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doi: 10.3934/mcrf.2021054
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## Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation

 Department of Mathematics, TU Dortmund University, Vogelpothsweg 87, 44227 Dortmund, Germany

* Corresponding author: Xenia Kerkhoff

Received  March 2021 Revised  July 2021 Early access October 2021

We consider one-dimensional distributed optimal control problems with the state equation being given by the viscous Burgers equation. We discretize using a space-time discontinuous Galerkin approach. We use upwind flux in time and the symmetric interior penalty approach for discretizing the viscous term. Our focus is on the discretization of the convection terms. We aim for using conservative discretizations for the convection terms in both the state and the adjoint equation, while ensuring that the approaches of discretize-then-optimize and optimize-then-discretize commute. We show that this is possible if the arising source term in the adjoint equation is discretized properly, following the ideas of well-balanced discretizations for balance laws. We support our findings by numerical results.

Citation: Xenia Kerkhoff, Sandra May. Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021054
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##### References:
Test 1: Results for gradient test. The $x-$axis denotes $\frac{1}{\rho}$ with $\rho$ being the step length in the difference quotient, the $y-$axis denotes the error of the gradient as given by (24)
Test 2: Computed solution $(q^h,u^h)$ for $p = 2$ and $N = 160$. Left column: control $q^h$, right column: state $u^h$. The second row shows the solutions $q^h(\cdot,t)$ and $u^h(\cdot,t)$ at time instances $t = 0.25$, $t = 0.5$, and $t = 0.75$ as a function of the space coordinate $x$
Test 1: Errors and orders of convergence for $\varepsilon = 10^{-3}$
 $p$ $N$ error $u$ order error $q$ order error $z$ order iter. gradient OD-wo 1 40 2.35e-04 – 1.93e-03 – 2.31e-04 – 25 4.66e-05 1 80 5.43e-05 2.11 5.09e-04 1.92 7.36e-05 1.65 28 1.97e-05 1 160 1.28e-05 2.08 1.44e-04 1.82 2.48e-05 1.57 31 7.11e-06 1 320 3.19e-06 2.01 4.05e-05 1.83 8.22e-06 1.59 36 2.81e-06 2 20 2.31e-05 – 1.67e-04 – 1.67e-05 – 49 2.18e-07 2 40 2.95e-06 2.97 2.05e-05 3.03 2.06e-06 3.02 52 3.00e-08 2 80 3.41e-07 3.11 2.61e-06 2.98 2.61e-07 2.98 54 9.51e-09 2 160 4.65e-08 2.87 3.93e-07 2.73 3.47e-08 2.91 54 9.51e-09 3 10 4.15e-06 – 4.97e-05 – 4.96e-06 – 54 9.51e-09 3 20 1.81e-07 4.52 3.02e-06 4.04 3.00e-07 4.05 54 9.51e-09 3 40 2.97e-08 2.61 2.86e-07 3.40 1.70e-08 4.14 54 9.51e-09 3 80 2.62e-08 0.18 2.07e-07 0.47 3.33e-09 2.35 54 9.51e-09 OD-w 1 40 2.31e-04 – 1.87e-03 – 1.87e-04 – 54 9.51e-09 1 80 5.05e-05 2.19 4.63e-04 2.02 4.63e-05 2.02 54 9.51e-09 1 160 9.91e-06 2.35 1.15e-04 2.01 1.15e-05 2.01 54 9.51e-09 1 320 1.72e-06 2.52 2.87e-05 2.01 2.87e-06 2.01 54 9.51e-09 2 20 2.26e-05 – 1.67e-04 3.07 1.67e-05 – 54 9.51e-09 2 40 2.88e-06 2.98 2.01e-05 3.06 2.01e-06 3.06 54 9.51e-09 2 80 3.23e-07 3.15 2.45e-06 3.04 2.44e-07 3.04 54 9.51e-09 2 160 4.35e-08 2.89 3.62e-07 2.76 3.01e-08 3.02 54 9.51e-09 3 10 4.16e-06 – 4.96e-05 – 4.96e-06 – 54 9.51e-09 3 20 1.73e-07 4.59 2.99e-06 4.05 2.98e-07 4.05 54 9.51e-09 3 40 2.66e-08 2.70 2.65e-07 3.49 1.72e-08 4.11 54 9.51e-09 3 80 2.59e-08 0.04 2.05e-07 0.37 3.52e-09 2.29 54 9.51e-09
 $p$ $N$ error $u$ order error $q$ order error $z$ order iter. gradient OD-wo 1 40 2.35e-04 – 1.93e-03 – 2.31e-04 – 25 4.66e-05 1 80 5.43e-05 2.11 5.09e-04 1.92 7.36e-05 1.65 28 1.97e-05 1 160 1.28e-05 2.08 1.44e-04 1.82 2.48e-05 1.57 31 7.11e-06 1 320 3.19e-06 2.01 4.05e-05 1.83 8.22e-06 1.59 36 2.81e-06 2 20 2.31e-05 – 1.67e-04 – 1.67e-05 – 49 2.18e-07 2 40 2.95e-06 2.97 2.05e-05 3.03 2.06e-06 3.02 52 3.00e-08 2 80 3.41e-07 3.11 2.61e-06 2.98 2.61e-07 2.98 54 9.51e-09 2 160 4.65e-08 2.87 3.93e-07 2.73 3.47e-08 2.91 54 9.51e-09 3 10 4.15e-06 – 4.97e-05 – 4.96e-06 – 54 9.51e-09 3 20 1.81e-07 4.52 3.02e-06 4.04 3.00e-07 4.05 54 9.51e-09 3 40 2.97e-08 2.61 2.86e-07 3.40 1.70e-08 4.14 54 9.51e-09 3 80 2.62e-08 0.18 2.07e-07 0.47 3.33e-09 2.35 54 9.51e-09 OD-w 1 40 2.31e-04 – 1.87e-03 – 1.87e-04 – 54 9.51e-09 1 80 5.05e-05 2.19 4.63e-04 2.02 4.63e-05 2.02 54 9.51e-09 1 160 9.91e-06 2.35 1.15e-04 2.01 1.15e-05 2.01 54 9.51e-09 1 320 1.72e-06 2.52 2.87e-05 2.01 2.87e-06 2.01 54 9.51e-09 2 20 2.26e-05 – 1.67e-04 3.07 1.67e-05 – 54 9.51e-09 2 40 2.88e-06 2.98 2.01e-05 3.06 2.01e-06 3.06 54 9.51e-09 2 80 3.23e-07 3.15 2.45e-06 3.04 2.44e-07 3.04 54 9.51e-09 2 160 4.35e-08 2.89 3.62e-07 2.76 3.01e-08 3.02 54 9.51e-09 3 10 4.16e-06 – 4.96e-05 – 4.96e-06 – 54 9.51e-09 3 20 1.73e-07 4.59 2.99e-06 4.05 2.98e-07 4.05 54 9.51e-09 3 40 2.66e-08 2.70 2.65e-07 3.49 1.72e-08 4.11 54 9.51e-09 3 80 2.59e-08 0.04 2.05e-07 0.37 3.52e-09 2.29 54 9.51e-09
Test 1: Errors and orders of convergence for $\varepsilon = 10^{-5}$
 $p$ $N$ error $u$ order error $q$ order error $z$ order iter. gradient OD-wo 1 40 2.79e-04 – 1.97e-03 – 2.47e-04 – 24 5.33e-05 1 80 7.51e-05 1.89 5.51e-04 1.84 9.20e-05 1.42 27 2.81e-05 1 160 2.28e-05 1.72 1.93e-04 1.51 4.08e-05 1.18 29 1.26e-05 1 320 8.15e-06 1.49 7.90e-05 1.29 1.99e-05 1.04 33 7.11e-06 2 20 2.52e-05 – 1.72e-04 – 1.72e-05 – 44 2.25e-07 2 40 3.98e-06 2.66 2.18e-05 2.98 2.18e-06 2.97 51 4.63e-08 2 80 6.28e-07 2.67 3.09e-06 2.82 3.10e-07 2.82 54 9.51e-09 2 160 1.01e-07 2.64 5.60e-07 2.46 5.42e-08 2.51 54 9.51e-09 3 10 4.63e-06 – 4.87e-05 – 4.87e-06 – 54 9.51e-09 3 20 2.76e-07 4.07 3.12e-06 3.96 3.08e-07 3.98 54 9.51e-09 3 40 3.58e-08 2.94 3.19e-07 3.29 1.87e-08 4.04 54 9.51e-09 3 80 2.68e-08 0.42 2.12e-07 0.59 2.94e-09 2.67 54 9.51e-09 OD-w 1 40 2.74e-04 – 1.89e-03 – 1.89e-04 – 54 9.51e-09 1 80 6.99e-05 1.97 4.68e-04 2.01 4.68e-05 2.01 54 9.51e-09 1 160 1.76e-05 1.99 1.17e-04 2.01 1.17e-05 2.01 54 9.51e-09 1 320 4.39e-06 2.00 2.91e-05 2.00 2.91e-06 2.00 54 9.51e-09 2 20 2.46e-05 – 1.72e-04 – 1.72e-05 – 54 9.51e-09 2 40 3.84e-06 2.68 2.12e-05 3.02 2.12e-06 3.02 54 9.51e-09 2 80 5.86e-07 2.71 2.65e-06 3.00 2.65e-07 3.00 54 9.51e-09 2 160 8.82e-08 2.73 3.89e-07 2.77 3.33e-08 2.99 54 9.51e-09 3 10 4.65e-06 – 4.85e-05 – 4.85e-06 – 54 9.51e-09 3 20 2.65e-07 4.13 3.05e-06 3.99 3.04e-07 4.00 54 9.51e-09 3 40 2.96e-08 3.16 2.75e-07 3.47 1.86e-08 4.03 54 9.51e-09 3 80 2.59e-08 0.19 2.05e-07 0.42 3.53e-09 2.40 54 9.51e-09
 $p$ $N$ error $u$ order error $q$ order error $z$ order iter. gradient OD-wo 1 40 2.79e-04 – 1.97e-03 – 2.47e-04 – 24 5.33e-05 1 80 7.51e-05 1.89 5.51e-04 1.84 9.20e-05 1.42 27 2.81e-05 1 160 2.28e-05 1.72 1.93e-04 1.51 4.08e-05 1.18 29 1.26e-05 1 320 8.15e-06 1.49 7.90e-05 1.29 1.99e-05 1.04 33 7.11e-06 2 20 2.52e-05 – 1.72e-04 – 1.72e-05 – 44 2.25e-07 2 40 3.98e-06 2.66 2.18e-05 2.98 2.18e-06 2.97 51 4.63e-08 2 80 6.28e-07 2.67 3.09e-06 2.82 3.10e-07 2.82 54 9.51e-09 2 160 1.01e-07 2.64 5.60e-07 2.46 5.42e-08 2.51 54 9.51e-09 3 10 4.63e-06 – 4.87e-05 – 4.87e-06 – 54 9.51e-09 3 20 2.76e-07 4.07 3.12e-06 3.96 3.08e-07 3.98 54 9.51e-09 3 40 3.58e-08 2.94 3.19e-07 3.29 1.87e-08 4.04 54 9.51e-09 3 80 2.68e-08 0.42 2.12e-07 0.59 2.94e-09 2.67 54 9.51e-09 OD-w 1 40 2.74e-04 – 1.89e-03 – 1.89e-04 – 54 9.51e-09 1 80 6.99e-05 1.97 4.68e-04 2.01 4.68e-05 2.01 54 9.51e-09 1 160 1.76e-05 1.99 1.17e-04 2.01 1.17e-05 2.01 54 9.51e-09 1 320 4.39e-06 2.00 2.91e-05 2.00 2.91e-06 2.00 54 9.51e-09 2 20 2.46e-05 – 1.72e-04 – 1.72e-05 – 54 9.51e-09 2 40 3.84e-06 2.68 2.12e-05 3.02 2.12e-06 3.02 54 9.51e-09 2 80 5.86e-07 2.71 2.65e-06 3.00 2.65e-07 3.00 54 9.51e-09 2 160 8.82e-08 2.73 3.89e-07 2.77 3.33e-08 2.99 54 9.51e-09 3 10 4.65e-06 – 4.85e-05 – 4.85e-06 – 54 9.51e-09 3 20 2.65e-07 4.13 3.05e-06 3.99 3.04e-07 4.00 54 9.51e-09 3 40 2.96e-08 3.16 2.75e-07 3.47 1.86e-08 4.03 54 9.51e-09 3 80 2.59e-08 0.19 2.05e-07 0.42 3.53e-09 2.40 54 9.51e-09
Test 2: Values and orders of convergence (computed using values from 3 subsequent meshes) for functional $J$ for $p = 1$
 OD-w OD-wo N value $J$ order iter. gradient value $J$ order iter. gradient 20 0.10640082692 - 44 8.58e-08 0.10641958284 - 11 2.10e-03 40 0.10818515778 - 44 7.94e-08 0.10819225166 - 14 1.02e-03 80 0.10855578246 2.27 43 8.03e-08 0.10855750188 2.28 18 3.44e-04 160 0.10862035158 2.52 43 8.00e-08 0.10862054051 2.53 22 1.09e-04
 OD-w OD-wo N value $J$ order iter. gradient value $J$ order iter. gradient 20 0.10640082692 - 44 8.58e-08 0.10641958284 - 11 2.10e-03 40 0.10818515778 - 44 7.94e-08 0.10819225166 - 14 1.02e-03 80 0.10855578246 2.27 43 8.03e-08 0.10855750188 2.28 18 3.44e-04 160 0.10862035158 2.52 43 8.00e-08 0.10862054051 2.53 22 1.09e-04
Test 2: Errors and orders of convergence for the control $q$ for $p = 1$
 OD-w OD-wo N error $q$ order iter. gradient error $q$ order iter. gradient 20 1.67e-02 - 44 8.58e-08 1.77e-02 - 11 2.10e-03 40 4.56e-03 1.87 44 7.94e-08 6.10e-03 1.54 14 1.02e-03 80 1.47e-03 1.64 43 8.03e-08 2.83e-03 1.11 18 3.44e-04 160 4.20e-04 1.80 43 8.00e-08 1.00e-03 1.50 22 1.09e-04
 OD-w OD-wo N error $q$ order iter. gradient error $q$ order iter. gradient 20 1.67e-02 - 44 8.58e-08 1.77e-02 - 11 2.10e-03 40 4.56e-03 1.87 44 7.94e-08 6.10e-03 1.54 14 1.02e-03 80 1.47e-03 1.64 43 8.03e-08 2.83e-03 1.11 18 3.44e-04 160 4.20e-04 1.80 43 8.00e-08 1.00e-03 1.50 22 1.09e-04
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