论文标题
线性传输方程的有限差异方法
Finite difference methods for linear transport equations
论文作者
论文摘要
Diperna-Lions（Invent。Math。1989）建立了具有Sobolev速度场的线性传输方程的存在和唯一性结果。本文提供了对具有无差异（无界）sobolev速度字段的有限域上的两种简单有限差异方法的数学分析。第一种方法是基于具有广义双曲线量表的宽松液化型显式方案，其中无界速度场的截断及其测量估计值可以确保该方案的单调性；该方法为$ l^p $ - 施加收敛。第二种方法基于具有$ l^2 $估计的隐式方案，其中离散速度字段的离散Helmholtz-Hodge分解是确保离散问题中无差异约束的重要作用；该方法是无尺度的，$ l^2 $ - 施加收敛。我们这两种方法的关键点是获得近似解决方案的精细$ l^2 $结合，这些解决方案趋向于二型狮子狮提供的确切解决方案的标准。最后，从涉及传输方程的级别集方法的角度，将显式方案应用于情况，并使用平滑的速度字段，其中讨论了严格的离散级别级别集合的几何量。
DiPerna-Lions (Invent. Math. 1989) established the existence and uniqueness results for linear transport equations with Sobolev velocity fields. This paper provides mathematical analysis on two simple finite difference methods applied to linear transport equations on a bounded domain with divergence-free (unbounded) Sobolev velocity fields. The first method is based on a Lax-Friedrichs type explicit scheme with a generalized hyperbolic scale, where truncation of an unbounded velocity field and its measure estimate are implemented to ensure the monotonicity of the scheme; the method is $L^p$-strongly convergent. The second method is based on an implicit scheme with $L^2$-estimates, where the discrete Helmholtz-Hodge decomposition for discretized velocity fields plays an important role to ensure the divergence-free constraint in the discrete problem; the method is scale-free and $L^2$-strongly convergent. The key point for both of our methods is to obtain fine $L^2$-bounds of approximate solutions that tend to the norm of the exact solution given by DiPerna-Lions. Finally, the explicit scheme is applied to the case with smooth velocity fields from the viewpoint of the level-set method involving transport equations, where rigorous discrete approximation of geometric quantities of level sets is discussed.
