On the Stability of Unconditionally Positive and Linear Invariants Preserving Time Integration Schemes

Higher-order time integration methods that unconditionally preserve the positivity and linear invariants of underlying differential equation system cannot belong to class general methods. This poses a major challenge for stability analysis such since new iterate depends nonlinearly on current iterate. Moreover, systems, existence is always associated with zero eigenvalues, so steady states continuous problem become nonhyperbolic fixed points numerical scheme. Altogether, requires investigation nonlinear iterations. Based center manifold theory maps we present theorem schemes applied problems whose form subspace. provides sufficient conditions both method local convergence iterates state initial value problem. then used prove unconditional MPRK22-family modified Patankar–Runge–Kutta when arbitrary positive conservative systems equations. The theoretical results are confirmed by experiments.