At each time step, the transient solver solves

0 = \boldsymbol{f}_0 = a_0 \boldsymbol{q}_0 + a_{-1} \boldsymbol{q}_{-1} + a_{-2} \boldsymbol{q}_{-2} + b_0 \boldsymbol{i}_0 + b_{-1} \boldsymbol{i}_{-1} + b_{-2} \boldsymbol{i}_{-2}

where \boldsymbol{f}_0 is the vector of net flux at each node in the mesh. The \boldsymbol{i}_x represents the time independent part of the semiconductor equations and \boldsymbol{q}_x represents the time derivatives terms. The subscript 0 denotes the current time step being solved. The subscripts -1 and -2 denote the previous two time steps.

At the beginning of each time step the components are copied in order so that:

\boldsymbol{i}_{-2} &= \boldsymbol{i}_{-1}

\boldsymbol{i}_{-1} &= \boldsymbol{i}_{0}

\boldsymbol{q}_{-2} &= \boldsymbol{q}_{-1}

\boldsymbol{q}_{-1} &= \boldsymbol{q}_{0}

This is a steady state solution with:

a_0 &= 1

a_1 &= 0

b_0 &= 1

b_1 &= 0

b_2 &= 0

and represent the DC steady state. This step can be used to initialize the initial time step so that the other transient methods can begin.

t_\Delta &= \gamma t_{step}

t_f &= \frac{1}{t_\Delta}

a_0 &= t_f

a_1 &= -t_f

b_0 &= 1

b_1 &= 0

b_2 &= 0

\gamma &= 1

t_{\Delta} &= (1 - \gamma) t_{step}

a_0       &= \frac{2 - \gamma}{t_{\Delta}}

a_1       &= \frac{-1}{\gamma t_{\Delta}}

a_2       &= \frac{1 - \gamma}{\gamma t_{step}}

b_0       &= 1

b_1       &= 0

b_2       &= 0

\gamma    &= 0.5

t_{\Delta} &= \gamma t_{step}

t_{f}     &= \frac{2}{t_{\Delta}}

a_0       &= t_{f}

a_1       &= -t_{f}

b_0       &= 1

b_1       &= 1

b_2       &= 0

Combination of 2 methods described in :cite:`bank1270142`.

\gamma &= 2 - \sqrt{2}

and use TR followed by BDF2

Calculate \boldsymbol{q}_0 as part of the solution process. Then compare with:

0 = i_1 + \frac{q_{proj} - q_1}{t_{\Delta}}

q_{proj} = - i_1 t_{\Delta} + q_1

Calculate error between projection and actual charge solution

Every time integration method, except for the Transient DC method in :numref:`sec_transientdc` requires a solution at a previous time step.

As an alternative the :meth:`devsim.set_initial_condition` command can be used to set \boldsymbol{i}_0 and \boldsymbol{q}_0. These vectors can be calculated from :meth:`devsim.get_matrix_and_rhs`, which makes it possible to initialize previous time steps from equations which were not solved self consistently.

For each transient time step, :eq:`transient_solve_eq` is solved. The previous time step data is shifted

\boldsymbol{i}_{-2} &= \boldsymbol{i}_{-1}

\boldsymbol{i}_{-1} &= \boldsymbol{i}_{0}

\boldsymbol{q}_{-2} &= \boldsymbol{q}_{-1}

\boldsymbol{q}_{-1} &= \boldsymbol{q}_{0}

The Newton method iterates to find a solution for and stores new values for \boldsymbol{i}_0 and \boldsymbol{q}_0.