论文标题
关于无条件积极和线性不变的稳定性保留时间整合方案
On the Stability of Unconditionally Positive and Linear Invariants Preserving Time Integration Schemes
论文作者
论文摘要
无条件地保留基础微分方程系统的阳性和线性不变性的高阶时间整合方法不能属于常规线性方法的类别。这对此类方法的稳定性分析构成了重大挑战，因为新的迭代非线性取决于当前的迭代。此外，对于线性系统，线性不变的存在始终与零特征值相关联，因此连续问题的稳态状态成为数值时间积分方案的非纤维固定点。总之，对此类方法的稳定性分析需要研究一般非线性迭代的非纤维固定点。基于地图的中心歧管理论，我们提出了一个定理，用于分析非纤维固定时间积分方案的稳定性，用于应用于稳态形成子空间的问题。该定理为方法的稳定性和迭代的局部收敛提供了足够的条件。然后，该定理用于证明MPRK22（$α$）的无条件稳定性 - 修改后的patankar-runge-kutta方案家族时，当应用于特定方程的任意正和保守的线性系统。理论结果通过数值实验证实。
Higher-order time integration methods that unconditionally preserve the positivity and linear invariants of the underlying differential equation system cannot belong to the class of general linear methods. This poses a major challenge for the stability analysis of such methods since the new iterate depends nonlinearly on the current iterate. Moreover, for linear systems, the existence of linear invariants is always associated with zero eigenvalues, so that steady states of the continuous problem become non-hyperbolic fixed points of the numerical time integration scheme. Altogether, the stability analysis of such methods requires the investigation of non-hyperbolic fixed points for general nonlinear iterations. Based on the center manifold theory for maps we present a theorem for the analysis of the stability of non-hyperbolic fixed points of time integration schemes applied to problems whose steady states form a subspace. This theorem provides sufficient conditions for both the stability of the method and the local convergence of the iterates to the steady state of the underlying initial value problem. This theorem is then used to prove the unconditional stability of the MPRK22($α$)-family of modified Patankar-Runge-Kutta schemes when applied to arbitrary positive and conservative linear systems of differential equations. The theoretical results are confirmed by numerical experiments.
