On the Stability of Modified Patankar Methods

Patankar schemes have attracted increasing interest in recent years because they preserve the positivity of the analytical solution of a production-destruction system (PDS) irrespective of the chosen time step size. Although they are now of great interest, for a long time it was not clear what stability properties such schemes have. Recently a new stability approach based on Lyapunov stability with an extension of the center manifold theorem has been proposed to study the stability properties of positivity preserving time integrators. In this work, we study the stability properties of the classical modified Patankar--Runge--Kutta schemes (MPRK) and the modified Patankar Deferred Correction (MPDeC) approaches. We prove that most of the considered MPRK schemes are stable for any time step size and compute the stability function of MPDeC. We investigate its properties numerically revealing that also most MPDeC are stable irrespective of the chosen time step size. Finally, we verify our theoretical results with numerical simulations.Comment: 34 pages, 14 Figure

Lyapunov Stability of First and Second Order GeCo and gBBKS Schemes

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.Comment: 31 pages, 7 figure

On the non-global linear stability and spurious fixed points of MPRK schemes with negative RK parameters

Recently, a stability theory has been developed to study the linear stability of modified Patankar--Runge--Kutta (MPRK) schemes. This stability theory provides sufficient conditions for a fixed point of an MPRK scheme to be stable as well as for the convergence of an MPRK scheme towards the steady state of the corresponding initial value problem, whereas the main assumption is that the initial value is sufficiently close to the steady state. Initially, numerical experiments in several publications indicated that these linear stability properties are not only local, but even global, as is the case for general linear methods. Recently, however, it was discovered that the linear stability of the MPDeC(8) scheme is indeed only local in nature. Our conjecture is that this is a result of negative Runge--Kutta (RK) parameters of MPDeC(8) and that linear stability is indeed global, if the RK parameters are nonnegative. To support this conjecture, we examine the family of MPRK22($\alpha$) methods with negative RK parameters and show that even among these methods there are methods for which the stability properties are only local. However, this local linear stability is not observed for MPRK22($\alpha$) schemes with nonnegative Runge-Kutta parameters.Comment: 19 pages, 3 figure

A study of the local dynamics of modified Patankar DeC and higher order modified Patankar–RK methods

Patankar schemes have attracted increasing interest in recent years because they preserve the positivity of the analytical solution of a production–destruction system (PDS) irrespective of the chosen time step size. Although they are now of great interest, for a long time it was not clear what stability properties such schemes have. Recently a new stability approach based on Lyapunov stability with an extension of the center manifold theorem has been proposed to study the stability properties of positivity-preserving time integrators. In this work, we study the stability properties of the classical modified Patankar–Runge–Kutta schemes (MPRK) and the modified Patankar Deferred Correction (MPDeC) approaches. We prove that most of the considered MPRK schemes are stable for any time step size and compute the stability function of MPDeC. We investigate its properties numerically revealing that also most MPDeC are stable irrespective of the chosen time step size. Finally, we verify our theoretical results with numerical simulations

A Stability Analysis of Modified Patankar–Runge–Kutta methods for a nonlinear Production–Destruction System

Gefördert im Rahmen des Projekts DEA

Recent Developments in the Field of Modified Patankar-Runge-Kutta-methods

Gefördert im Rahmen des Projekts DEA

On Lyapunov stability of positive and conservative time integrators and application to second order modified Patankar–Runge–Kutta schemes

Since almost twenty years, modified Patankar–Runge–Kutta (MPRK) methods have proven to be efficient and robust numerical schemes that preserve positivity and conservativity of the production-destruction system irrespectively of the time step size chosen. Due to these advantageous properties they are used for a wide variety of applications. Nevertheless, until now, an analytic investigation of the stability of MPRK schemes is still missing, since the usual approach by means of Dahlquist’s equation is not feasible. Therefore, we consider a positive and conservative 2D test problem and provide statements usable for a stability analysis of general positive and conservative time integrator schemes based on the center manifold theory. We use this approach to investigate the Lyapunov stability of the second order MPRK22(α) and MPRK22ncs(α) schemes. We prove that MPRK22(α) schemes are unconditionally stable and derive the stability regions of MPRK22ncs(α) schemes. Finally, numerical experiments are presented, which confirm the theoretical results

On the dynamics of first and second order {GeCo} and {gBBKS} schemes

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 C1-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

On the stability of strong-stability-preserving modified Patankar-Runge-Kutta schemes

In this paper, we perform a stability analysis for classes of second and third order accurate strong-stability-preserving modified Patankar–Runge–Kutta (SSPMPRK) schemes, which were introduced in [9, 10] and can be used to solve convection equations with stiff source terms, such as reactive Euler equations, with guaranteed positivity under the standard CFL condition due to the convection terms only. The analysis allows us to identify the range of free parameters in these SSPMPRK schemes in order to ensure stability. Numerical experiments are provided to demonstrate the validity of the analysis

Numerical Investigation of Ultrashort Laser-Ablative Synthesis of Metal Nanoparticles in Liquids Using the Atomistic-Continuum Model

International audienceWe present a framework based on the atomistic continuum model, combining the Molecular Dynamics (MD) and Two Temperature Model (TTM) approaches, to characterize the growth of metal nanoparticles (NPs) under ultrashort laser ablation from a solid target in water ambient. The model is capable of addressing the kinetics of fast non-equilibrium laser-induced phase transition processes at atomic resolution, while in continuum it accounts for the effect of free carriers, playing a determinant role during short laser pulse interaction processes with metals. The results of our simulations clarify possible mechanisms, which can be responsible for the observed experimental data, including the presence of two populations of NPs, having a small (5-15 nm) and larger (tens of nm) mean size. The formed NPs are of importance for a variety of applications in energy, catalysis and healthcare