An Ensemble Algorithm for Numerical Solutions to Deterministic and Random Parabolic PDEs
An Ensemble Algorithm for Numerical Solutions to Deterministic and Random Parabolic PDEs
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DOI：
10.1137/17m1131489
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发表时间：
2017-10
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影响因子：
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通讯作者：
Yan Luo;Zhu Wang
中科院分区：
文献类型：
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作者：
Yan Luo;Zhu Wang
在本文中，我们开发了一个基于集成的时间步进算法，有效地找到一组线性，二阶抛物型偏微分方程（PDE）的数值解。特别地，该组中的PDE模型可以受到不同的扩散系数、初始条件、边界条件和体积力的影响。该算法通过引入扩散系数函数的系综平均值和使用一种新的半隐式时间积分方法，将具有多个右侧向量的组转化为单个离散系统。该系统可以更有效地解决比多个线性系统与一个单一的右手侧向量。我们首先将该算法应用于确定性抛物型偏微分方程，并推导出严格的误差估计，表明该计划是一阶准确的时间和最佳准确的空间。然后，我们将其推广到随机系数抛物型偏微分方程的随机解，并提出了一个基于整体的Monte Carlo方法。通过理论分析证明了新方法的有效性。几个数值实验来说明我们的理论结果。
In this paper, we develop an ensemble-based time-stepping algorithm to efficiently find numerical solutions to a group of linear, second-order parabolic partial differential equations (PDEs). Particularly, the PDE models in the group could be subject to different diffusion coefficients, initial conditions, boundary conditions, and body forces. The proposed algorithm leads to a single discrete system for the group with multiple right-hand-side vectors by introducing an ensemble average of the diffusion coefficient functions and using a new semi-implicit time integration method. The system could be solved more efficiently than multiple linear systems with a single right-hand-side vector. We first apply the algorithm to deterministic parabolic PDEs and derive a rigorous error estimate that shows the scheme is first-order accurate in time and is optimally accurate in space. We then extend it to find stochastic solutions of parabolic PDEs with random coefficients and put forth an ensemble-based Monte Carlo method. The effectiveness of the new approach is demonstrated through theoretical analysis. Several numerical experiments are presented to illustrate our theoretical results.