半拉格朗日法廣泛應用於一階演化方程的數值計算，而半拉格朗日法是一種在空間固定網格上以拉格朗日描述法為基礎的時間離散化方法。在本文中，我們討論以有限差分法及半拉格朗日法求解一維平流方程及程函方程，其中平流方程描述曲線以給定的速度移動，程函方程則是描述曲線沿著它的法方向移動。此外我們討論有限差分法及半拉格朗日法之間的等價性。為了提升演算法的精度，我們介紹HJ-WENO方法計算導數及WENO插值方法近似函數。接著我們探討利用半拉格朗日法求解二維介面方程，其中此方程描述介面同時沿著速度場及法方向移動。我們也提出了類五階插值技術以降低插值所需的計算量。最終我們的數值結果與[7]的結果進行比較。

Semi-Lagrangian schemes are widely applied in numerical computation of first-order evolution equations. The solutions are discretized by Lagrangian formulation in time over a fixed grid in space. In this article, we discuss finite difference methods and semi-Lagrangian schemes for one dimensional scalar advection equation and eikonal equation. Note that the solution curve of the advection equation moves with a given velocity, and the solution curve of the eikonal equation moves in the normal direction. In addition, we illustrate the equivalence between semi-Lagrangian schemes and finite difference methods. To achieve higher-order accuracy, the HJ-WENO scheme is used to evaluate the spatial derivative in the finite difference method and the WENO interpolation is used to approximate the unknown function value. Then we study the semi-Lagrangian scheme for two dimensional level set equation where the interface moves in a flow velocity and a laminar velocity. We also propose a quasififth-order interpolation to reduce the computational cost. Finally, the numerical results are compared with the results in [7].

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