2.5 Temporal discretisation

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2.5 Temporal discretisation

Although we have described the discretisation of temporal derivatives in Sections 2.4.3 and 2.4.4, we need to consider how to treat the spatial derivatives in a transient problem. If we denote all the spatial terms as $Af \special {t4ht=$ where $A \special {t4ht=$ is any spatial operator, e.g. Laplacian, then we can express a transient PDE in integral form as

integral t+Dt[ integral integral ] @-- rf dV + Af dV dt = 0 t @t V V \special {t4ht=

(2.30)

Using the Euler implicit method of Equation 2.21, the first term can be expressed as

integral t+Dt [@ integral ] integral t+Dt(rP fP V)n - (rPfP V )o --- rf dV dt = ----------------------- dt t @t V t Dt (rPfP-V-)n---(rP-fP-V)o- = Dt Dt \special {t4ht=

(2.31)

The second term can be expressed as

integral t+Dt [ integral ] integral t+Dt Af dV dt = A*f dt t V t \special {t4ht=

(2.32)

where $A* \special {t4ht=$ represents the spatial discretisation of $A \special {t4ht=$ . The time integral can be discretised in three ways:

Euler implicit

uses implicit discretisation of the spatial terms, thereby taking current values $n f \special {t4ht=$ .

integral t+Dt A*f dt = A*fnDt t \special {t4ht=

(2.33)

It is first order accurate in time, guarantees boundedness and is unconditionally stable.

Explicit

uses explicit discretisation of the spatial terms, thereby taking old values $o f \special {t4ht=$ .

integral t+Dt A*f dt = A*foDt t \special {t4ht=

(2.34)

It is first order accurate in time and is unstable if the Courant number $Co \special {t4ht=$ is greater than 1. The Courant number is defined as

U •d Co = ---f--- |d|2Dt \special {t4ht=

(2.35)

where $Uf \special {t4ht=$ is a characteristic velocity, e.g. velocity of a wave front, velocity of flow.

Crank Nicholson

uses the trapezoid rule to discretise the spatial terms, thereby taking a mean of current values $n f \special {t4ht=$ and old values $o f \special {t4ht=$ .

integral t+Dt (fn + fo ) A*f dt = A* -------- Dt t 2 \special {t4ht=

(2.36)

It is second order accurate in time, is unconditionally stable but does not guarantee boundedness.

2.5.1 Treatment of temporal discretisation in OpenFOAM

At present the treatment of the temporal discretisation is controlled by the implementation of the spatial derivatives in the PDE we wish to solve. For example, let us say we wish to solve a transient diffusion equation

@f- = k\ ~/ 2f @t \special {t4ht=

(2.37)

An Euler implicit implementation of this would read

solve(fvm::ddt(phi) == kappa*fvm::laplacian(phi))

where we use the fvm class to discretise the Laplacian term implicitly. An explicit implementation would read

solve(fvm::ddt(phi) == kappa*fvc::laplacian(phi))

where we now use the fvc class to discretise the Laplacian term explicitly. The Crank Nicholson scheme can be implemented by the mean of implicit and explicit terms:

    solve
        (
        fvm::ddt(phi)
        ==
        kappa*0.5*(fvm::laplacian(phi) + fvc::laplacian(phi))
        )

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