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 are 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.

中文翻译：

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耦合多物理场系统的分区指数方法

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无处不在。这项工作为多物理场问题的时间积分开发了一种新的分区指数方法。在空间中进行可能的半离散化之后，所考虑的问题类别由常微分方程组建模，其中右侧是两个分量函数的总和，每个分量函数对应于一组给定的物理过程。本文提出的分区指数方法通过指数积分器演化系统的每个组件，分区之间的信息通过耦合项进行交换。构建指数方法的传统方法，基于常数变化公式，不能直接适用于分区系统。相当，我们开发新分区指数族的方法是基于方案的一般结构加法公式。考虑了两种方法公式，一种基于右手分量函数的线性-非线性分裂，另一种基于近似雅可比矩阵。本文基于特定的 T 树族和 B 系列理论，为分区指数方案开发了经典（非刚性）阶条件理论。构建了几种实用的三阶方法，扩展了指数积分器的 Rosenbrock 型和 EPIRK 系列。还讨论了特定于将这些方法应用于反应扩散系统的几种实施优化。