This paper addresses the question whether there are numerical schemes for constant-coefficient advection problems that can yield convergent solutions for an infinite time horizon. The motivation is that such methods may serve as building blocks for long-time accurate solutions in more complex advection-dominated problems. After establishing a new notion of convergence in an infinite time limit of numerical methods, we first show that linear methods cannot meet this convergence criterion. Then we present a new numerical methodology, based on a nonlinear jet scheme framework. We show that these methods do satisfy the new convergence criterion, thus establishing that numerical methods exist that converge on an infinite time horizon, and demonstrate the long-time accuracy gains incurred by this property.

翻译：本文探讨一个问题,即是否有关于常效高效的对冲问题的数字办法,能够产生无限时间范围内的趋同性解决办法。这种办法的动机是,这种方法可以作为长期精确解决更复杂的对冲主导问题的基础。在以无限的数值方法时限确立新的趋同概念之后,我们首先表明线性方法无法满足这一趋同标准。然后我们根据非线性喷气机计划框架提出新的数字方法。我们表明,这些方法确实符合新的趋同标准,从而确定存在着在无限时间范围内趋同的数字方法,并表明该属性的长期准确性收益。