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

Izgin, Thomas; Kopecz, Stefan; Meister, Andreas

doi:10.1002/pamm.202100027

Cited by 9 publications

(9 citation statements)

References 57 publications

(240 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…for |w n 1 | < , where U and u are given in (19). According to Theorem 2.6, the fixed point 0 ∈ R 2 of G is stable, if the fixed point 0 ∈ R is a stable fixed point of G. From (18) and (20) we see…”

Section: Now We Definementioning

confidence: 96%

“…Taking advantage of the transformation T, this is equivalent to proving that w n+1 = G(w n ) is locally convergent to 0, if the starting value w 0 is sufficiently close to 0 and satisfies w 0 1 = 0. Moreover, since G leaves the first component of its argument fixed, see (20), we only need to show w n 2 → 0 for n → ∞. According to Theorem 2.5 b) the distance of (w n 1 , w n 2 ) ∈ R 2 to the center manifold tends to zero for n → ∞, i. e.…”

Section: Now We Definementioning

confidence: 97%

“…The system (3) is also considered in [20], where it is used to study a linearization of second order MPRK schemes. Section 3 extends the results of [20] to the nonlinear case.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Izgin¹,

Kopecz²,

Meister³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

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

Section: Now We Definementioning

confidence: 96%

Section: Now We Definementioning

confidence: 97%

See 1 more Smart Citation

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

Izgin¹,

Kopecz²,

Meister³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Moreover, summation in (1.5) shows 𝑦 1 (𝑡) + 𝑦 2 (𝑡) = 𝑦 0 1 + 𝑦 0 2 for all 𝑡 ≥ 0, which confirms that the PDS is also conservative. The system (1.3) is also considered in [20], where it is used to study a linearization of second order MPRK schemes. Section 3 extends the results of [20] to the nonlinear case.…”

Section: Introductionmentioning

confidence: 99%

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

2022

Self Cite

View full text Add to dashboard Cite

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

“…which was also used in [21] for studying the linearization of MPRK22 schemes. Hence, to understand the stability behavior of such nonlinear schemes, one should directly investigate the system (1).…”

Section: Introductionmentioning

confidence: 99%

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

Izgin¹,

Kopecz²,

Meister³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

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

Cited by 9 publications

References 57 publications

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

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

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

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

Contact Info

Product

Resources

About