Substantial extension and unification of the theory of Patankar-type schemes by means of unified order analysis, first-time investigation of stability, time-step adaptation and dense-output formulas.

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Many applications can be described by positive and conservative ordinary differential equations and it is highly desirable to guarantee the positivity and conservativity also for the numerical solution. Standard methods such as Runge-Kutta (RK) methods preserve conservativity, but in general cannot guarantee positivity of the solution components. This has to be done by additional and costly postprocessing. A class of methods which guarantee not only conservativity but also unconditional positivity are the Patankar-type methods. This class is divided into BBKS and MPRK schemes and in the last three years several publications appeared to these promising schemes. In particular, since MPRK methods have proven excellent for the solution of stiff problems.In the proposed project, existing theory in the field of order analysis will be unified and theoretical gaps regarding stability, time adaptation and dense output formulas will be closed. All Patankar-type methods are based on the modification of explicit RK methods with the so-called Patankar trick. By formally considering them as perturbed RK schemes, a unified order analysis will be possible and facilitate the comparison of the different Patankar-type methods. The main goal of the project is to develop for the first time a stability analysis for Patankar-type methods. Although MPRK schemes in particular have been shown to be very stable in numerical calculations, theoretical investigations of this have been lacking up to now. A major reason for the lack of a stability theory is the nonlinear dependence of the iterates, which even occur when the methods are applied to linear systems. The project will be concerned with both local and global stability. For this purpose, the theory of nonlinear dynamical systems with several unknowns and parameters will be applied. This analysis will allow to derive conditions on the Patankar weights which guarantee stability. Patankar type methods use lower order methods to determine the required Patankar weights. These, in turn, can be used to estimate local error and select the time step size adaptively. Currently, there are no known adaptive Patankar-type methods that are competitive at low tolerances. Using the new stability analysis, efficient adaptive Patankar-type methods can be developed. Finally, dense output formulas (DOF) for Patankar-type methods are developed, which can be used to generate approximations of appropriate order for arbitrary times. A new feature here is that the DOF also guarantee positivity and conservativity at arbitrary times.