时间相关偏微分方程的新型时空并行方法理论与应用

The research of space-time parallel methods for time-dependent partial differential equations has been a hot spot in the world over the years due to its extensive demand in engineering applications. From the perspective of numerical solution of mathematical equations, most existing methods of parallel computing are characterized by the parallel solutions to the separation or independent of space and time directions, leading to the parallel performances cannot be fully exploited. In order to efficiently apply high-performance computational methods for true engineering computing, the current urgent need is to study new efficient parallel (time, space, space-time) methods that can save communication costs more fully. This project intends to study a novel class of space-time parallel methods for time-dependent partial differential equations. This kind of methods not only can make full uses of existing parallel methods to improve the parallelism in space-time directions, but also can overcome many shortcomings of the traditional numerical solutions. The new methods are highly parallel and scalable in time and space. Meanwhile, the project intends to use finite element analysis to improve the theoretical treatment of such methods to have a wider range of mathematical impacts: study the properties of the new methods such as the well-posedness, convergence behaviors, and error estimates, etc. Finally, a space-time parallel function library based on this kind of new methods will be developed and applied to the thermal performance models with real physical parameters to provide a fast numerical simulation platform for the field of building simulation, and to improve the research and the real applications of computational and applied mathematics in China.

时间相关偏微分方程的时空并行方法在工程领域具有广泛重要意义，一直是国际上研究热点。从数学方程数值求解角度来说，现有大多并行类计算方法都是时空方向分离或独立并行求解，这容易导致并行性能无法被充分发挥与利用。为实现高性能计算方法理论的本质与深刻利用，目前十分需要研究能够更加充分节约计算通信成本的新型高效并行（时间、空间、时空）方法。本项目拟研究时间相关偏微分方程一类新颖的时空并行方法，该类方法不仅能充分利用已有并行方法提高时空方向并行度，而且能克服传统求解方法的一些缺陷。新方法具有时间和空间上的高度并行性和可扩展性。同时，本项目拟利用有限元分析手段完善该类方法的理论分析使其具有更广泛的数学应用意义：研究新方法的适定性和收敛性等算法性质及其工程应用。最后，开发基于新型时空并行方法函数库，并运用于具有真实物性参数的热性能模型，为工程领域提供快捷数值模拟平台，大力提高我国数学研究与应用水平。

时间相关偏微分方程的时空并行方法在工程领域具有广泛重要意义，一直是国际上研究热点。从数学方程数值求解角度来说，现有大多并行类计算方法都是时空方向分离或独立并行求解，这容易导致并行性能无法被充分发挥与利用。为实现高性能计算方法理论的本质与深刻利用，目前十分需要研究能够更加充分节约计算通信成本的新型高效并行（时间、空间、时空）方法。本项目研究了时间相关偏微分方程一类新颖的时空并行方法，该类方法不仅能充分利用已有并行方法提高时空方向并行度，而且能克服传统求解方法的一些缺陷。新方法具有时间和空间上的高度并行性和可扩展性。针对几类时间相关偏微分方程研究了时空并行方法及相关高效数值离散方法理论。利用有限元分析手段完善方法的理论分析，研究了新方法的适定性和收敛性等算法性质。最后将新型并行方法运用于具有真实物性参数的建筑动态热性能模型，取得了很好的并行数值模拟结果。本项目发表了SIAM Journal on Scientific Computing, SIAM Journal on Matrix Analysis and Applications, IEEE Transactions on Automatic Control, IEEE Transactions on Circuits and Systems等国际知名SCI期刊论文，所得成果是对偏微分方程快速计算手段的扩充和深化，为相关工程领域模拟软件的开发提供了算法理论基础。