Unconditionally stable higher order semi-implicit level set method for advection equations

We present compact semi-implicit finite difference schemes on structured grids for numerical solutions of advection by an external velocity and by a speed in the normal direction that are applicable in level set methods. The recommended numerical scheme is third order accurate for the linear advection in the 2D case with a space dependent velocity. Using a combination of analytical and numerical tools in the von Neumann stability analysis, the third order scheme is claimed to be unconditionally stable. We also present a simple high-resolution scheme that gives a TVD (Total Variation Diminishing) approximation of the spatial derivative for the advected level set function in the 1D case. In the case of nonlinear advection, a semi-implicit discretization is proposed to linearize the problem. The compact implicit part of the stencil of numerical schemes contains unknowns only in the upwind direction. Consequently, algebraic solvers like the fast sweeping method can be applied efficiently to solve the resulting algebraic systems. Numerical tests to evolve a smooth and non-smooth interface and an example with a large variation of the velocity confirm the good accuracy of the third order scheme even in the case of very large Courant numbers. The advantage of the high-resolution scheme is documented for examples where the advected level set functions contain large jumps in the gradient.