4.3.1 Time Step Size Prediction

In MINIMOS-NT a quadratic extrapolation of the norm of the potential update is used to estimate the size of the next time step $\Delta$$\tilde{t}$. The last two step sizes $\Delta$t_{n - 1} and $\Delta$t_n and the last two potential update norms u_{n - 1} and u_n are used to calculate the coefficients of a quadratic extrapolation polynomial. From this function the step size $\Delta$$\tilde{t}$ corresponding to the maximum update norm u_max is determined (see Fig. 4.6). These calculations are performed using the L² norm and the L^$\infty$ norm of the potential update. The minimum of the results is used for further checks.

$$\Delta \tilde{t} \Delta \tilde{t}_{\mathrm{L^2}}^{} \Delta \tilde{t}_{\mathrm{L^\infty}}^{}$$ (4.3)

Then the ratio of the estimated step size $\Delta$$\tilde{t}$ and the previous step size $\Delta$t_n is limited to the sectio aurea constant ( ${\frac{1 + \sqrt{5}}{2}}$ = 1.618...), to limit the step size variations. With this restriction a quasi-uniform mesh is achieved, which gives a second order local truncation error.

$$\Delta \tilde{t} \leftarrow \left(\vphantom{\frac{\Delta\tilde{t}}{\Delta t_{n}},\frac{1 + \sqrt{5}}{2}}\right. {\frac{\Delta\tilde{t}}{\Delta t_{n}}} {\frac{1 + \sqrt{5}}{2}} \left.\vphantom{\frac{\Delta\tilde{t}}{\Delta t_{n}},\frac{1 + \sqrt{5}}{2}}\right)$$ (4.4)

Figure 4.6: Quadratic step size prediction.

In order to follow the pointwise predefined input signals accurately it is necessary to calculate a time step at the instances used to specify the input signals. Therefore the step size has to be reduced if it is larger than the difference of the instance t_k and the present time t_n (see Fig. 4.7).

$$\Delta \tilde{t} \leftarrow \left(\vphantom{\Delta\tilde{t}, t_{k} - t_{n}}\right. \Delta \tilde{t} \left.\vphantom{\Delta\tilde{t}, t_{k} - t_{n}}\right)$$ (4.5)

To avoid strong step size variations when reaching the next instance t_k a check is performed whether double the estimated step size is greater than the difference of the instance t_k and the actual time t_n. In this case the next two step sizes $\Delta$$\tilde{t}$ and $\Delta$$\hat{t}$ are chosen in such a way that the second step matches the instance and the step size ratio is the sectio aurea constant (see Fig. 4.8).

$$\Delta \tilde{t} \Delta \hat{t}$$ (4.6)

$${\frac{\Delta\hat{t}}{\Delta\tilde{t}}} {\frac{1 + \sqrt{5}}{2}}$$ (4.7)

Figure 4.7: To accurately follow the predefined input signal the step size has to be reduced to match the instance t_k.
Figure 4.8: When the double estimated step size is greater than the difference between the next instance t_k and the current time t_n the step size has to be reduced to avoid strong step size variations.

Finally there is a check whether the estimated step size $\Delta$$\tilde{t}$ is greater than a specified minimum step size $\Delta$t_min and less than a specified maximum step size $\Delta$t_max.

$$\Delta \tilde{t} \leftarrow \Delta \Delta \tilde{t}$$ (4.8)

$$\Delta \tilde{t} \leftarrow \Delta \tilde{t} \Delta$$ (4.9)