A new class of entropy stable schemes for hyperbolic systems: Finite element methods

Abstract:

In this work we propose a new class of entropy consistent schemes for hyperbolic systems of conservation laws (HCL). The schemes developed so far in the classic works of Tadmor, Johnson and their collaborators start from an appropriate entropy conservative formulation of the system. Then entropy diminishing schemes are obtained by adding appropriate artificial diffusion terms. This program was based on the formulation of the HCL using the entropy variables. In this work we propose an alternative approach which has as a starting point a new mixed reformulation of the hyperbolic system which retains the original variables but still allows for conservative discretisation. The original variables are approximated directly and significant flexibility is allowed in the design of the corresponding computational algorithms. New finite element schemes are introduced and analysed. It is shown that the resulting approximations are consistent at the limit to an entropy weak and when appropriate to an entropy measure valued solution.