Crank-Nicolson (CN)

Crank-Nicolson is an implicit time marching scheme. While it adds a measure of complexity, it is unconditionally stable. Note that this does not mean that accuracy is unaffected by time step size, degree of the operator (p = 1 or p = 2, etc.), or element size.

Selecting CN & Problem Setup

CN is selected with "run_type" => 20 in your problem's input options file. As this is an unsteady problem, you will also need to specify your time step size ("delta_t" option) and your final solution time ("t_max" option)

Forward-in-time

\begin{aligned} \underbrace{\left( \mathbf{I} - \frac{\Delta t}{2} \underbrace{\frac{\partial R\left(u^{i+1}\right)}{\partial u^{i+1}}}_{\text{jac from physicsJac}}
\right)}_{\text{cnJac}} \Delta u
= \ \underbrace{- \left(
\underbrace{u^{i+1}}_{\text{eqn_nextstep.q_vec}} - \ \frac{\Delta t}{2} \underbrace{R\left(u^{i+1}\right)}_{\text{eqn_nextstep.res_vec}} - \ \underbrace{u^i}_{\text{eqn.q_vec}} - \ \frac{\Delta t}{2} \underbrace{R\left(u^i\right)}_{\text{eqn.res_vec}}
\right)}_{\text{cnRhs}} \end{aligned}

This equation is solved with PDESolver's Newton's method.