强非线性偏微分方程基于梯度重构的新型算法

Abstract: Nonlinear partial differential equations is crucial for the development of physics and all sciences in large. Its study has extremely important scientific value and significance in economy. However, for some complex nonlinear problems, even with today’s high performance parallel supercomputers, the traditional way of overly relying on mesh refinement, which usually results in super-large scale massive computation, faces a bottleneck. The study for numerical approximation of nonlinear, especially fully nonlinear PDEs is under-developed comparing with linear PDEs, and there is still ample room for improvement in basic algorithms for fully nonlinear problems. This project uses the Monge Ampere equation and p-Laplace equation as models, combines the efforts of researchers from different disciplines such as mathematical modeling, computational mathematics, and software engineering from two research institutes, studies numerical methods for fully nonlinear PDES form a new angle. We utilize newly developed gradient recovery techniques to treat higher derivative terms as well as to post-process the computed data in fully nonlinear PDEs. Our goal is to establish scientifically sound and practically feasible mathematical model, construct problem driven self-adaptive grids combined with proper polynomial degree (the so-called hp-adaptive), design high accurate approximation algorithms, develop nonlinear iterative methods, raise overall ability of computation, and carry on high efficient numerical simulation.

非线性偏微分方程对于科学和工程的发展起着举足轻重的作用，其算法和理论研究具有极其重要的科学价值和经济意义。对完全非线性问题与高维大型耦合型非线性问题，传统的过度依赖网格加密导致超大规模计算的算法研究也已经遇到严重的瓶颈。必须从计算原理的新角度，研究非线性、尤其是完全非线性偏微分方程的共性基础算法与理论，并获得突破与提升。本项目以Monge-Ampere方程、p-Laplace 方程以及网络神经动力学方程组为研究对象，旨在通过国内参与单位（建模、计算数学与软件科学）科研人员的优势联合，探索一条具有我国自主特色的实用化的高效计算思路：利用近年发展起来的并被工程界采用的梯度重构技术，提升非线性项数值高阶导数的连续性并进行后处理, 匹配合理的自适应网格与多项式阶数，发展大时间步长稳定的高精度离散格式，设计非线性迭代算法，以提升问题求解的整体性能，并在高性能计算平台上实现高效数值模拟。

本项目以Monge-Ampère方程、非线性Ginzburg-Landau方程、非线性Hamilton方程等强非线性方程为对象，通过项目成员的精细分工与优势联合，发展了科学与工程计算界切实可用的基于梯度重构的新型计算方法，设计其高精度离散逼近格式及其高效数值求解算法，并现实在高性能计算平台上的数值模拟。项目组从偏微分方程数值解原理的新角度研究强非线性偏微分方程数值求解的基础理论，探索出了一条可实用化的高效计算思路。通过将超收敛理论扩展到有限体积法、DG方法、以及谱方法，实现了对各种偏微分方程计算方法的超收敛现象的系统深入研究。设计时间分数阶导数快速算法，计算复杂度和存储复杂度分别从O(N2)和O(N) 的量级减少至O(N log N)和O(log N)，其中N表示总的时间步数。该算法及相关理论在诸多方面有着重要应用，得到了业内的广泛关注。.项目组在若干前沿问题研究上取得了突破性进展，获得了一些原创性研究成果，完成了一系列高质量的学术论文，在国内外同行中收获了盛誉，并培养了众多优秀博士后与研究生，最终完成了本项目拟定的各项研究目标。

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