A Newton--Krylov Solver for Remapping-Based Volume-of-Fluid Methods

A new conservative remapping method is presented here for generating multi-dimensional fluxes used for the time integration of hyperbolic equations through single-stage unsplit advection. A solution to the Lagrangian mesh location is generated that converges the volumes of all remapped flux definitions to the volumes of equivalent one-dimensional flux definitions at their corresponding cell faces. The method is most feasible within localized subdomains of an Eulerian solution domain. The main utility of the method is the generation of fluxes in the vicinity of discontinuities and sharp gradients that does not introduce physically spuriously extrema while locally conserving mass. In generating solutions to Lagrangian mesh location by solving a nonlinear equation system, the new remap requires the anchoring of the Lagrangian mesh—removal of degrees of freedom to make the problem definite. The method is based on characteristic tracing along trajectories assumed to be linear. This paper investigates the convergence and sensitivity of the Lagrangian mesh solver to mesh and timestep size, in assessing its merit for application in transient CFD codes. For single-stage unsplit advection volume-of-fluid methods, the new remap suppresses the introduction of undershoots, overshoots, and wisps. In combination with a conservative remapping phase, the new method for fixing the location of the Lagrangian mesh eliminates the need for heuristic redistribution or clipping procedures for spurious extrema removal.

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