Optimal balance is a non-asymptotic numerical method to compute a point on the slow manifold for certain two-scale dynamical systems. It works by solving a modified version of the system as a boundary value problem in time, where the nonlinear terms are adiabatically ramped up from zero to the fully nonlinear dynamics. A dedicated boundary value solver, however, is often not directly available. The most natural alternative is a nudging solver, where the problem is repeatedly solved forward and backward in time and the respective boundary conditions are restored whenever one of the temporal end points is visited. In this paper, we show quasi-convergence of this scheme in the sense that the termination residual of the nudging iteration is as small as the asymptotic error of the method itself, i.e., under appropriate assumptions exponentially small. This confirms that optimal balance in its nudging formulation is an effective algorithm. Further, it shows that the boundary value problem formulation of optimal balance is well posed up at most a residual error as small as the asymptotic error of the method itself. The key step in our proof is a careful two-component Gronwall inequality.
翻译:最佳平衡是一种非简单的数字方法,用于计算某些两尺度动态系统慢速元数的点数。它的工作方式是作为边界值问题在时间上解决系统修改版本,因为非线性术语从零度上升为完全非线性动态。然而,一个专用的边界值求解器往往无法直接使用。最自然的替代方法是一个杂交求解答器,问题在时间上反复向前和向后解决,每当访问一个时间端点时恢复各自的边界条件。在本文中,我们显示了这个办法的准趋同性,其含义是,非线性迭代法的终止剩余与方法本身的无序错误一样小,也就是说,在适当的假设下,该方法本身的断线性误差是极小的。这证实了其编译法中的最佳平衡是一种有效的算法。此外,它表明边界值问题形成的最佳平衡在多数情况下构成的剩余错误,其大小与方法本身的防波性误差一样小。我们所要小心地证明的关键的G- 平方是谨慎的两步。