Positively invariant cones of dynamical systems under Runge-Kutta and Rosenbrock-type discretization
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Positively invariant cones of dynamical systems under Runge-Kutta and Rosenbrock-type discretization Zolt´ an Horv´ ath * Department of Mathematics, Sz´ echenyi Istv´ an University, 1 Egyetem square, 9028, Gy˝ or, Hungary. In this paper we consider positively invariant cones of finite dimensional dynamical systems and study conditions on the time step-size that guarantee the discrete positive invariance of these cones under Runge-Kutta and Rosenbrock-type methods. We conclude quite simple sufficient conditions, which involve the positivity (or absolute monotonicity) radius of the Runge-Kutta schemes and its generalization when the Rosenbrock-type methods are applied. 1 Introduction and basic definitions Many finite dimensional dynamical systems of practical importance possesses positively invariant sets which arise naturally from the physical meaning of the dynamical system. By positive invariance of a subset C of the state-space IR N we mean that each trajectory emanating from C remains completely in C . Such a C forms often the set of physically admissible states. For example, for reaction kinetics or properly semi-discretized advection-diffusion-reaction problems the components of u represent concentration-like quantities and C = [0, ∞) N is positively invariant (see e.g. [4]). Another example arises in control problems when prescribed inequalities of the states are maintained resulting the target set, C = {v ∈ V | Gv ≥ 0} with given G ∈ IR m×N positively invariant (see e.g. [6]). From the point of view of physical meaning, it is very natural and important to require that the discretization of the dynamical system should preserve the positive invariance of C ; this is often required by the stability of the exact solutions as well. In this section we give the necessary definitions and pose the problem for the Runge-Kutta and Rosenbrock-type methods. In Section 2 and 3 we present our main results (proofs will be published in subsequent papers). 1.1 Positively invariant cones for continuous dynamical systems We consider dynamical systems in the state space V := IR N generated by initial value problems of form u 0 (t)= f (u(t)), t ≥ 0, u(0) = u 0 (1) where f : V → V is a given continuous function and u 0 ∈ V . For the ease of presentation we assume that (1) has a unique solution for all u 0 ∈ V on the time domain [0, ∞). We call the non-empty subset C of V positively invariant w.r.t. (1) if for all u 0 ∈C and t ≥ 0 we have u(t) ∈C . In the rest of the paper we assume that C is a given cone, i.e. for all v 1 ,v 2 ∈C ,α 1 ,α 2 ∈ [0, ∞) we have α 1 v 1 + α 2 v 2 ∈C . Lemma 1.1 If ρ ∈ (0, ∞) s.t. ρu + f (u) ∈C for all u ∈C , C is positively invariant w.r.t. (1). Moreover, if M ∈ IR N×N and μ ∈ (0, ∞) s.t. (μI - M )C⊂C and Mu + f (u) ∈C for all u ∈C , C is again positively invariant w.r.t. (1). 1.2 Runge-Kutta methods The Runge-Kutta (RK) method (see e.g. [1]) determined by the arrays A := (a ij ) ∈ IR s×s and b =(b i ) ∈ IR s generate approximations to the exact solution u n ≈ u(t n ) recursively: u 0 is given by (1) and u n+1 = u n + h n ∑ s i=1 b i f (y n i ) where y n i are solutions of the system y n i = u n + h n ∑ s j=1 a ij f (y n j )(i =1,...,s) with given time grid t 0 =0 <t 1 < ... and step-sizes h n := t n+1 - t n . We assume in the paper that there is a constant H def = H def (f, (A, b)) > 0 such that the above system of algebraic equations has a uniqe solution y n i , i =1, ..., s, depending continuously on u n and h n whenever u 0 ∈ V is arbitrary and h n ≤ H def for all n. (Explicit formulas for such a H def in quite general situations can be found e.g. in [1].) 1.3 Rosenbrock-type methods Rosenbrock-type (Ros-type) methods (see e.g. [5]) are determined by the arrays of coefficients α =(α ij ), Γ=(γ ij ) with γ ii = γ> 0, b =(b i ) as u n+1 = u n + h n ∑ s i=1 b i k n i with k n i = f (y n i )+ h n Q n ∑ i j=1 γ ij k n j and y n i = u n + h n ∑ i-1 j=1 α ij k n j (i =1,...,s), where Q n := Q(u n ) equals f 0 (u n ) for the Ros-type methods and a suitable N -by-N matrix for the W-methods. Further, H def = H def (f,Q, (α, Γ,b)) is defined analogously to that for RK-methods. The research was partly supported by the Hungary-Austria Phare CBC Programme, under the contract No. 2002/000-317-02-20 and by the Hungarian Scientific Research Fund OTKA T043258. * E-mail: [email protected], Phone: +36 96 613 647, Fax: +36 96 503 461 PAMM · Proc. Appl. Math. Mech. 4, 688–689 (2004) / DOI 10.1002/pamm.200410325 © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Transcript of Positively invariant cones of dynamical systems under Runge-Kutta and Rosenbrock-type discretization
Positively invariant cones of dynamical systems under Runge-Kutta andRosenbrock-type discretization
Zolt an Horvath∗
Department of Mathematics, Szechenyi Istvan University, 1 Egyetem square, 9028, Gyor, Hungary.
In this paper we consider positively invariant cones of finite dimensional dynamical systems and study conditions on the timestep-size that guarantee the discrete positive invariance of these cones under Runge-Kutta and Rosenbrock-type methods. Weconclude quite simple sufficient conditions, which involve the positivity (or absolute monotonicity) radius of the Runge-Kuttaschemes and its generalization when the Rosenbrock-type methods are applied.
1 Introduction and basic definitions
Many finite dimensional dynamical systems of practical importance possesses positively invariant sets which arise naturallyfrom the physical meaning of the dynamical system. By positive invariance of a subsetC of the state-spaceIRN we mean thateach trajectory emanating fromC remains completely inC. Such aC forms often the set of physically admissible states.
For example, for reaction kinetics or properly semi-discretized advection-diffusion-reaction problems the components ofu represent concentration-like quantities andC = [0,∞)N is positively invariant (see e.g. [4]). Another example arises incontrol problems when prescribed inequalities of the states are maintained resulting the target set,C = {v ∈ V |Gv ≥ 0}with givenG ∈ IRm×N positively invariant (see e.g. [6]).
From the point of view of physical meaning, it is very natural and important to require that the discretization of thedynamical system should preserve the positive invariance ofC; this is often required by the stability of the exact solutionsas well. In this section we give the necessary definitions and pose the problem for the Runge-Kutta and Rosenbrock-typemethods. In Section 2 and 3 we present our main results (proofs will be published in subsequent papers).
1.1 Positively invariant cones for continuous dynamical systems
We consider dynamical systems in the state spaceV := IRN generated by initial value problems of form
u′(t) = f(u(t)), t ≥ 0, u(0) = u0 (1)
wheref : V → V is a given continuous function andu0 ∈ V . For the ease of presentation we assume that (1) has aunique solution for allu0 ∈ V on the time domain[0,∞). We call the non-empty subsetC of V positively invariant w.r.t.(1) if for all u0 ∈ C and t ≥ 0 we haveu(t) ∈ C. In the rest of the paper we assume thatC is a given cone, i.e. for allv1, v2 ∈ C, α1, α2 ∈ [0,∞) we haveα1v1 + α2v2 ∈ C.
Lemma 1.1 If ρ ∈ (0,∞) s.t.ρu + f(u) ∈ C for all u ∈ C, C is positively invariant w.r.t. (1). Moreover, ifM ∈ IRN×N
andµ ∈ (0,∞) s.t.(µI −M)C ⊂ C andMu + f(u) ∈ C for all u ∈ C, C is again positively invariant w.r.t. (1).
1.2 Runge-Kutta methods
The Runge-Kutta (RK) method (see e.g. [1]) determined by the arraysA := (aij) ∈ IRs×s andb = (bi) ∈ IRs generateapproximations to the exact solutionun ≈ u(tn) recursively:u0 is given by (1) andun+1 = un + hn
∑si=1 bi f(yn
i ) whereyn
i are solutions of the systemyni = un + hn
∑sj=1 aij f(yn
j )(i = 1, . . ., s) with given time gridt0 = 0 < t1 < . . . andstep-sizeshn := tn+1 − tn. We assume in the paper that there is a constantHdef = Hdef(f, (A, b)) > 0 such that the abovesystem of algebraic equations has a uniqe solutionyn
i , i = 1, ..., s, depending continuously onun andhn wheneveru0 ∈ Vis arbitrary andhn ≤ Hdef for all n. (Explicit formulas for such aHdef in quite general situations can be found e.g. in [1].)
1.3 Rosenbrock-type methods
Rosenbrock-type (Ros-type) methods (see e.g. [5]) are determined by the arrays of coefficientsα = (αij),Γ = (γij) withγii = γ > 0, b = (bi) asun+1 = un +hn
∑si=1 bi kn
i with kni = f(yn
i )+hnQn
∑ij=1 γijk
nj andyn
i = un +hn
∑i−1j=1 αij kn
j
(i = 1, . . ., s), whereQn := Q(un) equalsf ′(un) for the Ros-type methods and a suitableN -by-N matrix for the W-methods.Further,Hdef = Hdef(f,Q, (α, Γ, b)) is defined analogously to that for RK-methods.
The research was partly supported by the Hungary-Austria Phare CBC Programme, under the contract No. 2002/000-317-02-20 and by the HungarianScientific Research Fund OTKA T043258.
∗ E-mail: [email protected], Phone: +36 96 613 647, Fax: +36 96 503 461
PAMM · Proc. Appl. Math. Mech. 4, 688–689 (2004) / DOI 10.1002/pamm.200410325
© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1.4 Discrete positive invariant cones. Problem setting
Suppose thatC is positively invariant w.r.t. (1) and there is given a RK or a Ros-type method. We callC discrete positiveinvariant w.r.t. (1) under the given method if there existsH ∈ (0,Hdef ] depending only onf and the method parameters suchthathn ≤ H for all n impliesun ∈ C for all n whenever the method in question is applied to (1) withu0 ∈ C. Our aim is tofind a formula forH if the positive invariance ofC is ensured by Lemma 1.1.
Remark 1.2 The existence of such anH > 0 is not a consequence of other remarkable properties of the method. Forexample, we constructed a non-autonomous linear problem with anu0 ∈ [0,∞)N and an L- and algebraically stable RK-method (the Lobatto IIIC,s = 2 method) s.t.[0,∞)N is positively invariant w.r.t. the problem butu1 6∈ [0,∞)N for any smallpositive step-sizes, see [2].
Concerning RK methods, results in the literature exist only for the classical non-negativity preservation, i.e. whenC =[0,∞)N , for a survey see e.g. [4, 3] and it seems no general theory exists yet concerning Ros-type methods.
2 Results for the Runge-Kutta methods
Assumeρ > 0, p(u) := ρu + f(u) ∈ C for all u ∈ C, ensuringC positively invariant w.r.t. (1) by Lemma 1.1. Insertingf(yj) = −ρyj + p(yj) into the equations of the RK method (see Section 1.2) and solving formally toyi we obtain forn = 0
u1 = Kb(−ρh)u0 +∑
j
hJb(−ρh)jp(yj), yi = KA(−ρh)iu0 +∑
j
hJA(−ρh)ijp(yj), i = 1, ..., s
whereKA(z) := (I − zA)−1e, JA(z) := A(I − zA)−1, Kb(z) := 1 + bT z(I − zA)−1e, Jb(z) := bT (I − zA)−1
are the so-called scheme functions (z ∈ IR). Thusu1 and theyi vectors are linear combinations of vectorsu0 andp(yj)and the coefficients are nonnegative wheneverρh ∈ [0, R(A, b)] whereR(A, b) is the positivity radius of the scheme, i.e.the supremum of positive numbersr for which all the components of rational functionsKA, JA,Kb, Jb are nonnegative on[−r, 0]. These arguments make the following theorem plausible.
Theorem 2.1 Suppose thatρ > 0 andρu + f(u) ∈ C for all u ∈ C. ThenC is discrete positive invariant w.r.t. (1) underRK method with arrays(A, b) providedhn ≤ H := min{R(A,b)
ρ , Hdef} for all n.
3 Results for the Rosenbrock-type methods
We follow the arguments with RK methods closely, but now we assume thatM, µ fulfill the conditions of Lemma 1.1. Thus wehavep(u) := Mu + f(u) ∈ C for all ∈ C. Suppose in addition thatMQ = QM ; then insertingf(yj) = −Myj + p(yj) intothe equations defining the method (see Section 1.3) we get similar formulas as in the RK caseu1 = Kb(hQ,−hM)u0 +∑
j hJb(hQ,−hM)jp(yj), yi = KA(hQ,−hM)iu0 +∑
j hJA(hQ,−hM)ijp(yj)(i = 1, ..., s), whereKA(x, z) :=(I − Aros(x)z)−1e, JA(x, z) := (I − Aros(x)z)−1Aros(x), Kb(x, z) := 1 + bros(x)T z(I − Aros(x)z)−1e, Jb(x, z) :=bros(x)T (I − zAros(x))−1, the scheme functions of Ros-type methods withAros(x) := α(I − xΓ)−1, bros(x)T := bT (I −xΓ)−1. In the following theorem we deal with the special case whenM = dI − Q with some constantd. First ob-serve thatKb(hQ,−hM) = Kb(hQ, hQ − hdI) hence it is enough to consider the simplified scheme functions, namely
KA
δ(x) := KA(x, x − δ) and so on. It is advantegeous to introduce the simplified absolute monotonicity radius of the
Ros-type method,R(δ) as the supremum ofr > 0 values for which the components of the simplified scheme functions areabsolutely monotonic on[−r, 0] (hereδ is a fixed positive parameter for the computation ofR(δ)).
Theorem 3.1 Suppose that(dI −Q)u + f(u) ∈ C for all u ∈ C, (ρI + Q)C ⊂ C andhρ ≤ R(hd) for all h ≤ HR. ThenC is discrete positive invariant w.r.t. (1) under Ros-type method with arrays(α, Γ, b) providedhn ≤ H := inf{HR,Hdef}.
We remark that the conditions of the theorem are fulfilled for many problems of practical interest, e.g. the semi-discretizeddiffusion-reaction problem wheref(u) = Lu + g(u), L is a -M matrix,C = [0,∞)N , Qn = L, g is a scalar function withdη + g(η) ≥ 0 for all η ≥ 0. For this problem Theorem 3.1 results in that[0,∞)N is discrete positive invariant under theROS2 methods (see [4]) withH = 1/(d+ρ
√2) andH = 1/(d+4.2ρ) whenγ = 1−1/
√2 andγ = 1+1/
√2, respectively.
References
[1] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. (Springer-Verlag, Berlin Heidelberg, 1991).[2] Z. Horvath, Positivity of Runge-Kutta and diagonally split Runge-Kutta methods. Appl. Numer. Math.28, 309-326 (1998).[3] Z. Horvath, On the positivity step-size threshold of Runge-Kutta methods. (submitted to Appl. Numer. Math.)[4] W. Hundsdorfer and J. Verwer, Numerical solution of time-dependent advection-diffusion-reaction equations. (Berlin: Springer, 2003.)[5] K. Strehmel and R. Weiner, Linear-implizite Runge-Kutta-Methoden und ihre Anwendung. (Teubner, Stuttgart-Leipzig, 1992.)[6] S. Tarbouriech and C. Burgat, Positively invariant sets for constrained continuous-time systems with cone properties. IEEE Trans.
Automat. Contr.39, 401–405 (1994).
© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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