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Izgin, Thomas

Kopecz, Stefan

Meister, Andreas

Received: 5 October 2022 Accepted: 9 November 2022
DOI: 10.1002/pamm.202200083
A Stability Analysis of Modified Patankar–Runge–Kutta methods for a
nonlinear Production–Destruction System
Thomas Izgin1,∗, Stefan Kopecz1,∗∗, and Andreas Meister1,∗∗∗
1 Institut für Mathematik, Universität Kassel, Heinrich-Plett-Straße 40, 34132 Kassel, Hessen, Deutschland
Modified Patankar–Runge–Kutta (MPRK) methods preserve the positivity as well as conservativity of a production–destruction
system (PDS) of ordinary differential equations for all time step sizes. As a result, higher order MPRK schemes do not belong
to the class of general linear methods, i. e. the iterates are generated by a nonlinear map g even when the PDS is linear.
Moreover, due to the conservativity of the method, the map g possesses non-hyperbolic fixed points.
Recently, a new theorem for the investigation of stability properties of non-hyperbolic fixed points of a nonlinear iteration
map was developed. We apply this theorem to understand the stability properties of a family of second order MPRK methods
when applied to a nonlinear PDS of ordinary differential equations. It is shown that the fixed points are stable for all time step
sizes and members of the MPRK family. Finally, experiments are presented to numerically support the theoretical claims.
© 2023 The Authors. Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH.
1 Introduction
The mathematical modeling of numerous applications of natural sciences and engineering leads to positive and conservative
production-destruction systems (PDS)
y′i(t) =
N∑
j=1
(pij(y(t))− dij(y(t))), y(0) = y0 ∈ RN>0, pij(y), dij(y) ≥ 0, i = 1, . . . , N,
where y = (y1, . . . , yN )T denotes the vector of state variables. The terms pij , dij denote production and destruction terms of
the i-th constituent, respectively. A PDS is called conservative, if pij = dji, i. e.
∑N
i=1 yi(t) =
∑N
i=1 yi(0) is satisfied for all
t ≥ 0. The PDS is called positive, if y(t) > 0 holds for all t > 0 whenever y(0) > 0.
When applied to a positive and conservative PDS, a numerical method should satisfy these properties in a discrete manner.
This means that we call the method unconditionally positive if yn is positive for all y0 > 0, n ∈ N and ∆t > 0, and
unconditionally conservative if
∑N
i=1 y
n+1
i =
∑N
i=1 y
n
i holds true for all n ∈ N0 and ∆t > 0.
One class of unconditionally positive and conservative numerical methods of second order is given by the two-stage one-
parameter family of modified Patankar–Runge–Kutta methods denoted by MPRK22(α). This family of schemes is based on
the set of explicit two-stage Runge–Kutta (RK) methods with nonnegative parameters given by the Butcher tableau
0
α α
1− 1/(2α) 1/(2α)
, α ≥ 1
2
,
and defined by
y
(1)
i = y
n
i , y
(2)
i = y
n
i + α∆t
N∑
j=1
(
pij(y
(1))
y
(2)
j
y
(1)
j
− dij(y(1))
y
(2)
i
y
(1)
i
)
,
yn+1i = y
n
i +∆t
N∑
j=1
(((
1− 1
2α
)
pij(y
(1)) +
1
2α
pij(y
(2))
)
yn+1j
(y
(2)
j )
1/α(y
(1)
j )
1−1/α
−
((
1− 1
2α
)
dij(y
(1)) +
1
2α
dij(y
(2))
)
yn+1i
(y
(2)
i )
1/α(y
(1)
i )
1−1/α
)
,
for i = 1, . . . , N , see [7].
Since the numerical method should also be capable of replicating the stability properties of the underlying problem, steady
states of the differential equation should be fixed points of the numerical scheme with identical stability properties.
Definition 1.1 Let y∗ ∈ RN be a steady state solution of a differential equation y′ = f(y), that is f(y∗) = 0.
∗ E-mail: izgin@mathematik.uni-kassel.de, phone: +49 561 804 4316
∗∗ Corresponding author: E-mail: kopecz@mathematik.uni-kassel.de, phone: +49 561 804 2890
∗∗∗ E-mail: meister@mathematik.uni-kassel.de, phone: +49 561 804 2898
This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution
in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.
PAMM · Proc. Appl. Math. Mech. 2022;22:1 e202200083. www.gamm-proceedings.com 1 of 6
https://doi.org/10.1002/pamm.202200083 © 2023 The Authors. Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH.
2 of 6 Section 18: Numerical methods of differential equations
a) Then y∗ is called Lyapunov stable if, for any ϵ > 0, there exists a δ = δ(ϵ) > 0 such that ∥y(0) − y∗∥ < δ implies
∥y(t)− y∗∥ < ϵ for all t ≥ 0.
b) If in addition to a), there exists a constant c > 0 such that ∥y(0) − y∗∥ < c implies ∥y(t) − y∗∥ → 0 for t → ∞, we
call y∗ asymptotically stable.
c) A steady state solution that is not Lyapunov stable is said to be unstable.
In the following we will also briefly speak of stability instead of Lyapunov stability.
With these notions in mind let us consider the nonlinear test equation
y′(t) = f(y) =
(
y22−y21
y21−y22
)
(2)
together with the initial condition
y(0) = y0 =
(
y01
y02
)
> 0. (3)
The test problem (2) can be written as a PDS using
pij(y) = dji(y) = y
2
j (4)
for i, j = 1, 2 with i ̸= j and pii(y) = dii(y) = 0 for i = 1, 2. Using 1 = (1, 1)⊺, the set of positive steady states of (2) is
given by span(1) ∩ R2>0, and the solution of the associated initial value problem given by (2) and (3) can be written as
y(t) =
1
2
(y01 + y
0
2)1+
1
2
(
y0 −
(
y02
y01
))
e−2(y
0
1+y
0
2)t. (5)
Due to y0 > 0, the exponential term vanishes as t → ∞ and hence, limt→∞ y(t) = 12 (y01 + y02)1 ∈ span(1) ∩ R2>0. In
particular, given a positive initial condition, the exact solution monotonically approaches the positive steady state solution
y∗ =
1
2
(y01 + y
0
2)1 (6)
along the line y1 + y2 = y01 + y
0
2 in the y1-y2-coordinate system. As a result, ∥y(t) − y∗∥ < ∥y(0) − y∗∥ is true for all
t > 0, which means that we can choose δ = ϵ in Definition 1.1 to see that all positive steady states are stable. However, none
of them is asymptotically stable since in any neighborhood of a steady state there are infinitely many further steady states.
In total the numerical method should transfer stable but not asymptotically stable steady states to fixed points with similar
properties, which we define analogously to the continuous case.
Definition 1.2 Let y∗ ∈ RN be a fixed point of an iteration scheme yn+1 = g(yn), that is y∗ = g(y∗).
a) Then y∗ is called Lyapunov stable if, for any ϵ > 0, there exists a δ = δ(ϵ) > 0 such that ∥y0 − y∗∥ < δ implies
∥yn − y∗∥ < ϵ for all n ≥ 0.
b) If in addition to a), there exists a constant c > 0 such that ∥y0 − y∗∥ < c implies ∥yn − y∗∥ → 0 for n → ∞, we call
y∗ asymptotically stable.
c) A fixed point that is not Lyapunov stable is said to be unstable.
Analogous to steady states, we also speak of stability instead of Lyapunov stability in the case of fixed points.
So far, the stability of MPRK schemes, see [2, 3, 7–9] has been studied exclusively in the context of linear systems of
differential equations in [1, 4–6]. Here we want to show that the same approach can be used to investigate the stability of
MPRK schemes applied to the nonlinear problem (2).
To recap the stability theorem from [4], we introduce a matrix A ∈ RN×N such that n1, . . . ,nk with k ≥ 1 form a basis
of ker(A⊺). We also define the matrix
N =


n⊺1
...
n⊺k

 ∈ Rk×N
and the set
H = {y ∈ RN | Ny = Ny∗}. (7)
With this in mind, we can now state the following theorem and point out that it is a generalization of [5, Theorem 2.9] to
systems of arbitrary finite size.
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PAMM · Proc. Appl. Math. Mech. 22:1 (2022) 3 of 6
Theorem 1.3 ( [4, Theorem 2.9]) Let A ∈ RN×N such that ker(A) = span(v1, . . . ,vk) represents a k-dimensional
subspace of RN with k ≥ 1. Also, let y∗ ∈ ker(A) be a fixed point of g : D → D where D ⊆ RN contains a neighborhood
D of y∗. Moreover, let any element of ker(A) ∩ D be a fixed point of g and suppose that g
∣∣
D ∈ C1 as well as that the first
derivatives of g are Lipschitz continuous on D. Then Dg(y∗)vi = vi for i = 1, . . . , k and the following statements hold.
a) If the remaining N − k eigenvalues of Dg(y∗) have absolute values smaller than 1, then y∗ is stable.
b) Let H be defined by (7) and g conserve all linear invariants, which means that g(y) ∈ H ∩ D for all y ∈ H ∩ D. If
additionally the assumption of a) is satisfied, then there exists a δ > 0 such that y0 ∈ H ∩D and ∥y0 − y∗∥ < δ imply
yn → y∗ as n → ∞.
Remark 1.4 In order to use Theorem 1.3 to analyze MPRK22(α) applied to (2) we note that
a) the matrix A =
(
1 −1
−1 1
)
satisfies ker(A) = ker(A⊺) = span(1) ⊆ {y ∈ RN | f(y) = 0},
b) any positive steady state of (2) is a fixed point of MPRK22(α), see [10],
c) g conserves all linear invariants as the only linear invariant is conservativity, and
d) it is sufficient to prove g ∈ C2 rather than that g ∈ C1 has locally Lipschitz first derivatives, see [4, 5].
As a result of [4, Remark 2.10] it remains to prove that g ∈ C2 in a small enough neighborhood of a positive steady state y∗
and to analyze the eigenvalues of the Jacobian Dg(y∗).
We first apply the MPRK22(α) method to (2) obtaining
y
(2)
i = y
n
i + α∆t(y
n
j y
(2)
j − yni y
(2)
i ),
yn+1i = y
n
i +∆t
((
1− 12α
)
(ynj )
2 +
1
2α
(y
(2)
j )
2
)
yn+1j
(y
(2)
j )
1
α (ynj )
1− 1α
−∆t
((
1− 12α
)
(yni )
2 +
1
2α
(y
(2)
i )
2
)
yn+1i
(y
(2)
i )
1
α (yni )
1− 1α
.
Following the approach described in [1, 4, 6], we define the functions Φ2 and Φn+1 by
Φ2(y
n,y(2))i = −y(2)i + yni + α∆t(ynj y
(2)
j − yni y
(2)
i ), (8)
Φn+1(y
n,y(2),yn+1)i = −yn+1i + yni +∆t
((
1− 12α
)
(ynj )
2 +
1
2α
(y
(2)
j )
2
)
yn+1j
(y
(2)
j )
1
α (ynj )
1− 1α
−∆t
((
1− 12α
)
(yni )
2 +
1
2α
(y
(2)
i )
2
)
yn+1i
(y
(2)
i )
1
α (yni )
1− 1α
(9)
for i, j = 1, 2 and i ̸= j. The functions Φ2 = Φ2(u,v) and Φn+1 = Φn+1(u,v,w) are in C2 for u,v,w ∈ R2>0, so that we
can define
DnΦk =
∂Φk
∂u
, D2Φk =
∂Φk
∂v
for k = 2, n+ 1
as well as
Dn+1Φn+1 =
∂Φn+1
∂w
.
It is worth mentioning that the operator D∗ applied to Φk for k = 2, n+ 1 means that we plug in y∗ for all arguments, e. g.
D∗nΦ2 = DnΦ2(y
∗,y∗).
If the inverses of D∗n+1Φn+1 and D2Φ2 exist we can conclude
Dg(y∗) = −(D∗n+1Φn+1)−1(D∗nΦn+1 −D∗2Φn+1(D∗2Φ2)−1D∗nΦ2), (10)
see [6]. Moreover, in this case the implicit function theorem states that also g ∈ C2 in a small enough neighborhood of y∗. As
a result, all we have to do is prove that the inverses occurring in (10) exist and to investigate the spectrum of Dg(y∗).
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4 of 6 Section 18: Numerical methods of differential equations
Let us introduce the matrix
B =
(
−1 1
1 −1
)
,
so that (8) and y∗ = c1 with c = 12 (y
0
1 + y
0
2) yield
D∗nΦ2 =
(
1− α∆tc α∆tc
α∆tc 1− α∆tc
)
= I− α∆tcB
and
D∗2Φ2 =
(
−1− α∆tc α∆tc
α∆tc −1− α∆tc
)
= −I+ α∆tcB.
Note that σ(B) = {0,−2} and α∆tc > 0 imply that the inverse of D∗2Φ2 exists. Next, using (9) and the quotient rule, we
obtain the Jacobian
D∗nΦn+1 =


1−∆t 2(1−
1
2α )(y
∗
1 )
3−(y∗1 )3(1−
1
α )
(y∗1 )
2 ∆t(2(1− 12α )y∗2 − y∗2(1− 1α )
∆t(2(1− 12α )y∗1 − y∗1(1− 1α ) 1−∆t
2(1− 12α )(y
∗
2 )
3−(y∗2 )3(1−
1
α )
(y∗2 )
2


=
(
1− c∆t c∆t
c∆t 1− c∆t
)
= I+ c∆tB.
(11)
Furthermore, we have
D∗2Φn+1 =


∆t
1
α (y
∗
1 )
3−(y∗1 )3
1
α
(y∗1 )
2 ∆t(
1
αy
∗
2 − y∗2 1α )
∆t( 1αy
∗
1 − y∗1 1α )
1
α (y
∗
2 )
3−(y∗2 )3
1
α
(y∗2 )
2

 = 0 (12)
as well as the matrix
D∗n+1Φn+1 =
(
−1−∆ty∗1 ∆ty∗2
∆ty∗1 −1−∆ty∗2
)
= −I+∆tcB, (13)
which is nonsingular due to ∆tc ≥ 0. Altogether, we are able to use the formula (10), and due to (12) it simplifies to
Dg(y∗) = −(D∗n+1Φn+1)−1D∗nΦn+1.
Using (11) and (13) we obtain
Dg(y∗) = (I−∆tcB)−1(I+∆tcB) = 1
2∆tc+ 1
(
1 + ∆tc ∆tc
∆tc 1 + ∆tc
)(
1−∆tc ∆tc
∆tc 1−∆tc
)
=
1
2∆tc+ 1
(
1 2∆tc
2∆tc 1
)
.
In accordance with [5] we find that the eigenvectors are given by y∗ and ȳ = (1,−1)⊺ satisfying Dg(y∗)y∗ = y∗ and
Dg(y∗)ȳ =
1− 2∆tc
1 + 2∆tc
ȳ.
Hence, using Theorem 1.3 and substituting z = 2∆tc, we have to analyze the stability function
R(z) =
1− z
1 + z
. (14)
Indeed, R(0) = 1, limz→∞ R(z) = −1 and R′(z) = − 2(1+z)2 < 0 yield
|R(z)| < 1 for all z = 2∆tc > 0,
and hence, Theorem 1.3 implies the following results.
Corollary 1.5 Let y∗ be a positive steady state of the differential equation (2). Then y∗ is a stable fixed point of the
MPRK22(α) scheme for all ∆t > 0 and α ≥ 12 .
Corollary 1.6 Let the unique steady state y∗ of the initial value problem (2), (3) be positive. Then the iterates of
MPRK22(α) locally converge towards y∗ for all ∆t > 0 and α ≥ 12 .
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0 0.1 0.2 0.3 0.4 0.5
t
0
2
4
6
8
10
Reference solution
y
1
y
2
 y
i
Fig. 1: Analytical solution (5) of (2) with y0 = (9.98, 0.02)⊺.
2 Numerical Experiments
In this section we numerically validate the statements of Corollary 1.5 and Corollary 1.6. For this, we solve the test problem
(2) with initial condition y0 = (9.98, 0.02)⊺. The analytical solution is plotted in Figure 1, where one can see that the steady
state solution is approximately reached at time t = 0.4. For the numerical approximation we use a comparably large time step
size of ∆t = 2 to test the statements of Corollary 1.5 and Corollary 1.6. Indeed, Figure 2 reflects the theoretical results very
well. Numerical approximations can be seen in Figures 2a, 2c and 2e. The corresponding error plots are depicted in Figures
2b, 2d and 2f showing that the error is of magnitude 10−14 for t ≥ 700.
3 Summary and Conclusion
In this work we used Theorem 1.3 for the stability analysis of MPRK22(α) when applied to the nonlinear test problem (2).
This is the first time that the stability theory of [4] is used in the context of a nonlinear differential equation. We discovered
that all steady states of (2) are stable fixed points of the method regardless of the chosen time step size. Furthermore, we
proved that the iterates of MPRK22(α) locally converge towards the correct steady state solution of the initial value problem
(2), (3). The theoretical results are numerically validated and underline the robustness of the MPRK22(α) schemes.
Acknowledgements The author Th. Izgin gratefully acknowledges the financial support by the Deutsche Forschungsgemeinschaft (DFG)
through grant ME 1889/10-1. Open access funding enabled and organized by Projekt DEAL.
References
[1] J. Huang, Th. Izgin, S. Kopecz, A. Meister, and C.-W. Shu, arXiv:2205.01488 [math.NA], (2022).
[2] J. Huang, and C.-W. Shu, J. Sci. Comput. 78, 1811–1839 (2019).
[3] J. Huang, W. Zhao, and C.-W. Shu, J. Sci. Comput. 79, 1015–1056 (2019).
[4] Th. Izgin, S. Kopecz, and A. Meister, ESAIM: M2AN 56, 1053–1080 (2022).
[5] Th. Izgin, S. Kopecz, and A. Meister, arXiv:2202.11649 [math.NA], (2022).
[6] Th. Izgin, and P. Öffner, arXiv:2206.07371 [math.NA], (2022).
[7] S. Kopecz, and A. Meister, APNUM 123, 159–179 (2018).
[8] S. Kopecz, and A. Meister, BIT 58, 691–728 (2018).
[9] P. Öffner, and D. Torlo, APNUM 153, 15–34 (2020).
[10] D. Torlo, P. Öffner, and H. Ranocha, APNUM 182, 117–147 (2022).
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6 of 6 Section 18: Numerical methods of differential equations
0 20 40 60 80 100
t
0
2
4
6
8
10
MPRK22(0.500)
y
1
y
2
 y
i
y
1
 num
y
2
 num
 y
i
 num
(a)
0 200 400 600 800 1000
t
10
-15
10
-10
10
-5
10
0
MPRK22(0.500) - Error
err
1
err
2
(b)
0 20 40 60 80 100
t
0
2
4
6
8
10
MPRK22(1.000)
y
1
y
2
 y
i
y
1
 num
y
2
 num
 y
i
 num
(c)
0 200 400 600 800 1000
t
10
-15
10
-10
10
-5
10
0
MPRK22(1.000) - Error
err
1
err
2
(d)
0 20 40 60 80 100
t
0
2
4
6
8
10
MPRK22(2.000)
y
1
y
2
 y
i
y
1
 num
y
2
 num
 y
i
 num
(e)
0 200 400 600 800 1000
t
10
-15
10
-10
10
-5
10
0
MPRK22(2.000) - Error
err
1
err
2
(f)
Fig. 2: Numerical approximations of (2) using MPRK22(α) schemes for α ∈ {0.5, 1, 2}, see Figures 2a, 2c and 2e. There, the dashed lines
indicate the exact solution (5). In the Figures 2b, 2d and 2f the corresponding errors erri = |yni − yi(n∆t)| are plotted for i = 1, 2.
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A Stability Analysis of Modified Patankar–Runge–Kutta methods for a nonlinear Production–Destruction System

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