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Data assimilation for conservation laws. (English) Zbl 1114.35122

Data assimilation is important in meteorology and oceanography because it is a way to improve the models with newly measured data, statically or dynamically. It is a type of inverse problem for which the most popular solution method is least square with regularization and optimal control algorithm. In this paper the authors investigate the differentiated equation of some systems of conservation laws and show that the calculus of variation can be applied in a formal and rigorous manner provided that the principal values are defined at shocks and the equations are written in the sense of distribution theory. Numerical illustrations are given for the control of shocks for Burgers’ equation and for the shallow water equations in one space dimension.

MSC:

35L65 Hyperbolic conservation laws
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35L67 Shocks and singularities for hyperbolic equations
35R30 Inverse problems for PDEs
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