A hybrid Taylor–Galerkin variational multi-scale stabilization method for the level set equation

• The hybrid TG-VMS method for the level set equation was modified to an implicit formulation.
• Taylor series expansion up to fourth order were studied favoring a second order expansion.
• Results indicate good conservation characteristics compared to other methods available in literature.
• High speed advection, typical in in-flight icing, is easily handled.

A stabilized finite element formulation of the level set equation is proposed for the numerical simulation of water droplet dynamics for in-flight ice accretion problems. The variational multi-scale and Taylor–Galerkin approaches are coupled such that the temporal derivative in the weak Galerkin formulation is replaced with a Taylor series expansion improving the temporal accuracy of the scheme. The variational multi-scale approach is then applied to the semi-discrete equation, allowing the stabilization terms to appear naturally. Taylor series expansions up to the fourth order have been studied in terms of accuracy and convergence rates. A second order implicit expansion was found to provide a good trade-off between accuracy and computational cost when compared to a fourth order implicit expansion. Validation is done through a number of benchmark cases considering droplet stretching and high-speed advection. Results indicate good mass conservation characteristics compared to other methods available in the literature.