A Semi-Discrete Lagrangian-Eulerian formulation for three-dimensional hyperbolic systems and applications

In this work, we develop a three-dimensional Semi-Discrete formulation for hyperbolic systems of conservation laws in three-space dimensions on structured cubical and tetrahedral meshes, thereby extending the results presented in [1], which is based on the concept of no-flow curves introduced in [4]. A first numerical validation study of the scheme is presented and discussed, addressing several preliminary 3D numerical solutions for systems, namely: (1) compressible Euler flows with positivity of the density, (2) the nontrivial Orszag-Tang problem in magnetohydrodynamics, which is well-known to satisfy the notable involution-constrained partial differential equation ∇ · B = 0 (this condition is verified numerically by the proposed approach, i.e., without any imposition of an additional constraint in the formulation), and (3) a nonstrictly hyperbolic three-phase flow system in porous media with a resonance point (coincidence of eigenvalues). Due
to the no-flow framework, there is no need to employ/compute the eigenvalues (exact or approximate values) - in fact there is no need to construct the relevant Jacobian of the hyperbolic flux functions, and thus giving rise to an effective weak CFL-stability condition, which is feasible in the computing practice. Overall, the method is based on locally well-balanced properties, it is Riemann-solver-free and, hence, time-consuming field-by-field type decompositions are avoided in the case of multidimensional systems.