Improved Structural Methods for Nonlinear Differential-Algebraic Equations via Combinatorial Relaxation
Improved Structural Methods for Nonlinear Differential-Algebraic Equations via Combinatorial Relaxation
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DOI：
10.1145/3326229.3326236
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发表时间：
2019-07
期刊：
影响因子：
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通讯作者：
Taihei Oki
中科院分区：
文献类型：
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作者：
Taihei Oki
微分代数方程（DAE）被广泛用于动态系统的建模。在微分方程数值分析中，一致初始化和指数约简是数值积分前的重要预处理。现有的DAE求解器通常采用基于组合优化的结构化预处理方法。不幸的是，如果DAE具有数字或符号取消，则结构方法失败。对于这样的DAE，已经提出了将它们修改为结构方法适用的其他DAE的方法。修改方法是基于组合松弛技术。但是，现有的修改方法仅适用于一类线性或接近线性的DAE。本文提出了两种非线性微分方程的修正方法：替代法和增广法。这两种方法都是基于组合松弛方法，适用于一个大类的非线性DAE。代入法是基于隐函数定理对方程组进行符号化求解，求出方程组的某些导数，并将解代入到系统中。增广方法不是求解方程，而是通过添加新变量和方程来修改DAE。增广方法的优点是不需要求解方程，并保持了DAE的稀疏性。数值实验表明，这两种方法都成功地修改了MATLAB中DAE求解器无法处理的高指数DAE。
Differential-algebraic equations (DAEs) are widely used for modeling of dynamical systems. In numerical analysis of DAEs, consistent initialization and index reduction are important preprocessing prior to numerical integration. Existing DAE solvers commonly adopt structural preprocessing methods, which are based on combinatorial optimization. Unfortunately, the structural methods fail if the DAE has numerical or symbolic cancellations. For such DAEs, methods have been proposed to modify them to other DAEs to which the structural methods are applicable. The modification methods are based on the combinatorial relaxation technique. Existing modification methods, however, work only for a class of DAEs that are linear or close to linear. This paper proposes two modification methods for nonlinear DAEs: the substitution method and the augmentation method. Both methods are based on the combinatorial relaxation approach and are applicable to a large class of nonlinear DAEs. The substitution method symbolically solves equations for some derivatives based on the implicit function theorem and substitutes the solution back into the system. Instead of solving equations, the augmentation method modifies DAEs by appending new variables and equations. The augmentation method has advantages that the equation solving is not needed and the sparsity of DAEs is retained. It is shown in numerical experiments that both methods successfully modify high-index DAEs that the DAE solver in MATLAB cannot handle.