Asymptotic adaptive methods for multi-scale problems in fluid mechanics

This paper reports on the results of a three-year research effort aimed at investigating and exploiting the role of physically motivated asymptotic analysis in the design of numerical methods for singular limit problems in fluid mechanics. Such problems naturally arise, among others, in combustion, magneto-hydrodynamics, and geophysical fluid mechanics. Typically, they are characterized by multiple-space and/or -time scales and by the disturbing fact that standard computational techniques fail entirely, are unacceptably expensive, or both. The challenge here is to construct numerical methods which are robust, uniformly accurate, and efficient through different asymptotic regimes and over a wide range of relevant applications. Summaries of multiple-scales asymptotic analyses for low-Mach-number flows, magneto-hydrodynamics at small Mach and Alfvén numbers, and of multiple-scales atmospheric flows are provided. These reveal singular balances between selected terms in the respective governing equations within the considered flow regimes. These singularities give rise to problems of severe stiffness, stability, or to dynamic-range issues in straight-forward numerical discretizations. A formal mathematical framework for the multiple scales asymptotics is then summarized by use of the example of multiple-length-scale single-time-scale asymptotics for low-Mach-number flows. The remainder of the paper focuses on the construction of numerical discretizations for the respective full governing equation systems. These discretizations avoid the pitfalls of singular balances by exploiting the asymptotic results. Importantly, the asymptotics are not used here to derive simplified equation systems, which are then solved numerically. Rather, numerical integration of the full equation sets is aimed at and the asymptotics are used only to construct discretizations that do not deteriorate as a singular limit is approached. One important ingredient of this strategy is the numerical identification of a singular limit regime given a set of discrete numerical state variables. This problem is addressed in an exemplary fashion for multiple-length single-time-scale low-Mach-number flows in one space dimension. The strategy allows a dynamic determination of an instantaneous relevant Mach number, and it can thus be used to drive the appropriate adjustment of the numerical discretizations when the singular limit regime is approached.