Courant–Friedrichs–Lewy condition In mathematics, the convergence condition by Courant–Friedrichs–Lewy is a necessary condition for convergence while solving certain partial differential equations (usually hyperbolic PDEs) numerically. It arises in the numerical analysis of ex ...
.
* Von Neumann stability analysis
The term ''von'' () is used in German language surnames either as a nobiliary particle indicating a noble patrilineality, or as a simple preposition used by commoners that means ''of'' or ''from''.
Nobility directories like the ''Almanach de Go ...
.
* Finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
* Explicit and implicit methods
Explicit and implicit methods are approaches used in numerical analysis for obtaining numerical approximations to the solutions of time-dependent ordinary and partial differential equations, as is required in computer simulations of physical pro ...
* Chi-Wang Shu
References
{{reflist
Computer-aided engineering