A conservative, positivity preserving scheme for advection-diffusion-reaction equations in biochemical applications

Benz, Joachim; Meister, Andreas; Zardo, Philipp Andrea

doi:10.1090/psapm/067.2/2605235

Cited by 9 publications

(4 citation statements)

References 4 publications

(4 reference statements)

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“…The numerical treatment of the right-hand side function in the Method (3) is motivated by some of the discretization ideas of J. Benz et al [13]. In [13], the authors propose a numerical method that works for both conservative and non-conservative systems by using a term similar to equation (4) which switches based on the conservativity-property of the system.…”

Section: Construction Of the Numerical Methodsmentioning

confidence: 99%

A class of nonstandard numerical methods for autonomous dynamical systems

Wood

Kojouharov

2015

Applied Mathematics Letters

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Section: Construction Of the Numerical Methodsmentioning

confidence: 99%

A class of nonstandard numerical methods for autonomous dynamical systems

Wood

Kojouharov

2015

Applied Mathematics Letters

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“…Based on deferred correction methods and the idea in [10], arbitrary high order MPRK schemes where introduced in [34]. MPRK schemes are of considerable interest and widely used such as in the context of ecosystems [16,17,40,4,3,32] or ocean models [38,9]. Further applications can be found in the context of magneto-thermal winds [14] or warm-hot intergalactic mediums [24] as well as in that of the SIR epidemic model [41].…”

Section: Introductionmentioning

confidence: 99%

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

Izgin¹,

Kopecz²,

Meister³

2022

Preprint

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

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“…The second order MPRK scheme of [8] is applied to an ecosystem model for the simulation of the cyanobacteria life cycle [15,16] or that of dinoflagellates [36]. In [4,3,27] this scheme is also used to model the phosphor cycle in rivers and lakes. Moreover this scheme was found to be beneficial when applied to NPZD-models in [9] and is also implemented in the General Ocean Turbulence Model (GOTM) [7].…”

Section: Introductionmentioning

confidence: 99%

“…In the following, we introduce a framework to study the Lyapunov stability of positive and conservative time integrators, when applied to (3). Within this framework we analyze the stability of the second order MPRK22(α) and MPRK22ncs(α) schemes introduced in [22].…”

Section: Introductionmentioning

confidence: 99%

On Lyapunov Stability of Positive and Conservative Time Integrators and Application to Second Order Modified Patankar--Runge--Kutta Schemes

Izgin¹,

Kopecz²,

Meister³

2022

Preprint

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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.

show abstract

A conservative, positivity preserving scheme for advection-diffusion-reaction equations in biochemical applications

Cited by 9 publications

References 4 publications

A class of nonstandard numerical methods for autonomous dynamical systems

A class of nonstandard numerical methods for autonomous dynamical systems

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

On Lyapunov Stability of Positive and Conservative Time Integrators and Application to Second Order Modified Patankar--Runge--Kutta Schemes

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