Recently, an approach known as relaxation has been developed for preserving the correct evolution of a functional in the numerical solution of initial-value problems, using Runge-Kutta methods. We generalize this approach to multistep methods, including all general linear methods of order two or higher, and many other classes of schemes. We prove the existence of a valid relaxation parameter and high-order accuracy of the resulting method, in the context of general equations, including but not limited to conservative or dissipative systems. The theory is illustrated with several numerical examples.

翻译：最近,我们开发了一种称为“放松”的方法,以利用龙格-库塔方法,在初始价值问题的数字解决方案中保持功能的正确演变。我们将这一方法推广到多步骤方法,包括所有一般的二级或二级以上线性方法,以及许多其他类别的方案。我们证明,在一般方程中,存在有效的放松参数和由此产生的方法的高级准确性,包括但不限于保守或消散系统。这个理论用几个数字例子加以说明。