论文标题
保持阳性的自适应runge-kutta方法
Positivity-Preserving Adaptive Runge-Kutta Methods
论文作者
论文摘要
许多重要的微分方程模型的数量必须保持正值或保持在某些有限的间隔。当模型通过数值求解时，可能不会保留这些边界。我们建议通过应用runge-kutta集成来确保阳性或其他界限，在该集成中，在该集成中，将使用该方法权重以实施界限。在计算阶段衍生物后，在每个步骤中选择权重，以（并在可能的情况下）保留该方法的准确性顺序。权重的选择是通过线性程序的解决方案给出的。我们通过考虑增加进一步的约束来研究选择权重的不同方法。我们还提供了具有扰动权重的runge-kutta方法的性能的分析。数值示例证明了该方法的有效性，包括应用于僵硬和非Stift问题。
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.
