Patankar-type schemes are linearly implicit time integration methods designed to be unconditionally positivity-preserving. However, there are only little results on their stability or robustness. We suggest two approaches to analyze the performance and robustness of these methods. In particular, we demonstrate problematic behaviors of these methods that, even on very simple linear problems, can lead to undesired oscillations and order reduction for vanishing initial condition. Finally, we demonstrate in numerical simulations that our theoretical results for linear problems apply analogously to nonlinear stiff problems.
翻译:Patankar型计划是线性隐含的时间整合方法,旨在无条件保护积极性。然而,这些方法的稳定性或稳健性几乎没有什么结果。我们建议用两种方法分析这些方法的性能和稳健性。特别是,我们展示了这些方法的有问题的行为,即使存在非常简单的线性问题,也可能导致不理想的振荡和减少秩序,从而导致初始状态的消失。最后,我们在数字模拟中证明,我们线性问题的理论结果与非线性硬性问题类似。