Abstract

CHAPTER 2. TEMPORAL INTEGRATION 29
and the time-derivate G(y, t), and they return an approximate value of y at time t ∗ .
ROCK4 is an explicit fourth-order Runge-Kutta method [1]. ROCK4 uses variable
stages in its computations to create a stability region that contains a large portion of
the negative real axis. This stability region allows this method to solve stiff problems,
which are generally a challenge for an explicit method. Due to their stability regions,
implicit methods are used to handle stiff problems, but their computational cost per
time step is much higher than an explicit method. While the explicit time step is
cheaper to compute, the implicit methods are more stable than the explicit methods,
and therefore are allowed to take larger time steps. So if we can efficiently solve the
nonlinear equations that arise from the implicit method, it could easily be the faster
method.
VODEPK is an implicit ODE solver [6], [7]. VODEPK allows the user to se-
lect one of two numerical integration methods. The first is a BDF method [30],
VODEPK’s stiff method. BDF methods approximate the derivative of the solution
with the derivative of the interpolation polynomial constructed with previous com-
puted values of the solution. The second method is an implicit Adams method [30],
VODEPK’s non-stiff method. This method also uses an interpolation polynomial.
The polynomial, though, does not interpolate the solution, but it is an approxima-
tion of the time-derivative. The method then integrates the polynomial instead of the
actual time-derivative to get the approximated solution at the next time step. The