3.18 Second order time schemes

A Taylor’s series expansion between the current time eqn and ‘old’ time at eqn relates to the Euler implicit scheme by

 o 2 --------= @---- @---- --t+ O( t2)::: t @t @t2 2 \relax \special {t4ht=
The truncation error eqn, i.e. it is first order accurate in time when the time derivative relates to eqn at the current time, as shown in the figure below. Despite its low order, the Euler scheme is sufficiently accurate for many simulations since eqn is generally small when it corresponds to eqn.

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 eqn.

Backward scheme

In Eq. (3.25 ) we can replace eqn by the values eqn at ‘old-old’ time eqn. 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=
Substituting Eq. (3.26 ) into Eq. (3.25) gives the backward scheme which is second order accurate, using values at three time levels eqn, eqn and eqn:
@---- 3 ----4---o +---oo @t ! 2 t \relax \special {t4ht=

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 eqn, the Crank-Nicolson method solves

@ 1 1 ---- + --[A jb ] + --[A jb ]o o = 0; @t Euler 2 2 \relax \special {t4ht=
where eqn is calculated using old time values eqn.

A modern version of the scheme replaces the two eqn factors by eqn and eqn, introducing the ‘offset coefficient’ eqn, where eqn corresponds to Euler implicit and eqn is Crank-Nicolson Eq. (3.28 ). If eqn 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=
In practice, eqn 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