Skip to main content

Euler implicit time scheme

Properties​

  • Implicit
  • First order in time
  • Transient
  • Unconditionally stable for linear problems

Discretisation​

The Euler implicit scheme evaluates all terms at the new time level. For the temporal derivative alone, on a static mesh with uniform time step Δt\Delta t, the rate is represented as:

∂∂t(ϕ)≈ϕ−ϕoΔt \ddt{\phi} \approx \frac{\phi - \old{\phi}}{\Delta t}

Unlike Crank-Nicolson, no auxiliary field is stored; the discretisation uses only the current and old-time field values.

Implicit and explicit contributions​

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

ContributionDiscrete form
Implicit (matrix diagonal)1Δt V ϕ\displaystyle\frac{1}{\Delta t}\, V\,\phi
Explicit (source)1Δt ϕo V\displaystyle\frac{1}{\Delta t}\,\old{\phi}\, V

where VV 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         Euler;
    ddt(phi)        Euler;
}

Euler implicit is the most robust first-order scheme and forms the basis for Crank-Nicolson (with off-centring coefficient α=0\alpha = 0) and the start-up behaviour of higher-order schemes. For second-order accuracy in time, consider backward differencing or Crank-Nicolson instead.

Further information​

Source code