TR/BDF2

  Although backward Euler has properties appropriate for robustly integrating the stiff systems of ODEs arising from semiconductor diffusion models, it is only first-order accurate. The TR/BDF2 method, discussed in [1], provides a second order accurate A-stable and L-stable single step method comprised of a trapezoid step over part of the time-step followed by a second order backward difference step over the remainder of the time step.

Let and be the computed solution at . The TR substep goes from to via

 

The BDF2 substep goes from to via

 

The constant multiplying the truncation error is minimized if . This value of gamma also produces identical jacobians for both the TR and BDF2 steps.

The local truncation error (LTE) can be used to estimate a reasonable next time step. One approach is to use the divided difference formula

 

is used to estimate the LTE. Note that C is the constant multiplying the highest order truncation term in the method

The LTE is compared to an allowable error tolerance to produce a RMS measure of the normalized pointwise error. Let

and

Then, the RMS measure of the normalized pointwise error is given by

 

A candidate next time step is chosen as

If r is reasonably small (say less than 2), the error in the timestep is assumed to be acceptable and the next time step is chosen to be . The 2 times cap on increases is to avoid stepsize oscillations. If r is too large, the current timestep is repeated with .

A second timestep estimator is based on a LTE estimate computed using a Milne's device. A second TR step is taken from to . The solutions of the BDF2 step and the second TR step are compared and used to estimate the truncation error. See [4] for more details.