A Backward Point Shifted Levelset Method for Highly Accurate Interface Computation

Abstract

We propose a technique that could significantly improve the accuracy of the levelset method and has the potential for fully conservative treatment. Level set method uses the levelset function, usually an approximate signed distance function Φ to indirectly represent the interface by the zero set of Φ. When Φ is advanced to the next time level by an advection equation, it is no longer a signed distance function any more, therefore the uneven numerical dissipation associated with the discretization of the advection equation could distort the interface particularly in places where the radius of curvature of the interface changes dramatically or two segments of the interface are getting close. Also an auxiliary equation is usually solved at each time level to restore Φ into a signed distance function, which could further shift the interface position. We address the second problem by using an analytic point shifted algorithm to locally perturb the mesh without geometric reconstruction of the interface so that the zero set of Φ is located at grid nodes and therefore solving the auxiliary equation will not move the interface. Our strategy for solving the first problem is that when applying the advection equation for Φ, it does not initiate from a signed distance function, but ends up with one, which can be achieved by solving the advection equation backward in time. These two techniques are combined with an iterative procedure.