Since almost twenty years, modified Patankar–Runge–Kutta (MPRK) methods have proven to be efficient and robust numerical schemes that preserve positivity conservativity of the production-destruction system irrespectively time step size chosen. Due these advantageous properties they are used for a wide variety applications. Nevertheless, until now, an analytic investigation stability MPRK is still missing, since usual approach by means Dahlquist’s equation not feasible. Therefore, we consider...

.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...

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 Huang Shu [ J. Sci. Comput. 78 (2019) 1811–1839] et al . 79 1015–1056] can be used to solve convection equations with stiff source terms, such as reactive Euler equations, guaranteed positivity under the standard CFL condition due terms only. The allows us identify range free parameters these...

We present a framework based on the atomistic continuum model, combining Molecular Dynamics (MD) and Two Temperature Model (TTM) approaches, to characterize growth of metal nanoparticles (NPs) under ultrashort laser ablation from solid target in water ambient. The model is capable addressing kinetics fast non-equilibrium laser-induced phase transition processes at atomic resolution, while it accounts for effect free carriers, playing determinant role during short pulse interaction with...

Patankar schemes have attracted increasing interest in recent years because they preserve the positivity of analytical solution a production–destruction system (PDS) irrespective chosen time step size. Although are now great interest, for long it was not clear what stability properties such have. Recently new approach based on Lyapunov with an extension center manifold theorem has been proposed to study positivity-preserving integrators. In this work, we classical modified...

Abstract Modified Patankar–Runge–Kutta (MPRK) methods preserve the positivity as well conservativity of a production–destruction system (PDS) ordinary differential equations for all time step sizes. As result, higher order MPRK schemes do not belong to class general linear methods, i. e. iterates are generated by nonlinear map g even when PDS is linear. Moreover, due method, possesses non‐hyperbolic fixed points. Recently, new theorem investigation stability properties points iteration was...

Abstract Modified Patankar‐Runge‐Kutta (MPRK) schemes are numerical one‐step methods for the solution of positive and conservative production‐destruction systems (PDS). They adapt explicit Runge‐Kutta in a way to ensure positivity conservation approximation irrespective chosen time step size. Due nonlinear relationships between next current iterate, stability analysis such is lacking. In this work, we introduce strategy analyze MPRK22(α)‐schemes case PDS. Thereby, point out that usual based...

Abstract Recently, a stability theory has been developed to study the linear of modified Patankar–Runge–Kutta (MPRK) schemes. This provides sufficient conditions for fixed point an MPRK scheme be stable as well convergence towards steady state corresponding initial value problem, whereas main assumption is that sufficiently close state. Initially, numerical experiments in several publications indicated these properties are not only local but even global, case general methods. however, it was...

The work deals with two major topics concerning the numerical analysis of Runge-Kutta-like (RK-like) methods, namely their stability and order convergence. RK-like methods differ from additive RK in that coefficients are allowed to depend on solution step size. As a result this, we also refer them as non-standard (NSARK) methods. first part this thesis is dedicated providing tool for deriving conditions NSARK proposed approach may yield implicit conditions, which can be rewritten explicit...

In this work modified Patankar-Runge-Kutta (MPRK) schemes up to order four are considered and equipped with a dense output formula of appropriate accuracy. Since these time integrators conservative positivity preserving for any step size, we impose the same requirements on corresponding formula. particular, discover that there is an explicit first However, develop boot-strapping technique propose use implicit formulae which naturally fit into framework MPRK schemes. if lower used construct...

In recent years, many positivity-preserving schemes for initial value problems have been constructed by modifying a Runge--Kutta (RK) method weighting the right-hand side of system differential equations with solution-dependent factors. These include classes modified Patankar--Runge--Kutta (MPRK) and Geometric Conservative (GeCo) methods. Compared to traditional RK methods, analysis accuracy stability these methods is more complicated. this work, we provide comprehensive unifying theory...

Modified Patankar--Runge--Kutta (MPRK) methods are linearly implicit time integration schemes developed to preserve positivity and a linear invariant such as the total mass in chemical reactions. MPRK naturally equipped with embedded yielding local error estimate similar Runge--Kutta pairs. To design good step size controllers using these estimates, we propose use Bayesian optimization. In particular, novel objective function that captures important properties tolerance convergence...

In this paper we investigate the stability properties of so-called gBBKS and GeCo methods, which belong to class nonstandard schemes preserve positivity as well all linear invariants underlying system ordinary differential equations for any step size. A investigation these are outside general is challenging since iterates always generated by a nonlinear map even problems. Recently, theorem was derived presenting criteria understanding such schemes. For analysis, applied proven be $\mathcal...

Recently, a stability theory has been developed to study the linear of modified Patankar--Runge--Kutta (MPRK) schemes. This provides sufficient conditions for fixed point an MPRK scheme be stable as well convergence towards steady state corresponding initial value problem, whereas main assumption is that sufficiently close state. Initially, numerical experiments in several publications indicated these properties are not only local, but even global, case general methods. however, it was...

Since almost twenty years, modified Patankar--Runge--Kutta (MPRK) methods have proven to be efficient and robust numerical schemes that preserve positivity conservativity of the production-destruction system irrespectively time step size chosen. Due these advantageous properties they are used for a wide variety applications. Nevertheless, until now, an analytic investigation stability MPRK is still missing, since usual approach by means Dahlquist's equation not feasible. Therefore, we...

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 non-hyperbolic fixed points numerical scheme. Altogether, requires...

In this paper, we perform stability analysis for a class of second and third order accurate strong-stability-preserving modified Patankar Runge-Kutta (SSPMPRK) schemes, which were introduced in [4,5] can be used to solve convection equations with stiff source terms, such as reactive Euler equations, guaranteed positivity under the standard CFL condition due terms only. The allows us identify range free parameters these SSPMPRK schemes ensure stability. Numerical experiments are provided...

Patankar schemes have attracted increasing interest in recent years because they preserve the positivity of analytical solution a production-destruction system (PDS) irrespective chosen time step size. Although are now great interest, for long it was not clear what stability properties such have. Recently new approach based on Lyapunov with an extension center manifold theorem has been proposed to study positivity-preserving integrators. In this work, we classical modified...

Modified Patankar-Runge-Kutta (MPRK) methods preserve the positivity as well conservativity of a production-destruction system (PDS) ordinary differential equations for all time step sizes. As result, higher order MPRK schemes do not belong to class general linear methods, i.e. iterates are generated by nonlinear map $\mathbf g$ even when PDS is linear. Moreover, due method, possesses non-hyperbolic fixed points. Recently, new theorem investigation stability properties points iteration was...

Modified Patankar (MP) schemes are conservative, linear implicit and unconditionally positivity preserving time-integration constructed for production-destruction systems. For such schemes, a classical stability analysis does not yield any information about the performance. Recently, two different techniques have been proposed to investigate properties of MP schemes. In Izgin et al. [ESAIM: M2AN, 56 (2022)], inspired from dynamical systems, Lyapunov investigated, while in Torlo [Appl. Numer....