Going Further

Going Further#

Crank-Nicolson#

The Crank-Nicolson method uses a time-centered discretization of the diffusion term:

\frac{ϕ_{i}^{n + 1} - ϕ_{i}^{n}}{Δ x} = \frac{k}{2} (\frac{ϕ_{i + 1}^{n} - 2 ϕ_{i}^{n} + ϕ_{i - 1}^{n}}{Δ x^{2}} + \frac{ϕ_{i + 1}^{n + 1} - 2 ϕ_{i}^{n + 1} + ϕ_{i - 1}^{n + 1}}{Δ x^{2}})

This is still an implicit update, but the coefficients and righthand side in the linear system will be different

State-dependent transport coefficients#

If we have a non-constant conductivity that depends on $ϕ$ itself, then our discretization is more complicated:

\frac{ϕ_{i}^{n + 1} - ϕ_{i}^{n}}{Δ t} = \frac{{k \nabla ϕ}_{i + 1 / 2} - {k \nabla ϕ}_{i - 1 / 2}}{Δ x}

Now we need $k$ at the interfaces. There are several ways to do this, which should be motivated by the physics of what the coefficient $k$ represents:

k_{i + 1 / 2} = \frac{1}{2} (k_{i} + k_{i + 1})

or

\frac{1}{k_{i + 1 / 2}} = \frac{1}{2} (\frac{1}{k_{i}} + \frac{1}{k_{i + 1}})

Then a C-N update appears as:

\frac{ϕ_{i}^{n + 1} - ϕ_{i}^{n}}{Δ t} = \frac{1}{2} {\nabla \cdot [k (ϕ^{n}) \nabla ϕ^{n}]_{i} + \nabla \cdot [k (ϕ^{n + 1}) \nabla ϕ^{n + 1}]_{i}}

We can update this using a predictor-corrector scheme:

\frac{ϕ_{i}^{⋆} - ϕ_{i}^{n}}{Δ t} = \frac{1}{2} {\nabla \cdot [k (ϕ^{n}) \nabla ϕ^{n}]_{i} + \nabla \cdot [k (ϕ^{n}) \nabla ϕ^{⋆}]_{i}}

\frac{ϕ_{i}^{n + 1} - ϕ_{i}^{n}}{Δ t} = \frac{1}{2} {\nabla \cdot [k (ϕ^{n}) \nabla ϕ^{n}]_{i} + \nabla \cdot [k (ϕ^{⋆}) \nabla ϕ^{n + 1}]_{i}}

This will be second-order accurate in time.

Going Further

Contents

Going Further#

Crank-Nicolson#

State-dependent transport coefficients#