In computational mechanics, substantial advances have been made toward the development of structured integrators -- numerical methods which, at the algorithmic level, preserve invariants of motion like energy and momenta and respect the geometric structure of the underlying differential equations of motion. In sharp contrast, computational fluid dynamics has until recently seen little in the realm of structure preservation, favoring numerical analytic approaches over geometric methods.
This poster will discuss recent advances in the design of structured integrators for 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. Among the hallmarks of the numerical methods derived under this framework 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.