In this paper we investigate the stability properties of the so-called gBBKS and GeCo methods, which belong to the class of nonstandard schemes and preserve the positivity as well as all linear invariants of the underlying system of ordinary differential equations for any step size. A stability investigation for these methods, which are outside the class of general linear methods, is challenging since the iterates are always generated by a nonlinear map even for linear problems. Recently, a stability theorem was derived presenting criteria for understanding such schemes. For the analysis, the schemes are applied to general linear equations and proven to be generated by $\mathcal C^1$-maps with locally Lipschitz continuous first derivatives. As a result, the above mentioned 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：本文研究了属于非标准格式类的gBBKS与GeCo方法的稳定性性质。这类方法在任何步长下均能保持常微分方程组正性及所有线性不变量。由于即便针对线性问题，迭代过程始终由非线性映射生成，因此对这类不属于一般线性方法类的方法进行稳定性分析具有挑战性。近期，一项稳定性定理给出了理解此类格式的判据。分析中，将该格式应用于一般线性方程，并证明其可由具有局部Lipschitz连续一阶导数的$\mathcal C^1$类映射生成。由此，上述稳定性定理可通过分析生成映射对应的Jacobian矩阵谱，研究数值方法非双曲不动点的Lyapunov稳定性。此外，若不动点被证明稳定，该定理保证迭代值局部收敛于此点。对于一阶与二阶gBBKS格式，其稳定域与对应Runge-Kutta方法一致。值得注意的是，一阶GeCo格式可将所有有限尺寸线性测试问题的稳态转换为任意步长下的稳定不动点，而二阶GeCo格式在测试问题中具有有界稳定域。最后，稳定性分析的所有理论预测均通过数值实验得到验证。