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Center for Nonlinear Studies |
In this talk, I will start by revisiting the celebrated Lax Wendro theorem that states that schemes in conservation form are the right candidate for the numerical approximation of conservation law. In the multidimensional case, there is however more exibility, since, in the conservation law: ut + div f(u) = 0; f = (f(1); ... ; f(d)) one can also exploit the intrinsic nature of the divergence operator: of course it is the sum of the partial derivative f(j) xj , it is mainly the trace of the gradient of the tensor f. In this talk I will describe another way to translate conservation, and the 1D Lax Wendro case is a particular case of, as well as the multidimensional versions, see [1]. Under standard assumptions, it is possible to show that numerical solutions will converge to a weak one. Going further, I will also show how one can exploit this remark to incorporate additional conservation laws. For example, one can construct systematically numerical approximation that are also entropy conservative, or entropy dissipative with a controllable dissipation of entropy [5]. One can also exploit this idea to construct numerical schemes that are locally conservative, though one is directly integrating a non conservative version of: this has been exploited in Lagrangian hydrodynamics with S. Tokareva [3], but also in the Eulerian setting with P. Bacigaluppi and S. Tokareva [2]. This idea can also be exploited to construct schemes that locally preserve the kinetic momentum [4], etc. This is a joint work with P. Bacigaluppi, M. Veiga (both at University of Zurich) and S. Tokareva (LANL).
Host: Svetlana Tokareva