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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 *u*^{d} 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.