A Stability Analysis of Modified PatankarRungeKutta methods for a nonlinear ProductionDestruction System Thomas Izgin1 Stefan Kopecz1 and Andreas Meister1

2025-04-30 0 0 527.82KB 7 页 10玖币
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A Stability Analysis of Modified Patankar–Runge–Kutta
methods for a nonlinear Production–Destruction System
Thomas Izgin1, Stefan Kopecz1, and Andreas Meister1
1Department of Mathematics, University of Kassel, Germany
1izgin@mathematik.uni-kassel.de & kopecz@mathematik.uni-kassel.de &
meister@mathematik.uni-kassel.de
Abstract
Modified Patankar–Runge–Kutta (MPRK) methods preserve the positivity as well as con-
servativity 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 geven when
the PDS is linear. Moreover, due to the conservativity of the method, the map gpossesses
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.
1 Introduction
The mathematical modeling of numerous applications of natural sciences and engineering leads to
positive and conservative production-destruction systems (PDS)
y0
i(t) =
N
X
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)Tdenotes 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. PN
i=1 yi(t) = PN
i=1 yi(0) is satisfied for all t≥0. The PDS is called positive, if y(t)>0
holds for all t > 0whenever 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
ynis positive for all y0>0,n∈Nand ∆t > 0, and unconditionally conservative if PN
i=1 yn+1
i=
PN
i=1 yn
iholds true for all n∈N0and ∆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,
1
arXiv:2210.11845v2 [math.NA] 28 Oct 2022
and defined by
y(1)
i=yn
i, y(2)
i=yn
i+α∆t
N
X
j=1pij (y(1))y(2)
j
y(1)
j
−dij (y(1))y(2)
i
y(1)
i,
yn+1
i=yn
i+ ∆t
N
X
j=11−1
2αpij (y(1)) + 1
2αpij (y(2))yn+1
j
(y(2)
j)1/α(y(1)
j)1−1/α
−1−1
2αdij (y(1)) + 1
2αdij (y(2))yn+1
i
(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∗∈RNbe a steady state solution of a differential equation y0=f(y), that
is f(y∗) = 0.
a) Then y∗is called Lyapunov stable if, for any  > 0, there exists a δ=δ()>0such that
ky(0) −y∗k< δ implies ky(t)−y∗k<  for all t≥0.
b) If in addition to a), there exists a constant c > 0such that ky(0) −y∗k< c implies
ky(t)−y∗k → 0for 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
y0(t) = f(y) = y2
2−y2
1
y2
1−y2
2(2)
together with the initial condition
y(0) = y0=y0
1
y0
2>0.(3)
The test problem (2) can be written as a PDS using
pij (y) = dji(y) = y2
j(4)
for i, j = 1,2with i6=jand 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(y0
1+y0
2)1+1
2y0−y0
2
y0
1e−2(y0
1+y0
2)t.(5)
Due to y0>0, the exponential term vanishes as t→ ∞ and hence, limt→∞ y(t) = 1
2(y0
1+y0
2)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(y0
1+y0
2)1(6)
along the line y1+y2=y0
1+y0
2in the y1-y2-coordinate system. As a result, ky(t)−y∗k<ky(0)−y∗k
is true for all t > 0, which means that we can choose δ=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.
2
摘要：

AStabilityAnalysisofModiedPatankarRungeKuttamethodsforanonlinearProductionDestructionSystemThomasIzgin1,StefanKopecz1,andAndreasMeister11DepartmentofMathematics,UniversityofKassel,Germany1izgin@mathematik.uni-kassel.de&kopecz@mathematik.uni-kassel.de&meister@mathematik.uni-kassel.deAbstractModi...

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