Geometric, variational discretization of continuum theories

• Published in: Physica. D 2011-10, Vol.240 (21), p.1724-1760
• Main Authors: Gawlik, E.S., Mullen, P., Pavlov, D., Marsden, J.E., Desbrun, M.
• Format: Article
• Language: English
• Subjects: Complex fluids; Computational fluid dynamics; Continuums; Discretization; Exact sciences and technology; Fluid dynamics; Fluid flow; Fluids; Geometric discretization; Magnetohydrodynamics; Mathematical analysis; Mathematical models; Physics; Structure-preserving schemes
• ISSN: 0167-2789; 1872-8022
• DOI: 10.1016/j.physd.2011.07.011

This study derives geometric, variational discretization of continuum theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the dynamics of complex fluids. A central role in these discretizations is played by the geometric formulation of fluid dynamics, which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume-preserving diffeomorphisms of the fluid domain. Inspired by this framework, we construct a finite-dimensional approximation to the diffeomorphism group and its Lie algebra, thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group. The extension to MHD and complex fluid flow is then made through an appeal to the theory of Euler–Poincaré systems with advection, which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters. Upon deriving a family of structured integrators for these systems, we test their performance via a numerical implementation of the update schemes on a cartesian grid. Among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries, automatic satisfaction of solenoidal constraints on vector fields, good long-term energy behavior, robustness with respect to the spatial and temporal resolution of the discretization, and applicability to irregular meshes. ► Introduction of a discrete diffeomorphism group and its Lie algebra. ► Construction of geodesics on the group of discrete volume-preserving diffeomorphisms. ► Generalization to handle advected parameters, e.g., MHD and complex fluids. ► Numerical and mathematical proof of cross-helicity preservation for ideal MHD. ► Comparison of the resulting schemes to other low-order schemes.