Well-Balanced Schemes for Hyperbolic Kinetic Relaxation

Abstract

This work presents a novel family of well-balanced numerical schemes for hyperbolic systems of balance laws based on the kinetic relaxation approach. The method begins by transforming the original non-linear system into a linearized kinetic system with an increased number of variables. In this framework, non-linearities are shifted to the source term, and the connection to the macroscopic variables is maintained via a projection operator related to the Maxwellian equilibrium states. These relaxed systems are typically solved using splitting techniques, where the evolution is decomposed into two distinct steps: first, the transport stage solving the source-free kinetic system, and second, the relaxation stage governed by an ordinary differential equation where the time derivative is balanced by the kinetic source term. The primary objective is to design schemes that accurately preserve the steady-state solutions of the original macroscopic equations, noting that these do not necessarily coincide with the trivial equilibria of the relaxed system. This well-balanced property is achieved through an appropriate formulation of the kinetic system combined with consistent reconstruction operators. This approach ensures that the balance is maintained throughout both the transport stage and the projection stage, where the treatment of the macroscopic source term is carefully integrated. We demonstrate that this framework can be implemented using both Finite Volume (FV) and Semi-Lagrangian (SL) techniques in the transport step. We develop and analyze these schemes for the Finite Volume case in both explicit and implicit formulations. Results are validated through numerical benchmarks, including the Burgers equation with a source term, 1D Shallow Water Equations (SWE) with bathymetry preserving non-trivial moving water equilibria and the Euler equations under external potentials.