math.MP — Mathematical Physics
Stability and Convergence of a Regularized Immersed Boundary Method under Dynamic Cartesian Refinement. Part III: Fluid-Driven Rigid-Body Motion
We study a conforming regularized immersed boundary discretization of the two-dimensional Eulerian Stokes–Newton system for one freely moving rigid body. Regularized interior moments complete the surface constraints used for prescribed motion. Positive excess mass and inertia define a fixed fluid–body kinetic inner product. A skew inertial correction is derived from the fictitious rigid interior; it has zero power but is necessary for the physical translational momentum equation. The method advances the body configuration with the preceding computed rigid velocity and then solves a linear joint fluid–body step on a dynamically enriched Cartesian space. Exact orthogonal transfer gives an energy identity independent of the number of mesh changes. Interior and surface moment defects control the discrepancy between the discrete background energy and physical kinetic energy. A volume-preserving geometry comparison and a moving coarse recovery family yield conditional joint convergence of the fluid velocity, rigid velocity, and body configuration on an interval with positive wall clearance. The error bound distinguishes the energy norm after geometry alignment from the generally weaker estimate at the same Eulerian points. The analysis assumes exact integration and explicit strong-solution regularity; it does not cover arbitrary density ratios or collisions. Manufactured coupled computations, independent residual checks, and a Lean algebraic companion document the numerical and formal verification scope.