高性能谱/谱元方法研究及其在多相复杂流体中的应用

There are many difficult and challenging issues for Spectral methods although it is now applied in many areas widely. In order to increase the adaptability and flexibility, the spectral element methods obtain some ideas of domain decomposition from the Finite Element methods. Thus it is very suitable for the architecture development of modern parallel computers, and in particular, of the large-scale CPU-GPU/MIC heterogeneous systems. The free interface problem of immiscible and incompressible fluid systems are very common in nature and industry. Its mathematical model is usually a highly nonlinear and coupled multi-physical system with essential discontinuities. It is very challenging to capture the sharp interface of multiphase complex flows accurately. Spectral and spectral element methods have already shown their advantages when comparing to the lower order methods, and become a hot topic in the area of numerical simulations of multiphase complex flows. Oriented from the phase field method for 3-D multiphase complex fluid flow systems, this proposal will first develop numerical schemes, algorithms and theory of the spectral methods on some non-tensorial 3-D fundamental domains. Then the high-performance spectral element methods on unstructured grids will be developed. Some parallel preconditioned iterative solvers will be proposed; and the p- and hp-adaptive algorithms will be designed. Finally, the proposed algorithms on spectral and spectral element methods will be applied to the multiphase complex fluid system to obtain high-performance simulations on large large-scale heterogeneous systems.

谱方法的应用日益广泛，但一直面临着诸多困难和挑战。为了克服传统谱方法适应性和灵活性不足的缺点，谱元方法则借鉴了有限元方法区域分解的思想，符合当今大型异构并行计算机系统的结构特点和发展趋势。自然界和工业过程中广泛存在着不相掺混的两相或多相流体的界面问题，其研究对象是非线性、多物理量耦合且本质上强间断的偏微分方程组所描述的复杂系统。如何正确描述精细而锐利的界面一直是多相流数值模拟领域的一个难点，谱/谱元方法的高精度优势在这里大有作为。本项目针对三维多相复杂流体这一具体应用问题，主要研究三维基本非张量积区域单元上的新型谱方法的格式、算法和理论，发展非结构化网格上的高性能谱方法及其并行预条件子迭代求解算法，设计其自适应（特别是提高/降低基函数次数的p-自适应）算法，并最终应用于多相复杂流体的研究，最终实现大型异构系统下的高性能数值模拟。

本项目研究非张量积基本区域上的新型谱方法的格式、算法和理论，发展非结构化网格上的高性能谱元方法，设计其自适应算法，实现高性能数值模拟，并应用于多相复杂流体相场方法的研究。本项目主要在以下方面取得了突破:(1)发展基本区域上Galerkin谱方法特别是基于Sobolev正交多项式与函数的谱方法的格式、快速求解算法和数值分析理论；(2) 发展了复杂区域上二阶椭圆方程与Stokes方程的高效的并行可扩展的广义Koornwinder多项式三角谱元方法，特别是证明了其LBB常数的下确界; (3) 发展了反幂奇异势函数薛定谔方程特别是凹角区域上薛定谔方程的高效的并行可扩展谱元方法的离散格式、并行求解算法和数值分析理论; (4) 构造了四阶方程C1协调三角形与四边形谱元；(5) 发展了不可压缩粘性流三相相场模型并构造了能量稳定解耦离散格式。本项目研究成果发表于 Mathematics of Computation, SIAM Journal on Scientific Computing, Applied and Computational Harmonic Analysis 等学术刊物，它极大拓展了谱方法的区域适应性与问题适用性。