单调方案对保守律的二阶收敛性
Second-order convergence of monotone schemes for conservation laws
AI总结：
本文证明了在初始数据递减且由有限分段常数组成时，一类单调的W₁合同方案在Wasserstein距离下以Δx²速率收敛。Lax-Friedrichs、Enquist-Osher和Godunov方案被证明为W₁合同方案，且为非线性保守律不连续解的二阶收敛性首次证明。
摘要：
我们证明了一类单调的W₁合同方案用于标量保守律时，在Wasserstein距离下以Δx²的速率收敛，当初始数据递减且由有限个分段常数组成时。证明了Lax-Friedrichs、Enquist-Osher和Godunov方案为W₁合同方案。展示了数值实验以说明主要结果。据我们所知，这是首次证明任何数值方法对非线性保守律不连续解的二阶收敛性。
英文摘要：
We prove that a class of monotone, \emph{$W_1$-contractive} schemes for scalar conservation laws converge at a rate of $Δx^2$ in the Wasserstein distance ($W_1$-distance), whenever the initial data is decreasing and consists of a finite number of piecewise constants. It is shown that the Lax--Friedrichs, Enquist--Osher and Godunov schemes are $W_1$-contractive. Numerical experiments are presented to illustrate the main result. To the best of our knowledge, this is the first proof of second-order convergence of any numerical method for discontinuous solutions of nonlinear conservation laws.
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