New Exponential Integrators and Applications

在绝大多数科学和工程领域中，需要数值求解大型刚性常微分方程组的初值问题。传统上，隐式求解器被用来克服时间步长的稳定性限制，并与显式格式相比提高计算效率。然而，最近，指数积分器作为常用技术的一种有效替代方案出现。虽然已经提出了几个这样的积分器，但需要进行大量的研究工作来构建、分析和优化这些积分器。PI提出了一类新的指数传播Runge-Kutta型迭代格式(EPIRK)，旨在最大限度地提高求解超大型刚性方程组的计算效率。这项由这笔拨款赞助的研究将产生新的EPIRK方法，旨在在并行高性能计算平台上具有良好的可扩展性。对新积分器的研究将产生构造有效指数格式的方法，并探索这些方法的性质。该项目的一个重要部分是开发算法的自适应版本，并实施新的积分器，作为串联和并行平台的通用软件包。该软件将被用于研究等离子体物理和生物建模中的几个应用问题，并将被广泛使用。在各种科学和工程应用中，研究人员希望预测在广泛的时间和空间尺度上演化的复杂系统的行为。例如，这类系统出现在许多地质工程应用中，如储存温室气体、从高渗透率和裂隙介质中提取石油和天然气或管理地下水资源。另一个多尺度问题的例子是建模磁重联，这是天体物理和实验室等离子体中最基本的过程之一，它支配着诸如太阳耀斑、地球上的磁亚暴？S磁层和磁聚变实验的动力学等重要现象。计算机建模已成为研究这类系统的重要工具。然而，为了能够在计算机上模拟这些系统的行为，必须开发在高性能计算平台上提供非凡效率的先进数学工具。该项目将产生的数值技术将允许在标准方法无法获得的参数范围内预测具有科学和工程意义的广泛的复杂系统的行为。

The need to numerically solve initial value problems for large stiff systems of ordinary differential equations (ODEs) arises in an overwhelming majority of scientific and engineering fields. Traditionally implicit solvers have been used to overcome the stability restrictions on the time step and improve computational efficiency compared to explicit schemes. Recently, however, exponential integrators emerged as an efficient alternative to commonly used techniques. While several of such integrators have been proposed, significant research efforts are needed to construct, analyze and optimize these integrators. The PI proposed a class of new exponential propagation iterative schemes of Runge-Kutta type (EPIRK) which are designed to maximize computational efficiency for solving very large stiff systems. The research sponsored by this grant will produce new EPIRK methods designed to have good scalability on parallel high-performance computing platforms. The study of the new integrators will produce methodologies for construction of efficient exponential schemes and explore properties of these methods. An important part of the project is the development of adaptive versions of the algorithms and implementation of new integrators as a general-use software package for both serial and parallel platforms. The software will be used to study several application problems in plasma physics and biomodeling and will be made widely available.In a variety of scientific and engineering applications researchers want to predict behavior of complex systems which evolve on a wide range of temporal and spatial scales. Such systems, for instance, arise in many geo-engineering applications such as storing greenhouse gases, extraction of oil and gas from highly porous and fractured media or managing groundwater resources. Another example of a multiscale problem is modeling magnetic reconnection, one of the most fundamental processes in astrophysical and laboratory plasmas that governs such important phenomena as solar flares, magnetic substorms in the Earth?s magnetosphere and dynamics of magnetic fusion experiments. Computer modeling has become an essential tool in studying such systems. However, in order to be able to simulate the behavior of these systems on a computer advanced mathematical tools that offer exceptional efficiency on high-performance computing platforms have to be developed. The numerical techniques that will result from this project will allow prediction of the behavior of a wide range of complex systems of scientific and engineering interest over the parameter regimes inaccessible to standard methods.