Backward time scheme

Properties

• Implicit
• Second order in time
• Transient
• Boundedness not guaranteed
• Conditionally stable

Discretisation

The backward scheme uses second-order backward differencing in time, requiring the current and two previous time-level field values. On a static mesh with uniform time step $\Delta t$, the temporal rate is represented as:

$$\ddt{\phi} \approx \frac{1}{\Delta t} \left(\frac{3}{2}\phi - 2\old{\phi} + \frac{1}{2}\oldold{\phi}\right)$$

For non-uniform time steps, OpenFOAM uses the general coefficients $\mathrm{coeff}_t$, $\mathrm{coeff}_{t0}$ and $\mathrm{coeff}_{t00}$ derived from $\Delta t$ and the previous step size $\Delta t^{o}$:

$$\ddt{\phi} \approx \frac{1}{\Delta t} \left( \mathrm{coeff}_t\,\phi - \mathrm{coeff}_{t0}\,\old{\phi} + \mathrm{coeff}_{t00}\,\oldold{\phi} \right)$$

where

$$\mathrm{coeff}_t = 1 + \frac{\Delta t}{\Delta t + \Delta t^{o}}, \quad \mathrm{coeff}_{t00} = \frac{\Delta t^{2}}{\Delta t^{o}\left(\Delta t + \Delta t^{o}\right)}, \quad \mathrm{coeff}_{t0} = \mathrm{coeff}_t + \mathrm{coeff}_{t00}$$

On the first time step the scheme reduces to Euler implicit until two previous time levels are available.

Implicit and explicit contributions

When fvm::ddt(phi) is used, the temporal term contributes to the linear system for $\phi$ as:

Contribution	Discrete form
Implicit (matrix diagonal)	$\displaystyle\frac{\mathrm{coeff}_t}{\Delta t}\, V\,\phi$
Explicit (source)	$\displaystyle\frac{1}{\Delta t}\left(\mathrm{coeff}_{t0}\,\old{\phi} - \mathrm{coeff}_{t00}\,\oldold{\phi}\right) V$

where $V$ is the cell volume. The corresponding explicit evaluation from fvc::ddt(phi) is the difference between these two parts.

Usage

The scheme is specified using:

ddtSchemes
{
    default         backward;
    ddt(phi)        backward;
}

Compared with Crank-Nicolson, the backward scheme is generally more robust and provides formal second-order accuracy in time, but requires old-old-time field values in the matrix assembly and does not offer the same opportunity for flux limiting at the mid-point of the time step.

Further information

Source code

• Foam::fv::backwardDdtScheme