Preserving Constraints of Differential Equations by Numerical Methods Based on Integrating Factors
Chein-Shan Liu

doi:10.3970/cmes.2006.012.083
Source CMES: Computer Modeling in Engineering & Sciences, Vol. 12, No. 2, pp. 83-108, 2006
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Keywords Nonlinear dynamical system, Preserving constraints, Integrating factors, Cones, Minkowski space, Group preserving scheme
Abstract The system we consider consists of two parts: a purely algebraic system describing the manifold of constraints and a differential part describing the dynamics on this manifold. For the constrained dynamical problem in its engineering application, it is utmost important to developing numerical methods that can preserve the constraints. We embed the nonlinear dynamical system with dimensions$n$ and with$k$ constraints into a mathematically equivalent$n+k$-dimensional nonlinear system, which including$k$ integrating factors. Each subsystem of the$k$ independent sets constitutes a Lie type system of$\mathaccentV {dot}05F{\bf X}_i={\bf A}_i{\bf X}_i$ with${\bf A}_i \in so (n_i,1)$ and$n_1+\cdots +n_k=n$. Then, we can apply the exponential mapping technique to integrate the augmented systems and use the$k$ freedoms to adjust the$k$ integrating factors such that the$k$ constraints are satisfied. A similar procedure is also applied to the case when one integrates the$k$ augmented systems by the fourth-order Runge-Kutta method. Since all constraints are included in the newly developed integrating schemes, it is guaranteed that all algebraic equations that describe the manifold are satisfied up to an accuracy that is used to integrate these dynamical equations and hence a drift from the solution manifold can be avoided. Several numerical examples, including differential algebraic equations (DAEs), are investigated to confirm that the new numerical methods are effective to integrate the constrained dynamical systems by preserving the constraints.
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