Runge-Kutta methods for dissipative and gradient dynamical systems
Runge-Kutta methods for dissipative and gradient dynamical systems
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DOI：
10.1137/0731075
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发表时间：
1994-10
影响因子：
2.9
通讯作者：
A. R. Humphries;A. Stuart
中科院分区：
文献类型：
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作者：
A. R. Humphries;A. Stuart
考虑了用固定时间步长Runge-Kutta方法对耗散初值问题的数值逼近，并比较了数值解和精确解的渐近性质。研究了一类通过内积诱导耗散性的常微分方程。这一类自然出现在许多有限维应用中（如洛伦兹方程），也来自应用数学中各种偏微分方程的空间离散化。结果表明，由代数稳定的方法定义的数值解具有吸收集，因此对于任何固定的步长h > 0是耗散的。当h足够小，从而存在全局吸引子A_h时，数值解将动力系统定义在吸收集上，并建立了A_h在h = 0时的上连续性，表明当h很小时，数值吸引子上的每一点都接近于真实全局吸引子A上的一点.在全局Lipschitz问题的附加假设下，证明了当h充分小时，任何带正权的方法在整个空间上都定义了一个耗散动力系统，并再次建立了A_h在h = 0处的上连续性.对于具有全局Lipschitz向量场的梯度系统，证明了任何Runge-Kutta方法在h足够小时保持梯度结构。对于一般的耗散梯度系统，它示出代数稳定的方法保持吸收集内的梯度结构为h足够小。数值吸引子的收敛性进行了研究，并为一个耗散梯度系统的双曲型平衡，在h = 0下的连续性。因此，对于这样的系统，当h → 0时，A_h收敛于Hausdorff度量中的A。
The numerical approximation of dissipative initial value problems by fixed time-stepping Runge–Kutta methods is considered and the asymptotic features of the numerical and exact solutions are compared. A general class of ordinary differential equations, for which dissipativity is induced through an inner product, is studied throughout. This class arises naturally in many finite dimensional applications (such as the Lorenz equations) and also from the spatial discretization of a variety of partial differential equations arising in applied mathematics. It is shown that the numerical solution defined by an algebraically stable method has an absorbing set and is hence dissipative for any fixed step-size h > 0. The numerical solution is shown to define a dynamical system on the absorbing set if h is sufficiently small and hence a global attractor A_h exists; upper-semicontinuity of A_h at h = 0 is established, which shows that, for h small, every point on the numerical attractor is close to a point on the true global attractor A. Under the additional assumption that the problem is globally Lipschitz, it is shown that if h is sufficiently small any method with positive weights defines a dissipative dynamical system on the whole space and upper semicontinuity of A_h at h = 0 is again established. For gradient systems with globally Lipschitz vector fields it is shown that any Runge–Kutta method preserves the gradient structure for h sufficiently small. For general dissipative gradient systems it is shown that algebraically stable methods preserve the gradient structure within the absorbing set for h sufficiently small. Convergence of the numerical attractor is studied and, for a dissipative gradient system with hyperbolic equilibria, lower semicontinuity at h = 0 is established. Thus, for such a system, A_h converges to A in the Hausdorff metric as h → 0.