Many important differential equations model quantities whose value must remain positive or stay in some bounded interval. These bounds may not be preserved when the model is solved numerically. We propose to ensure positivity or other bounds by applying Runge-Kutta integration in which the method weights are adapted in order to enforce the bounds. The weights are chosen at each step after calculating the stage derivatives, in a way that also preserves (when possible) the order of accuracy of the method. The choice of weights is given by the solution of a linear program. We investigate different approaches to choosing the weights by considering adding further constraints. We also provide some analysis of the properties of Runge-Kutta methods with perturbed weights. Numerical examples demonstrate the effectiveness of the approach, including application to both stiff and non-stiff problems.

翻译：许多重要的差别方程式模型数量,其价值必须保持正数或保持一定的界限。当模型用数字方式解决时,这些界限可能无法保留。我们提议通过应用龙格-库塔集成法确保积极性或其他界限,在这种集成法中调整方法重量以强制执行界限。权重在计算阶段衍生物后的每一步选择,同时(在可能时)保持方法的准确性。权重的选择由线性程序的解决办法来决定。我们研究选择权重的不同方法,考虑增加进一步的限制。我们还对龙格-库塔方法的特性进行了一些分析,并附有边际重量。数字实例显示了该方法的有效性,包括对僵硬和非硬问题的应用。