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Partial differential Equations, Numerics and Control

Optimal control of scalar conservation laws in large time horizons

Alejandro Pozo

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It is known (see [IPZ]) that numerical viscosity introduced by a numerical flux can dramatically modify the large-time profile of the inviscid Burgers equation. Moreover, it can also affect the metastable states of the viscous Burgers equation, mainly when the viscosity is small.

This pathology can have a big influence in other problems involving numerical solvers for the mentioned equations. For instance, in this simulation we show the different performances of Engquist-Osher (top) and modified Lax-Friedrichs (bottom) schemes in the following optimal control problem:


where ud is a given target function, v=10-6 and T=100. Note that the solution by Engquist-Osher recovers successfully the target, while the one by modified Lax-Friedrichs needs to introduce spurious oscillations.

Reference:
N. Allahverdi, A. Pozo A., E. Zuazua, Numerical aspects in large-time optimal control of Burgers equation, preprint.

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Eusko Jaurlaritza - Gobierno Vasco ikerbasque - Basque Foundation for Science Bizkaia xede. Bizkaiko Foru Aldundia innobasque - Agencia vasca de la innovación Universidad del PaÌs Vasco (UPV/EHU)