Matched alternating direction implicit (ADI) schemes for solving the nonlinear Poisson-Boltzmann equation with complex dielectric interfaces

该项目的目标是开发嵌入交替方向隐式(ADI)框架的二阶界面方法，用于求解具有复杂介电界面的三维非线性Poisson-Boltzmann(PB)方程。效率和精度是数值求解非线性PB方程的两大难点。这种对效率的关注源于在苛刻的应用中求解PB方程的需要，例如具有大空间自由度的系统的一次性解，和/或动态模拟中的百万次解。精度的问题是由于PB模型的各种具有挑战性的特征，包括分段定义的介电常数、强非线性、奇点电荷和复杂的介电界面。如果不解决这些功能，就必须使用精细网格进行可靠的模拟，这反过来又会降低效率。在这个项目中，将基于适当的正则化设置构造一个新的伪瞬变延拓公式，以便对奇异电荷进行解析表示。PB方程的非线性项将通过时间分裂技术进行精确积分。为了处理分段介电常数，将对3D界面条件进行张量积分解，以基本上导出一维跳跃条件，从而使介电界面可以沿每个笛卡尔方向以交替的方式被容纳。将开发快速代数求解器来求解每个笛卡尔方向的矩阵。因此，所提出的匹配ADI方法不仅保持了笛卡尔网格的简单性和Thomas算法的效率，而且在空间上达到了求解复杂介电界面的二阶精度。静电相互作用不仅对于分子水平上的生物和化学体系和过程的研究，而且对于纳米尺度的半导体器件的设计都是至关重要的。通过平均力方法隐式计算静电相互作用的PB模型，可以令人惊讶地很好地描述带电系统的静电学。该模型在科学和工程中有着广泛的应用，如在界面和胶体科学中模拟带电聚合物和表面活性剂，在纳米技术中研究超大规模集成电路(VLSI)半导体器件上的晶体管，在分子生物学中分析包括蛋白质和DNA在内的溶剂化生物分子的结构、功能和动力学。拟议的数学建模、算法开发和数值计算将解决涉及计算数学、化学、生物和电气工程的跨学科领域的关键科学挑战。计划中的研究活动将为计算数学带来新的进步，并为各种物理、化学和生物系统/设备的静电分析带来可靠的模拟工具。此外，该项目将为追求科学和工程职业的学生提供跨学科的研究和培训机会。

The goal of the proposed project is to develop second order interface methods embedded in the alternating direction implicit (ADI) framework for solving the 3D nonlinear Poisson-Boltzmann (PB) equation with complex dielectric interfaces. Efficiency and accuracy are known to be the two major difficulties for solving the nonlinear PB equation numerically. The efficiency concern stems from the needs for solving the PB equation in demanding applications, such as one-time solution to systems with large spatial degrees of freedom, and/or million-time solutions in dynamical simulations. The accuracy concern is due to various challenging features of the PB model, including piecewisely-defined dielectric constants, a strong nonlinearity, singular point charges, and complex dielectric interfaces. Without addressing these features, fine meshes have to be used for a reliable simulation, which in turn impairs efficiency. In this project, a new pseudo-transient continuation formulation will be constructed based on a suitable regularization setting so that the singular charges are represented analytically. The nonlinear term of the PB equation will be integrated exactly with time splitting techniques. To deal with piecewise dielectric constants, a tensor product decomposition of 3D interface conditions will be carried out to derive essentially 1D jump conditions so that the dielectric interface can be accommodated along each Cartesian direction in an alternating manner. Fast algebraic solvers will be developed for solving matrices of each Cartesian direction. Consequently, the proposed matched ADI approaches not only maintain both the simplicity of Cartesian grids and the efficiency of the Thomas algorithm, but also achieve spatially second order accuracy in resolving complex dielectric interfaces.The electrostatic interactions are vital not only for the study of biological and chemical systems and processes at the molecular level, but also for the design of semiconductor devices at the nanoscale. The PB model, in which the electrostatic interactions are computed implicitly via a mean force approach, can surprisingly well describe the electrostatics of a charged system. This model finds broad applications in science and engineering, such as modeling the charged polymers and surfactants in interface and colloid science, studying transistors on very large scale integration (VLSI) semiconductor devices in nanotechnology, and analyzing structure, function, and dynamics of solvated biomolecules including proteins and DNAs in molecular biology. The proposed mathematical modeling, algorithm development, and numerical computations will address key scientific challenges in interdisciplinary fields involving computational mathematics, chemistry, biology, and electrical engineering. The planned research activities will bring new advances to computational mathematics and lead to reliable simulation tools for the electrostatic analysis of various physical, chemical, and biological systems/devices. In addition, this project will provide interdisciplinary research and training opportunities for students pursing careers in science and engineering.