Error Estimation Approach for Controlling the Communication Step-Size for Explicit Co-simulation Methods

Abstract

In this paper, an approach for controlling the communication-step size in connection with explicit co-simulation methods is suggested. In the framework of the proposed communication-step size controller, each subsystem integration is carried out with two different explicit co-simulation methods. By comparing the variables for both integrations, an error estimator for the local error can be constructed. Making use of the estimated local error, a step-size controller for the communication step-size can be implemented. Examples are presented demonstrating the applicability and accuracy of the proposed communication-step size controller.

Appendix: Alternative Co-simulation Approach with Improved Convergence Order

In Sect. 11.5.2, we have seen, that the local error of the position variables converges with order $k+3$. However, the velocity variables converge only with order $k+2$, which causes that the global error converges only with order $k+1$. Combining the two methods described in Sect. 11.4 yields an alternative method (method 3), which may increase the convergence order of the local error of the velocity variables. Thus, the local error may converge with order $k+3$ and consequently the global error may converge with order $k+2$. We define the new polynomial

$$\begin{aligned} \mathbf {p}_{N+1}^\alpha \!\left( t\right)= & {} \alpha \mathbf {p}_{N+1}^{\mathrm {int}}\!\left( t\right) + \left( 1- \alpha \right) \mathbf {p}_{N+1}^{\mathrm {ext}}\!\left( t\right) , \end{aligned}$$

(11.55)

where $\alpha$ is an arbitrary real number. For analyzing the consistency, we define

$$\begin{aligned} \mathbf {y}_{N+1}^\alpha:= & {} \mathbf {y}\!\left( T_N\right) + \int \limits _{T_N}^{T_{N+1}} \mathbf {F}^{\mathrm {co}}\!\left( t,\mathbf {y}\!\left( t\right) ,\mathbf {p}_{N+1}^\alpha \!\left( t\right) \right) \mathrm dt\, . \end{aligned}$$

We have

$$\begin{aligned} \begin{array}{rcl} \mathbf {y}_{N+1}^\alpha -\mathbf {y}\!\left( T_{N+1}\right) &{}=&{} \int \limits _{T_N}^{T_{N+1}}\mathbf {F}^{\mathrm {co}}\!\left( t,\mathbf {y}\!\left( t\right) ,\mathbf {p}_{N+1}^\alpha \!\left( t\right) \right) - \mathbf {F}^{\mathrm {co}}\!\left( t,\mathbf {y}\!\left( t\right) ,\mathbf {p}\!\left( t\right) \right) \mathrm dt\\ &{}=&{} \int \limits _{T_N}^{T_{N+1}}\mathbf {F}_{\mathbf {p}}^{\mathrm {co}}\!\left( t,\mathbf {y}\!\left( t\right) ,\mathbf {p}\!\left( t\right) \right) \left( \mathbf {p}_{N+1}^\alpha \!\left( t\right) - \mathbf {p}\!\left( t\right) \right) \mathrm dt+\mathcal O\!\left( H^{k+3}\right) \\ &{}=&{}\mathbf {F}_{\mathbf {p}}^{\mathrm {co}}\!\left( T_N,\mathbf {y}_N,\mathbf {u}_N\right) \int \limits _{T_N}^{T_{N+1}} \mathbf {p}_{N+1}^\alpha \!\left( t\right) - \mathbf {p}\!\left( t\right) \mathrm dt+\mathcal O\!\left( H^{k+3}\right) \\ &{}=&{}\mathbf {F}_{\mathbf {p}}^{\mathrm {co}}\!\left( T_N,\mathbf {y}_N,\mathbf {u}_N\right) \int \limits _{T_N}^{T_{N+1}} \mathbf {p}_{N+1}^\alpha \!\left( t\right) - \hat{\mathbf {p}}\!\left( t\right) \mathrm dt+\mathcal O\!\left( H^{k+3}\right) . \end{array} \end{aligned}$$

(11.56)

With the constants

$$\begin{aligned} D^i_{N+1} :=\int _{T_N}^{T_{N+1}}L^i_{N+1}\!\left( t\right) \mathrm dt \qquad \!\left( i\in \left\{ k,k+1\right\} \right) \end{aligned}$$

(11.57)

and with Eqs. (11.41) and (11.42), we obtain

$$\begin{aligned} \int \limits _{T_N}^{T_{N+1}} \mathbf {p}_{N+1}^\alpha \!\left( t\right) - \hat{\mathbf {p}}\!\left( t\right) \mathrm dt= & {} D^{k+1}_{N+1}- \alpha D^k_{N+1}\, , \end{aligned}$$

(11.58)

which vanishes for $\alpha = \frac{D^{k+1}_{N+1}}{D^k_{N+1}}$. Together with Eq. (11.56), we get

$$\begin{aligned} \frac{1}{H}\left\| \mathbf {y}_{N+1}^\alpha -\mathbf {y}\!\left( T_{N+1}\right) \right\|= & {} \mathcal O\!\left( H^{k+2}\right) . \end{aligned}$$

(11.59)

Fig. 11.15

Convergence of the local error of method 3 on position and velocity level

Fig. 11.16

Convergence of the global error of method 3 on position and velocity level

Figure 11.15 shows the convergence of the local error on position and velocity level for method 3. The convergence plots confirm that the local error converges with order $k+3$ on position and on velocity level. As expected, the global error converges with order $k+2$, which can be seen in Fig. 11.16. To derive an error estimator, it seems straightforward to use method 1 as reference. Then, the error is estimated by

$$\begin{aligned} \hat{\varepsilon }_{N+1}^\alpha :=\frac{1}{\alpha }\!\left( 1-\frac{C^{k+1}_{N+1}}{C^k_{N+1}}\right) \left\| \mathbf {q}_{N+1}^\alpha -\mathbf {q}_{N+1}^{\mathrm {ext}}\right\| , \end{aligned}$$

(11.60)

where $\mathbf {q}_{N+1}^\alpha$ is the numerical solution of the position variables of the simulations carried out with the polynomials $\mathbf {p}_{N+1}^\alpha \!\left( t\right)$. Unfortunately, the local error is too badly estimated, if the predicted values $\mathbf {u}_{N+1}^{\mathrm {pre}}$ (see Sect. 11.4.2) are computed with only $k+2$ sampling points. A similar problem was already mentioned in Remark 11.2 (Sect. 11.7.2). Figure 11.17 shows the local error and the estimated local error. The simulations have been carried out with quadratic approximation polynomials ($k=2$). The predicted coupling values are extrapolated with four supporting points.

Fig. 11.17

Comparison of the estimated local error to the real local error of method 3 carried out with quadratic approximation polynomials ($k=2$) and predicted values $\mathbf {u}_{N+1}^{\mathrm {pre}}$ calculated with 4 sampling points; reference solution is carried out with method 1

To improve the error estimator, method 3 is modified by increasing the extrapolation order for calculating the predicted coupling values. As can be seen in Fig. 11.18, the error estimator is improved, if the predicted values are extrapolated by five sampling point.

Fig. 11.18

Comparison of the estimated local error to the real local error of method 3 carried out with quadratic approximation polynomials ($k=2$) and predicted values $\mathbf {u}_{N+1}^{\mathrm {pre}}$ calculated with 5 sampling points; reference solution is carried out with method 1