Index-aware model order reduction for differential-algebraic equations

Figures & data

Figure 1. Frequency response.

Figure 2. Comparison of the output solutions, $u(t)=10sin(t),t\in \left[0,2\pi \right]$.

Figure 3. Approximation error.

Table 1. Dimension of IMOR model.

Decoupled system		Reduced Model
# differential equations	# algebraic equations	# differential equations	# algebraic equations
1257	10,428	400	401

Figure 4. Sparsity of matrix pencil $(E,A)$.

Figure 5. Sparsity of matrix pencil $(\tilde{E},\tilde{A})$.

Figure 6. Sparsity of matrix pencil $({\tilde{E}}_r,{\tilde{A}}_r)$ of IMOR model.

Figure 7. Sparsity of matrix pencil $(E_r,A_r)$ of PRIMA model.

Table 2. Computational time (in seconds).

	Original model		Reduced model
Relative tolerance	DAE	Decoupled system	PRIMA model	IMOR model
$RelTol={10}^{-6}$	−	14	30	5
$RelTol={10}^{-3}$	−	12	25	4

Figure 8. Frequency response and its error: (a) Frequency response. (b) frequency response error.

Figure 9. Output Solutions, input function $u(t)=sin(t),t\in \left[0,\pi \right]$. (a) Output solution $y(t)$. (b) approximation error.

Table 3. Dimension of IMOR model.

Decoupled system		Reduced model
# differential equations	# algebraic equations	# differential equations	# algebraic equations
606	6529	260	264

Figure 10. Sparsity of matrix pencil $(E,A)$.

Figure 11. Sparsity of matrix pencil $(\tilde{E},\tilde{A})$.

Figure 12. Sparsity of matrix pencil $({\tilde{E}}_r,{\tilde{A}}_r)$ of IMOR model.

Figure 13. Sparsity of matrix pencil $(E_r,A_r)$ of PRIMA model.

Figure 14. Frequency response and its error. (a) Frequency response (b) frequency response error.

Figure 15. Output solutions of the index-aware model and original model. (a) $y_1(t)$, (b) $y_2(t)$, (c) $y_3(t)$, (d) $y_4(t)$.

Figure 16. Approximation error.

Table 4. Computational time (in seconds).

	Original model		Reduced model
Relative tolerance	DAE	Decoupled system	PRIMA model	Index-aware model
$RelTol={10}^{-6}$	−	19.5	−	2.4
$RelTol={10}^{-3}$	−	14.3	−	1.9

Figure 17. Transfer function from bleed actuation to average throat Mach number for supersonic diffuser. (a) Frequency response, (b) phase lag.

Figure 18. Approximation error.

Figure 19. Frequency response and its error. (a) Frequency response, (b) phase response.

Figure 20. Approximation error.