Embedding General Conservation Constraints in Discretizations of Hyperbolic Systems on Arbitrary Meshes: A Multidimensional Framework

The purpose of this review is to discuss the notion of conservation in hyperbolic systems and how one can formulate it at the discrete level depending on the representation of the solution on the mesh. Since it is impossible to have a fully general theory, we discuss several alternatives possibilities: cases where the solution is represented by average in volumes; cases where the mesh is staggerred (i.e. the components of the solution are not localised at the same places); cases where the solution is solely represented by point values; and an example where all the previous options are mixed. We show how each configuration can provide, or not, enough flexibility. Though the discussion could be adapted to any hyperbolic system endowed with an entropy, we focus on compressible fluid mechanics, it its Eulerian and Lagrangian formulations. On a given mesh, the unifying element is that we systematically express the update of conserved variables as u n+1 = u n − Δt δu, where the functional u → δu depends on the value of u at the current degree of freedom and its values from a set of degrees of freedom. This set of define the stencil of the scheme. From the stencil, one can naturally define a graph connecting the states that appears in δu. The notion of local conservation can be defined from this graph. We are aware of only two possible situations: either the graph is constructed from the faces of the mesh elements (or the dual mesh), or it is defined from the mesh itself. Two notions of local conservation then emerge: either we define a numerical flux, or we define a “residual” attached to elements and the degrees of freedom within the element. We show that this two notions are in a way equivalent, but the one with residual allows much more flexibility, especially if additional algebraic constraints must be satisfied. Examples of specific additional conservation constraints are provided to illustrate this flexibility. We also show that this notion of conservation gives a very clear framework for the design of scheme in the Lagrangian framework. In the ending section we will provide a number of ongoing research avenues strongly related to the formulation discussed, and we highlight some open questions which will be explored in the future