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：我们提出了结构化网格上的紧致半隐式有限差分格式，用于平流方程（包括外部速度平流和法向速度平流）的数值求解，该方法适用于水平集方法。其中最具技术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，且满足冯·诺依曼稳定性分析意义上的无条件稳定。我们还提出了一种简单的高分辨率格式，该格式对平流水平集函数的空间导数给出了TVD（总变差减小）近似。在非线性平流情形下，我们采用半隐式离散化对问题进行线性化。数值格式中隐式模板的紧致形式仅包含迎风方向未知量，这使得快速扫瞄法等高效代数求解器得以应用。针对光滑与非光滑界面演化以及速度大变化情形的数值试验表明，即使在库朗数极大时，该方法仍能保持良好的精度，且代数求解器具有快速收敛性。