带跳随机偏微分方程分裂算法收敛性研究  

ON CONVERGENCE OF SPLITTING-UP ALGORITHM FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH JUMP

作　　者：张凤山 杨祖豪 邹永魁[1] Zhang Fengshan;Yang Zuhao;Zou Yongkui(JLU,School of Mathematics,Jilin University,Changchun 130012,China)

机构地区：[1]JLU,吉林大学数学学院,长春130012

出　　处：《计算数学》2023年第4期401-414,共14页Mathematica Numerica Sinica

基　　金：吉林省科技发展计划项目(20210201015GX,20210201078GX);国家自然科学基金项目(12171199,11971198);国家重点研发计划项目(2020YFA0714101,2020YFA0713601)资助.

摘　　要：本文对一类由维纳过程和泊松过程驱动的随机偏微分方程的数值求解方法进行了研究.我们应用分裂算法的思想将方程分裂为三个简单的子方程,并利用它们的解算子构造了分裂近似解,同时研究了其收敛性和收敛阶.之后我们用有限元方法和有限差分方法分别对空间变量和时间变量进行了离散化,结合分裂算法构造了求解跳跃随机偏微分方程的全离散分裂近似解,给出了误差分析结果.最后我们用数值实验验证了算法的收敛阶.In this paper,a new numerical method for solving a class of stochastic partial differential equations driven by Wiener process and Poisson process is derived and analyzed.By means of a splitting-up technique we decompose the stochastic partial differential equation into three simple sub-equations,and construct a splitting-up approximate solution based on three solution operators.We also investigate the convergence and convergence rate of the approximate solution.Then we discretize the spatial and temporal variables with the finite element method and the finite difference scheme,respectively.Combining with the splittingup method,we set up a fully discretized splitting-up approximate solution for solving the stochastic partial differential equations and present its convergence property.Finally,we provide some numerical experiments to verify the theoretical convergence order.

关 键 词：维纳过程 泊松过程 分裂算法 收敛阶 

分 类 号：O241.82[理学—计算数学]