In this paper we investigate the stability properties of fixed points of the so-called gBBKS and GeCo methods, which belong to the class of non-standard schemes and preserve the positivity as well as all linear invariants of the underlying system of ordinary differential equations for any step size. The schemes are applied to general linear test equations and proven to be generated by $\mathcal C^1$-maps with locally Lipschitz continuous first derivatives. As a result, a recently developed stability theorem can be applied to investigate the Lyapunov stability of non-hyperbolic fixed points of the numerical method by analyzing the spectrum of the corresponding Jacobian of the generating map. In addition, if a fixed point is proven to be stable, the theorem guarantees the local convergence of the iterates towards it. In the case of first and second order gBBKS schemes the stability domain coincides with that of the underlying Runge--Kutta method. Furthermore, while the first order GeCo scheme converts steady states to stable fixed points for all step sizes and all linear test problems of finite size, the second order GeCo scheme has a bounded stability region for the considered test problems. Finally, all theoretical predictions from the stability analysis are validated numerically.
翻译:在本文中,我们调查所谓的GBBKS和Geco方法的固定点的稳定性特性,这些固定点属于非标准办法的类别,并保存普通差异方程基础系统的正反性和所有直线变异性,适用于一般线性测试方程,并被证明是用当地Lipschitz第一级连续衍生物生成的$mathcal C ⁇ 1$-maps与当地Lipschitz第一级连续生成物生成的。因此,可以应用最近开发的稳定性理论来调查数字方法非双向固定点的Lyapunov稳定性,方法是分析生成地图对应的Jacobian的频谱。此外,如果一个固定点被证明是稳定的,则该理论保证了循环方程与它之间的本地趋同。在第一和第二顺序gBBBKS计划下,稳定性域与Runge-Kutta方法的基底部方法相吻合。此外,第一道,Geco方案将稳定点转换为所有步骤的固定点和所有线性测试范围的测试问题。此外,第二个测定点的GeCo计划从考虑的数值测测为最终的稳定度。