Highly efficient and accurate numerical schemes for nonlinear gradient flows with energy stability

本课题的目标是研究有效且稳健的工具来数值模拟某些非线性梯度流动。这些数值格式将为研究物理、材料工程和生物科学中不同模型的复杂长期动力学提供高效的求解器。这项工作预计将对许多科学学科产生直接和立即的影响。这些非线性梯度流的大时间尺度模拟对于理解材料在原子和纳米尺度上的相变、生物生长发育的复杂过程以及两相流中复杂的拓扑变化等具有重要意义。PI开发的一些数值格式已有效地应用于大型、多学科的科学项目；该项目的合作部分将利用肿瘤生长模拟血管生成分析方面的专业知识，并有望在医学科学界产生重大影响。PI将在公共领域提供计算工具，以便研究人员可以直接访问一些已开发的算法。通过该项目的工作，研究生将在高性能科学计算、数值数学和建模方面得到广泛的训练。在所提出的梯度流模型中，物理能量可以分解为纯凸和纯凹两部分。PI考虑凸分裂（CS）数值格式，包括时间上的一阶和二阶精确分裂，以及空间上的有限差分、有限元和伪谱近似。一个主要的挑战是为这些高度非线性的CS方案设计真正有效的求解器。例如，可以应用直接非线性多网格求解器。作为一种替代方法，在这项工作中提出了线性迭代算法，并期望这些基于线性迭代的求解器具有收缩映射性质。换句话说，虽然所表述的CS格式是非线性的，但是基于线性迭代的算法可以近似这种高度非线性系统的解，并且具有几何收敛率。本文还将对非线性求解器和线性迭代算法进行详细的比较。

The goal of this project is to study efficient and robust tools to numerically simulate certain nonlinear gradient flows. These numerical schemes will yield highly efficient solvers to study the complicated long-time dynamics of different models in physics, materials engineering, and biological sciences. This work is expected to have a direct and immediate impact on many scientific disciplines. The large time scale simulation of these nonlinear gradient flows is vital for understanding phase transformations of materials at the atomic and nanometer scales, the complex processes in biological growth and development, and the complicated topological change involved in two-phase flows, etc. Some numerical schemes developed by the PI have been efficiently applied in large scale, multi-discipline scientific projects; a collaborative part of the project will leverage expertise in angiogenesis analysis of tumor growth simulation and is expected to lead to significant impact in the medical sciences community. The PI will make the computational tools available in the public domain so that researchers will have direct access to some of the developed algorithms. Through work in the project, a graduate student will receive extensive training in high-performance scientific computing, numerical mathematics, and modeling. In the proposed gradient flow models, the physical energy can be decomposed into purely convex and concave parts. The PI considers convex splitting (CS) numerical schemes, including both the 1st and 2nd order accurate splittings in time, and finite difference, finite element and pseudospectral approximations in space. A major challenge is designing truly efficient solvers for these highly nonlinear CS schemes. For example, a direct nonlinear multigrid solver can be applied. As an alternate approach, linear iterative algorithms are proposed in this work, and a contraction mapping property is expected for these linear iteration based solvers. In other words, although the formulated CS scheme is nonlinear, a linear iteration based algorithm can be used to approximate solutions of this highly nonlinear system with a geometric convergence rate. A detailed comparison between a nonlinear solver and linear iteration algorithm will also be conducted.