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Numerical methods

Discretization in time

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 (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

φ = a scalar quantity.
n + 1 = value at the next time level, t + Δt.
n = value at the current time level, t.
n − 1 = value at the previous time level, t − Δt.

The function F() is evaluated using implicit- and explicit-time integration.

 

Methods for evaluating function F()
After discretizing the time derivative, function F({\displaystyle \varphi }) remains to be evaluated. The function is now evaluated using implicit and explicit-time integration.

Implicit-time integration
This methods evaluates the function F(phi ) at a future time.

Explicit-time integration
This methods evaluates the function F(phi ) at the current time.

 

Semi-Implicit Method for Pressure Linked Equations

In computational fluid dynamics (CFD), the SIMPLE algorithm is a widely used numerical procedure to solve the Navier-Stokes equationsSIMPLE is an acronym for Semi-Implicit Method for Pressure Linked Equations.

The SIMPLE algorithm was developed by Prof. Brian Spalding and his student Suhas Patankar at Imperial College, London in the early 1970s. Since then it has been extensively used by many researchers to solve different kinds of fluid flow and heat transfer problems.

Many popular books on computational fluid dynamics discuss the SIMPLE algorithm in detail. A modified variant is the SIMPLER algorithm (SIMPLE Revised), that was introduced by Patankar in 1979.

The algorithm is iterative. The basic steps in the solution update are as follows:

 

Splitting Fluid PDE

 

Courant–Friedrichs–Lewy condition

The principle behind the condition is that, for example, if a wave is moving across a discrete spatial grid and we want to compute its amplitude at discrete time steps of equal duration, then this duration must be less than the time for the wave to travel to adjacent grid points. As a corollary, when the grid point separation is reduced, the upper limit for the time step also decreases. In essence, the numerical domain of dependence of any point in space and time (as determined by initial conditions and the parameters of the approximation scheme) must include the analytical domain of dependence (wherein the initial conditions have an effect on the exact value of the solution at that point) to assure that the scheme can access the information required to form the solution.

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