Relaxation schemes for hyperbolic multi-scale flows

Multi-scale problems are omnipresent in environmental and industrial processes, posing a challenge to classical numerical solvers given that the propagation speeds of information span several orders of magnitude. In hyperbolic systems, the absolute fastest wave speed remains finite, but in classical, explicitly integrated numerical schemes, it determines the time step that ensures stability. This means in particular that it may lead to vanishing time increments in the presence of fast processes caused for instance by high pressure or strong magnetic fields. Therefore, a common approach, which will be also applied here, is to use implicit or semi-implicit time integrators, combined with a suitable approximation of spatial derivatives in the implicitly treated systems. This allows scale-independent artificial dissipation and stability even with large time steps. As the evolution of the modeled variables is described by a nonlinear flux function, treating it fully or partially implicitly involves solving nonlinear systems. Depending on the problem, these systems can be large, coupled and ill-posed, for which solvers such as the Newton method may converge very slowly or not at all. To avoid nonlinear implicit systems, we apply a flux linearization using a relaxation technique while keeping the systems compatible with crucial properties such as asymptotic limits, conservation of mass, momentum and energy, and necessary vector-calculus identities. In this talk, the construction of such numerical schemes in a 2D finite volume framework will be illustrated at the example of a model of continuum mechanics consisting of at least three scales including a convective and an acoustic sub-system.