耦合多物理场系统的分区指数方法
Partitioned Exponential Methods for Coupled Multiphysics Systems
摘要：
涉及两个或多个耦合物理现象的多物理场问题在科学和工程中普遍存在。本文提出了一种新的多物理场时间积分的分区指数方法。在空间中进行可能的半离散化之后，所考虑的一类问题由常微分方程系统建模，其中右侧是两个分量函数的总和，每个分量函数对应于给定的一组物理过程。本文提出的分区指数方法通过指数积分器来演化系统的每个组件，并且分区之间的信息通过耦合项来交换。基于常数变分公式构造指数方法的传统方法不能直接应用于分区系统。相反，我们的方法来开发新的分区指数的家庭是基于一般结构添加剂配方的计划。两种方法配方被认为是，一个基于线性-非线性分裂的右手分量函数，另一个基于近似雅可比矩阵。基于T-树和B-级数理论，发展了经典的（非刚性）分块指数格式的序条件理论。几个实用的方法，三阶构造扩展Rosenbrock型和EPIRK族的指数积分。具体到这些方法的反应扩散系统的应用程序的几个实施优化进行了讨论。数值实验表明，在某些问题上，新的分块指数方法比传统的未分块指数方法有更好的性能。
英文摘要：
Multiphysics problems involving two or more coupled physical phenomena are ubiquitous in science and engineering. This work develops a new partitioned exponential approach for the time integration of multiphysics problems. After a possible semi-discretization in space, the class of problems under consideration is modeled by a system of ordinary differential equations where the right-hand side is a summation of two component functions, each corresponding to a given set of physical processes. The partitioned-exponential methods proposed herein evolve each component of the system via an exponential integrator, and information between partitions is exchanged via coupling terms. The traditional approach to constructing exponential methods, based on the variation-of-constants formula, is not directly applicable to partitioned systems. Rather, our approach to developing new partitioned-exponential families is based on a general-structure additive formulation of the schemes. Two method formulations are considered, one based on a linear-nonlinear splitting of the right hand component functions, and another based on approximate Jacobians. The paper develops classical (non-stiff) order conditions theory for partitioned exponential schemes based on particular families of T-trees and B-series theory. Several practical methods of third order are constructed that extend the Rosenbrock-type and EPIRK families of exponential integrators. Several implementation optimizations specific to the application of these methods to reaction-diffusion systems are also discussed. Numerical experiments reveal that the new partitioned-exponential methods can perform better than traditional unpartitioned exponential methods on some problems.
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