带记忆的分段确定性过程的有限差分方法
A finite difference method for Piecewise Deterministic Processes with memory
AI总结：
针对带记忆的分段确定性过程的Liouville-Master方程，提出结合迎风格式和直接求积的一阶有限差分方法，并证明数值解向平衡态的收敛性。
摘要：
本文考虑带记忆的分段确定性过程的时间相关分布函数的Liouville-Master方程的数值逼近。这些方程是具有非常数系数的线性双曲型偏微分方程，其边界条件依赖于积分域内部的积分。我们通过结合偏微分方程的迎风格式和边界条件的直接求积，构造了一阶有限差分方法。我们分析了分布函数向平衡态演化时数值解的收敛性。两个具有封闭解析解的问题的数值结果说明了理论发现。
英文摘要：
In this paper the numerical approximation of solutions of Liouville-Master Equations for time-dependent distribution functions of Piecewise Deterministic Processes with memory is considered. These equations are linear hyperbolic PDEs with non-constant coefficients, and boundary conditions that depend on integrals over the interior of the integration domain. We construct a finite difference method of the first order, by a combination of the upwind method, for PDEs, and by a direct quadrature, for the boundary condition. We analyse convergence of the numerical solution for distribution functions evolving towards an equilibrium. Numerical results for two problems, whose analytical solutions are known in closed form, illustrate the theoretical finding.
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