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Partial differential Equations, Numerics and Control

Recovering shocks in 1D Burgers' equation

Mario Morales-Hernández

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Inverse problems related to the identification of an initial condition given a final objective function are ill-conditioned when dealing with non-linear problems and discontinuities.

In this work, the focus is put on trying to recover an initial data at t=0 that gives a shock as objective after t=T in the 1D inviscid Burgers' equation.

It is well known that multiple initial conditions satisfy the same objective function. In particular, the conventional adjoint methodology provides only one (the smoothest) of the multiple solutions, which is shown in Video 1 .

In order to recover a pure shock as initial condition, Castro et al. [1] proved that it is feasible to locate the original shock and to apply an internal boundary condition to define it.

Instead, a modification in the functional is proposed in this work. Since the L2 norm of the pure shock is greater than the bending solution shown in Video 1 , it is possible to penalize the functional by adding an extra term regarding the norm of the solution. This modification allow us to recover a family of initial profiles that gives a shock at t=T. In particular, if applying the Maximum Principle, we will recover the pure shock after some iterations (see Video 2 ). If not, a N-wave type solution with larger norm is achieved (see Video 3 )

[1] C. Castro, F. Palacios y E. Zuazua., An alternating descent method for the optimal control of the inviscid Burgers equation in the presence of shocks . Mathematical Models and Methods in Applied Sciences 18, pp. 369–416

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Video 1: Recovering a smooth solution without penalising the functional

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Video 2: Recovering the initial condition by penalising the functional and applying the Maximum Principle

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Video 3: Recovering the initial condition by penalising the functional without applying the Maximum Principle

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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 PaÃŒs Vasco (UPV/EHU)