We present compact semi-implicit finite difference schemes on structured grids for numerical solutions of the advection by an external velocity and by a speed in normal direction that are applicable in level set methods. The most involved numerical scheme is third order accurate for the linear advection with a space dependent velocity and unconditionally stable in the sense of von Neumann stability analysis. 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 case of nonlinear advection, the semi-implicit discretization is proposed to linearize the problem. The compact form of implicit stencil in numerical schemes containing unknowns only in the upwind direction allows applications of efficient algebraic solvers like fast sweeping methods. Numerical tests to evolve a smooth and non-smooth interface and an example with a large variation of velocity confirm the good accuracy of the methods and fast convergence of the algebraic solver even in the case of very large Courant numbers.

翻译：我们提出在结构化网格上的紧凑半隐式有限差分格式，用于外部速度及法向速度驱动的平流数值求解，这些格式可应用于水平集方法。其中最具复杂性的数值格式在空间依赖速度的线性平流问题中达到三阶精度，并在冯·诺依曼稳定性分析意义上无条件稳定。我们还提出一种简单的高分辨率格式，为平流水平集函数提供空间导数的TVD（总变差缩减）近似。针对非线性平流问题，采用半隐式离散化方案实现问题线性化。数值格式中隐式模板的紧凑形式仅包含逆风方向未知量，使得快速扫掠法等高效代数求解器得以应用。对光滑与非光滑界面演化案例以及速度大梯度变化算例的数值测试表明，即使在大库朗数条件下，该方法仍具有良好精度且代数求解器收敛迅速。