具有点延迟的一阶双曲型偏微分方程的有限差分逼近
Finite difference approximations for the first-order hyperbolic partial differential equation with point-wise delay
AI总结：
本文构造了基于Lax-Friedrichs和Leap-Frog有限差分格式的显式数值方法，用于求解具有空间点延迟或超前的一阶双曲型偏微分方程，并讨论了其相容性、稳定性和收敛性。
摘要：
构造了基于Lax-Friedrichs和Leap-Frog有限差分逼近的显式数值方法，用于求解具有点延迟或超前（即空间平移）的一阶双曲型偏微分方程的数值解。涉及点延迟和超前的微分方程模拟了连续神经元放电之间时间间隔的分布。我们构造了高阶数值逼近，并讨论了它们的相容性、稳定性和收敛性。本文构造的数值逼近在CFL条件下是相容的、稳定的且收敛的。我们还将方法推广到更高空间维度。包含一些测试实例以说明我们的方法。这些实例验证了理论估计，并显示了点延迟对解的影响。
英文摘要：
Explicit numerical methods based on Lax-Friedrichs and Leap-Frog finite difference approximations are constructed to find the numerical solution of the first-order hyperbolic partial differential equation with point-wise delay or advance, i.e., shift in space. The differential equation involving point-wise delay and advance models the distribution of the time intervals between successive neuronal firings. We construct higher order numerical approximations and discuss their consistency, stability and convergence. The numerical approximations constructed in this paper are consistent, stable under CFL condition, and convergent. We also extend our methods to the higher space dimensions. Some test examples are included to illustrate our approach. These examples verify the theoretical estimates and shows the effect of point-wise delay on the solution.
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