Many time-dependent differential equations are equipped with invariants. Preserving such invariants under discretization can be important, e.g., to improve the qualitative and quantitative properties of numerical solutions. Recently, relaxation methods have been proposed as small modifications of standard time integration schemes guaranteeing the correct evolution of functionals of the solution. Here, we investigate how to combine these relaxation techniques with efficient step size control mechanisms based on local error estimates for explicit Runge-Kutta methods. We demonstrate our results in several numerical experiments including ordinary and partial differential equations.

翻译：许多含时微分方程具有不变量。在离散化过程中保持这些不变量对于提高数值解的定性和定量性质至关重要。近年来，松弛方法被提出作为标准时间积分方案的小型修正，以确保解中泛函的正确演化。本文研究了如何将这些松弛技术与基于局部误差估计的显式龙格-库塔方法的高效步长控制机制相结合。我们通过包括常微分方程和偏微分方程在内的多个数值实验验证了所得结果。