Some remarks about conservation for residual distribution schemes
Provides a theoretical unification and clarification of conservation properties for known schemes, but is incremental as it extends prior work.
The paper unifies various residual distribution schemes for steady hyperbolic problems into a common framework, showing they all have a flux formulation and are locally conservative. It also reinterprets Tadmor's entropy stability condition within this framework.
We are interested in the discretisation of the steady version of hyperbolic problems. We first show that all the known schemes (up to our knowledge) can be rephrased in a common framework. Using this framework, we first show all the known schemes have a flux formulation, with an explicit construction of the flux, and thus are locally conservative. This is well known for the finite volume schemes or the discontinuous Galerkin ones, much less known for the continuous finite element methods. We also show that Tadmor's entropy stability formulation can naturally be rephrased in this framework as an additional conservation relation discretisation, and using this, we show some conenction with the recent papers [1, 2, 3, 4]. This contribution is an enhanced version of [5].