3.6.1 Time Step Adaption



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3.6.1 Time Step Adaption

 

The solution of parabolic PDEs (3.6-1) requires for time integration a suitable time step size . The selection of an appropriate is closely connected to the method applied for time integration, in our case the backward Euler method, and the according error estimation. Here, we will discuss three details of the time step selection: (1) The estimation of an initial time step, (2) the time step adaption and (3) some considerations for the last time step.

 

For the estimation of the initial time step we permit a local change (3.6-2) of the concentration of a few percent (3.6-3). Local means, we estimate the change for each quantity for each grid point .

 

 

If the local calculation (3.6-3) is not successful, we try a global estimation (3.6-4) and if this also fails, we take a heuristic value .

 

The time step adaption is based on an accuracy estimation. Let us consider the concentration at one specific grid point with respect to time, then we get (3.6-6) from Taylor series expansion (3.6-5).

 

 

From the concentrations at time steps , and we estimate the second derivative using divided differences (see Section 3.4.1). Three criteria are evaluated: (1) The local truncation error of the time discretization (3.6-7), (2) the influence of higher order terms on the concentration (3.6-8) and (3) a comparison of the scaled higher order terms with the Newton accuracy (3.6-9). The weakest of these criteria determines the accuracy .

 

 

 

 

The time step has to be rejected if the accuracy is too small (typically ). For accepted time steps, the new time step size is predicted by (3.6-11). We have introduced this nonlinear damping function to avoid oscillations of the time step size and to limit the time step increase to a factor (typically ).

 

This estimated time step size is truncated to a given minimum time step and is adjusted to the end time, if necessary (3.6-12) (see Figure 3.6-1).

 

 



next up previous contents
Next: 3.6.2 Adaption of the Up: 3.6 Transient Integration Algorithm Previous: 3.6 Transient Integration Algorithm



Martin Stiftinger
Wed Oct 19 13:03:34 MET 1994