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Automatic time stepping

numgeo uses an automatic time stepping scheme to advance the solution from time \(t\) to \(t + \Delta t\). The size of the time increment \(\Delta t\) is not constant, but can increase or decrease during the calculation depending on how well or poorly the solution converges. Details on this approach are given in the two following sections.

Automatic time stepping

  • If a solution converges in less than four iterations (\(n_i < 7\)), the time increment is increased by 25 %. If a solution converges in more than eight iterations (\(n_i > 12\)), the time increment is reduced by 25 %.

    \[ \Delta t^{i+1} = \begin{cases} 1.25 \cdot \Delta t^i & \mbox{if:} ~~ n_i < 7 \\ 0.75 \cdot \Delta t^i & \mbox{if:} ~~ n_i > 12 \\ \end{cases} \]

  • If a solution does not converge at a given increment \(i\), the analysis is brought back to the previous (converged) increment \(i-1\), the time increment is reduced by half (\(\Delta t^{i}_{new} = 0.5 \cdot \Delta t^i\)) and a new attempt for convergence of increment \(i\) is started. This procedure is repeated until convergence is achieved or the time increment is smaller than the minimum time increment specified (\(\Delta t^{i}_{new} < \Delta t^{min}\)).

    If such a measure has been taken, this is indicated in the statistics file (*.sta) with the addition <--- cutback in the corresponding iteration. Corresponding information is also provided in the message file (.msg).

Maximum solution increment

After a successful Newton iteration, the new time step for the next increment is computed. Per default, numgeo increases the time step size if convergence was found within a maximum number of iterations, or decreases the time step size if the number of required iterations exceeds some threshold value.

For transient problems, it might be beneficial to prescribe a tolerance controling the level of accuracy required in the approximate time integration of transient effects that have a physical time scale. In numgeo this can be achieved by considering the solution increment \(\Delta_{dof}\) in the calculation of the next time step size:

\[ \Delta t = \max\left(s^{max},\left|\dfrac{\tilde{\Delta}_{dof}}{\Delta_{dof}}\right|\right) \Delta t_0 \]

Therein, \(s^{max}\) is a maximum scaling parameter limiting the increase of the time step size. \(\tilde{\Delta}_{dof}\) is a user-specified target increment size an \(\Delta_{dof}\) the current solution increment of the degree of freedom (<dof>) to be controlled.

Currently this option is only available for the pore water pressure (\(p^w\),pw) degree of freedom, see Reference Manual.

Prescribing a maximum solution increment will override the default automatic time stepping as layed out in above section Automatic time stepping