Marker Redistancing/Level Set Method for High-Fidelity Implicit Interface Tracking

• Authors: Robert Nourgaliev;Samet Kadioglu;Vincent Mousseau
• Venue: SIAM Journal on Scientific Computing
• Year: 2010

A hybrid of the front tracking (FT) and the level set (LS) methods is introduced, combining advantages and removing drawbacks of both methods. The kinematics of the interface is treated in a Lagrangian (FT) manner, by tracking markers placed at the interface. The markers are not connected—instead, the interface topology is resolved in an Eulerian (LS) framework, by wrapping a signed distance function around Lagrangian markers each time the markers move. For accuracy and efficiency, we have developed a high-order “anchoring” algorithm and an implicit PDE-based redistancing. We have demonstrated that the method is 3rd-order accurate in space, near the markers, and therefore 1st-order convergent in curvature; this is in contrast to traditional PDE-based reinitialization algorithms, which tend to slightly relocate the zero level set and can be shown to be nonconvergent in curvature. The implicit pseudo-time discretization of the redistancing equation is implemented within the Jacobian-free Newton-Krylov (JFNK) framework combined with ILU(k) preconditioning. Due to the LS localization, the bandwidth of the Jacobian matrix is nearly constant, and the ILU preconditioning scales as $\sim N\log(\sqrt{N})$ in two dimensions, which implies efficiency and good scalability of the overall algorithm. We have demonstrated that the steady-state solutions in pseudo-time can be achieved very efficiently, with $\approx10$ iterations ($\mathrm{CFL}\approx10^4$), in contrast to the explicit redistancing which requires hundreds of iterations with $\mathrm{CFL}\leq1$.