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Compact integration factor methods in high spatial dimensions

机译：高空间维度的紧凑积分因子方法

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摘要

The dominant cost for integration factor (IF) or exponential time differencing (ETD) methods is the repeated vector-matrix multiplications involving exponentials of discretization matrices of differential operators. Although the discretization matrices usually are sparse, their exponentials are not, unless the discretization matrices are diagonal. For example, a two-dimensional system of N x N spatial points, the exponential matrix is of a size of N-2 x N-2 based on direct representations. The vector-matrix multiplication is of O(N-2), and the storage of such matrix is usually prohibitive even for a moderate size N. In this paper, we introduce a compact representation of the discretized differential operators for the IF and ETD methods in both two- and three-dimensions. In this approach, the storage and CPU cost are significantly reduced for both IF and ETD methods such that the use of this type of methods becomes possible and attractive for two- or three-dimensional systems. For the case of two-dimensional systems, the required storage and CPU cost are reduced to O(N-2) and O(N-3), respectively. The improvement on three-dimensional systems is even more significant. We analyze and apply this technique to a class of semi-implicit integration factor method recently developed for stiff reaction-diffusion equations. Direct simulations on test equations along with applications to a morphogen system in two-dimensions and an intra-cellular signaling system in three-dimensions demonstrate an excellent efficiency of the new approach. (C) 2008 Elsevier Inc. All rights reserved.
机译：积分因子（IF）或指数时间微分（ETD）方法的主要成本是重复的矢量矩阵乘法，其中涉及差分算子的离散化矩阵的指数。尽管离散化矩阵通常是稀疏的，但它们的指数不是，除非离散化矩阵是对角的。例如，在N x N个空间点的二维系统中，基于直接表示，指数矩阵的大小为N-2 x N-2。向量矩阵乘积为O（N-2），即使对于中等大小的N，这种矩阵的存储通常也是禁止的。在本文中，我们为IF和ETD方法引入了离散差分算子的紧凑表示在二维和三维上。在这种方法中，IF和ETD方法的存储和CPU成本都大大降低，从而使这种方法的使用成为可能，并且对二维或三维系统具有吸引力。对于二维系统，所需的存储空间和CPU成本分别降低为O（N-2）和O（N-3）。三维系统的改进甚至更加重要。我们分析此技术并将其应用于最近针对刚性反应扩散方程式开发的一类半隐式积分因子方法。对测试方程式的直接仿真以及在二维中对形态发生子系统的应用以及在三维中对细胞内信号系统的应用证明了这种新方法的出色效率。 （C）2008 Elsevier Inc.保留所有权利。

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