We generalize the idea of relaxation time stepping methods in order to preserve multiple nonlinear conserved quantities of a dynamical system by projecting along directions defined by multiple time stepping algorithms. Similar to the directional projection method of Calvo et. al., we use embedded Runge-Kutta methods to facilitate this in a computationally efficient manner. Proof of the accuracy of the modified RK methods and the existence of valid relaxation parameters are given, under some restrictions. Among other examples, we apply this technique to Implicit-Explicit Runge-Kutta time integration for the Korteweg-de Vries equation and investigate the feasibility and effect of conserving multiple invariants for multi-soliton solutions.

翻译：我们推广了松弛时间步进方法的思想，通过沿由多个时间步进算法定义的方向进行投影，以保持动力系统的多个非线性守恒量。类似于卡尔沃等人的方向投影方法，我们利用嵌入式龙格-库塔方法以计算高效的方式实现这一目标。在若干限制条件下，证明了修正的龙格-库塔方法的精度及有效松弛参数的存在性。作为示例，我们将该技术应用于科特韦格-德弗里斯方程的隐式-显式龙格-库塔时间积分，并研究了多孤子解中守恒多个不变量的可行性与效果。