Notes on Computational Fluid Dynamics: General Principles

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3.18 Second order time schemes

A Taylor’s series expansion between the current time $\text{[math]}$ and ‘old’ time at $\text{[math]}$ relates to the Euler implicit scheme by

o 2 --------= @---- @---- --t+ O( t2)::: t @t @t2 2 \relax \special {t4ht=

(3.25)

The truncation error $\text{[math]}$ , i.e. it is ﬁrst order accurate in time when the time derivative relates to $\text{[math]}$ at the current time, as shown in the ﬁgure below. Despite its low order, the Euler scheme is suﬃciently accurate for many simulations since $\text{[math]}$ is generally small when it corresponds to $\text{[math]}$ .

Nevertheless, a second order time scheme may be required for simulations that demand higher temporal accuracy or to enable greater computational eﬃciency by running with larger $\text{[math]}$ .

Backward scheme

In Eq. (3.25 ) we can replace $\text{[math]}$ by the values $\text{[math]}$ at ‘old-old’ time $\text{[math]}$ . Subtracting the expression from Eq. (3.25 ) and rearranging terms gives the following relation for the second derivative

@2 2 o + oo --2- = --------2------+ O( t2)::: @t t \relax \special {t4ht=

(3.26)

Substituting Eq. (3.26 ) into Eq. (3.25) gives the backward scheme which is second order accurate, using values at three time levels $\text{[math]}$ , $\text{[math]}$ and $\text{[math]}$ :

@---- 3 ----4---o +---oo @t ! 2 t \relax \special {t4ht=

(3.27)

Crank-Nicolson scheme

$PICT\relax \special {t4ht=$

An implicit solution expresses the terms in an equation, e.g. advection, Laplacian, at the current time. The Crank-Nicolson method,¹³ expresses the terms at the midpoint between the current and old times, to make the Euler time scheme second order accurate. Denoting discretised terms except the time derivative by $\text{[math]}$ , the Crank-Nicolson method solves

@ 1 1 ---- + --[A jb ] + --[A jb ]o o = 0; @t Euler 2 2 \relax \special {t4ht=

(3.28)

where $\text{[math]}$ is calculated using old time values $\text{[math]}$ .

A modern version of the scheme replaces the two $\text{[math]}$ factors by $\text{[math]}$ and $\text{[math]}$ , introducing the ‘oﬀset coeﬃcient’ $\text{[math]}$ , where $\text{[math]}$ corresponds to Euler implicit and $\text{[math]}$ is Crank-Nicolson Eq. (3.28 ). If $\text{[math]}$ is discretised implicitly (as normal), the Crank-Nicolson scheme can be represented as a time derivative discretised by

@---- -------o o o @t ! (1 + ) t + [A jb] : \relax \special {t4ht=

(3.29)

In practice, $\text{[math]}$ is generally used to ensure solution stability.

¹³John Crank and Phyllis Nicolson, A practical method for numerical evaluation of solutions of partial diﬀerential equations of the heat conduction type, 1947.

Notes on CFD: General Principles - 3.18 Second order time schemes