A numerical method for a nonlocal diffusion equation with additive noise
A numerical method for a nonlocal diffusion equation with additive noise
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DOI：
10.1007/s40072-022-00262-w
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发表时间：
2021-08
期刊：
影响因子：
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通讯作者：
G. Medvedev;G. Simpson
中科院分区：
文献类型：
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作者：
G. Medvedev;G. Simpson
我们考虑一个非局部演化方程表示连续极限的一个大的相互作用粒子的合奏图强迫噪声。连续介质模型的两个主要成分是一个非局部项和一个Q-Wiener过程，分别描述网络中粒子之间的相互作用和随机强迫。网络的连通性由一个平方可积函数给出，称为graphon。我们证明了连续介质模型的初值问题是适定的。此外，我们构造半离散（空间离散和时间连续）和全离散计划的非局部模型。前者是通过一个不连续的Galerkin方法和后者是基于进一步离散时间使用Euler-Maruyama方法。我们证明了收敛性，并估计在每种情况下的收敛速度。对于半离散格式，收敛速度表示在正则性的graphon，Q-Wiener过程，和初始数据。我们在广义Lipschitz空间中工作，这使得我们可以处理具有较低正则性数据的模型。这对于应用程序很重要，因为许多有趣的连接类型，包括小世界和幂律，都是由不光滑的图子表示的。另一方面，全离散方案的误差分析表明，对于应用科学中常见的一些模型，Euler-Maruyama方法的标准估计比预测的收敛速度更快。收敛速度分析补充了详细的数值实验，这与我们的分析结果是一致的。作为一个副产品，这项工作提出了一个严格的理由采取连续极限的一大类相互作用的动力系统的图受噪声。
We consider a nonlocal evolution equation representing the continuum limit of a large ensemble of interacting particles on graphs forced by noise. The two principle ingredients of the continuum model are a nonlocal term and a Q-Wiener process describing the interactions among the particles in the network and stochastic forcing, respectively. The network connectivity is given by a square integrable function called a graphon. We prove that the initial value problem for the continuum model is well-posed. Further, we construct semidiscrete (discrete in space and continuous in time) and fully discrete schemes for the nonlocal model. The former is obtained by a discontinuous Galerkin method and the latter is based on further discretizing time using the Euler–Maruyama method. We prove convergence and estimate the rate of convergence in each case. For the semidiscrete scheme, the rate of convergence is expressed in terms of the regularity of the graphon, the Q-Wiener process, and the initial data. We work in generalized Lipschitz spaces, which allows us to treat models with data of lower regularity. This is important for applications as many interesting types of connectivity, including small-world and power-law, are expressed by graphons that are not smooth. The error analysis of the fully discrete scheme, on the other hand, reveals that for some models common in applied science, one has a higher speed of convergence than that predicted by the standard estimates for the Euler–Maruyama method. The rate of convergence analysis is supplemented with detailed numerical experiments, which are consistent with our analytical results. As a by-product, this work presents a rigorous justification for taking continuum limit for a large class of interacting dynamical systems on graphs subject to noise.