On the positivity step size threshold of Runge–Kutta methods

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Applied Numerical Mathematics 53 (2005) 341–356www.elsevier.com/locate/apnum

On the positivity step size threshold of Runge–Kutta method

Zoltán Horváth

Department of Mathematics, Széchenyi István University, 1 Egyetem square, H-9026, Gy˝or, Hungary

Available online 27 October 2004

Abstract

In the first part of this paper we determine the largest step size of Runge–Kutta (RK) methods for whcorresponding numerical approximations are positive (component-wise non-negative) for arbitrary positivvector, whenever the underlying initial value problem (IVP) possesses the related positivity preserving pWe prove that step size thresholds for certain classes of positive IVPs guaranteeing positivity that wein a former paper are strict for irreducible and non-confluent RK methods. Investigating the strict positivisize thresholds we can see that these are rather small if at all positive: often they are, roughly speakingproportional to the Lipschitz constant of the problem.

However, for certain (stiff) IVPs with someparticular initial vectors, e.g., for some “smooth” vectors in semdiscretized diffusion problems, we experience preservation of positivity with much larger step sizes than tpositivity step size threshold. To catch this phenomenon, in the second part of the paper we construct pinvariant sets of positive vectors and derive step size thresholds for the discrete version of the positive invThe resulting threshold for discrete positive invariance is, roughly speaking, inverse proportional to the onLipschitz constant only and is shown in good accordance with some displayed computational experiments 2004 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords:Runge–Kutta methods; Positivity; Numerical methods; Contractivity; TVD-property; Invariant cone; Generapositivity

1. Introduction

There are many problems of practical interest that can be modelled by positive initial value pro(IVPs) for ordinary differential equations. Here we mean by a positive IVP that the componen

E-mail address:[email protected] (Z. Horváth).

0168-9274/$30.00 2004 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.apnum.2004.08.026

342 Z. Horváth / Applied Numerical Mathematics 53 (2005) 341–356

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non-negativity of the initial vector, shortly its positivity, is preserved in time for the exact solution oIVP. As typical examples we mention reaction, diffusion and advection problems or a mixture ofwhere the unknowns are concentration-like quantities being positive also from their physical nawell. Very often the numerical solution of these is required to reflect the physical property rigorouwe need to investigate the numerical methods applied to these problems whether they fulfill the popreserving property. In the literature we can find several papers devoted to discussing this propexamples we mention the papers [2,3,10,12,14,15].

In this paper we study Runge–Kutta (RK) methods. The main result of the first part of theSections 2 and 3, is the proof that the positivity step size formulas given in [12] for arbitrary RK meare strict for irreducible and non-confluent methods. In Section 3 we prove this by direct construintroducing a suitable quite simple linear test problem set. In this part, we discuss incidentally the rof positive problem sets used in the literature (see Section 2.4) and the consequence of strict pstep size threshold to the TVD threshold given in [17], namely we show that this TVD threshpositive only if the RK method is positive (see Remark 10).

Investigating the resulted strict positivity step size threshold we can see that the requiremein the first part, i.e., positivity preservation starting from arbitrary positive initial vector leads to srestrictions on the method (see Remark 9), which are violated by most methods of practical iHence, in the second part of the paper (Section 4) we consider the problem how to define a sufwide setS of positive vectors as initial values such that the method be positive starting fromS and,starting from any positive vector outside ofS , do enter thisS after finite steps and then remain there.other words, we are finding positively invariantS which attracts all positive initial vectors, in the discreprocess as well. As a nice surprise, the time step size restrictions are often much weaker to gthe discrete positive invariance and attractivity ofS than were in the first part, explaining some actnumerical experiments (see Remarks 18, 19, and 23).

2. Positivity thresholds for the classical problems

2.1. Basic definitions

First, we introduce some definitions and notations, which are the same as those employedmerely to ensure an easy adaptation of the referred results.

We shall consider IVPs of form

U ′(t) = f(t,U(t)

), t � t0, U(t0) = u0, (1)

wheret0 ∈ R, n is a positive integer,u0 ∈ Rn andf :R × R

n → Rn. We assume tacitly throughout th

paper thatf is continuous and (1) has a unique uncontinuable solution for allt0 ∈ R andu0 ∈ Rn (i.e.,

there exists the largestt∗ ∈ (t0,∞] with the property that (1) has a unique solution on[t0, t∗)).We call the IVP (1) positive ifU(t) � 0 holds for all t ∈ [t0, t∗) whenevert0 ∈ R and u0 � 0 are

arbitrary. Notice that in the paper� and> are meant entry-wise for matrices or vectors. We denotP the set of functionsf for which the corresponding IVPs of form (1) are positive. A sufficientnecessary condition onf to belong toP can be found in [12]. Namely,

f ∈P iff for all k, t andv � 0 with v = 0 we havef (t, v) � 0. (2)
k k

Z. Horváth / Applied Numerical Mathematics 53 (2005) 341–356 343

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P

For the numerical solution of the IVP (1) we consider the Runge–Kutta (RK) methods (see[4,11]). These methods approximateU(tm), the exact solution value attm by the recursively generateum ∈ R

n wheret0 < t1 < · · · < tm < · · ·, hm := tm − tm−1 and there holds subsequently form = 1,2, . . .

um = um−1 + hm

s∑i=1

bif(tm−1 + cihm, y

(m)i

), (3)

y(m)i = um−1 + hm

s∑j=1

aijf(tm−1 + cjhm, y

(m)j

), i = 1, . . . , s. (4)

Here s, aij , bi, ci = ∑j aij are the parameters of the method collected to the arraysA = (aij ) ∈ R

s×s ,b = (bi), c = (ci) ∈ R

s . This method is denoted by RK(A,b) and the pair(A,b) is called its scheme.In general, for the implicit methods (4) forms a system of algebraic equations which has to be

first for y(m)i , the stage values and then we can computeum from (3). If (4) admits a unique solution for a

hm � H andum−1 ∈ Rn with a certainH ∈ [0,∞], and this holds even in the case whenum−1 is replaced

by an arbitraryu(i)

m−1 ∈ Rn in the ith equation of (4) (i = 1, . . . , s) and the resultingy(m)

i values depend

continuously onu(i)m−1 andhm, we call the method well-defined on (1) with step sizes less than or e

to H (even for perturbations of the initial value of the time-step).

Definition 1. Let there be given∅ �= F ⊂ P , (A,b) a scheme of an RK method andH ∈ (0,∞]. Wecall RK(A,b) positive onF with positivity step size thresholdH if the method is well-defined on (1with step sizes less than or equal toH andum � 0 for anym, f ∈ F , t0 ∈ R, u0 � 0 and finite stepshm ∈ [0,H ]. If H is a positivity step size threshold and there is no greater positivity step size threthanH , we callH the strict positivity step size threshold of RK(A,b) w.r.t.F .

2.2. Positive problem sets

In [12] we derived positivity thresholds w.r.t. certainF ⊂ P . Several of theseF sets are defined banother qualitative property of the IVP, the dissipativity. The connection between positivity and dtivity seems quite natural in many problems, especially when (1) arises as a semi-discretization odifferential equations of parabolic or first order hyperbolic type (see, e.g., [9]).

We call the IVP (1) dissipative in a given norm‖ · ‖ (see [11,16]) if for any two solutions of thdifferential equation of (1), sayU andU , the functiont → ‖U(t)−U(t)‖, t � t0 is monotone decreasinTo define subclasses of dissipative problems one often uses the so-called circle condition. We saf ,the right side function of the IVP (1) fulfills the circle condition in the norm‖ · ‖ with constantρ > 0 ifthere holds∥∥ρ(v − v) + (

f (t, v) − f (t, v))∥∥ � ρ‖v − v‖ for all t ∈ R, v, v ∈ R

n. (5)

We shall denote byD∗ and, for anyρ > 0,D∗(ρ) the set off functions for which the corresponding IV(1) is dissipative in someLp-norm and respectively fulfills the circle condition in someLp-norm withp ∈ [1,∞] with constantρ. Using these sets we define the positive and dissipative setsF∗ := P ∩ D∗andF∗(ρ) := P ∩D∗(ρ).


, which

in the

r,

aboveal func-a

n-linear

The circle condition w.r.t. the maximum norm,‖ · ‖∞ and the weighted maximum norms‖ · ‖∞,g

can be checked easily whenf is continuously differentiable. Here‖v‖∞,g := maxgi |vi | with a g ∈ Rn,

g > 0. Namely, we consider the sets

F∗∞(ρ) :=

{f ∈P | f is dissipative in‖ · ‖∞ and

∂fk(t, v)

∂vk

� −ρ ∀k, t, v

},

andF∗∞,w(ρ) := ⋃g>0F∗∞,g(ρ) where

F∗∞,g(ρ) :=

{f ∈ P | f is dissipative in‖ · ‖∞,g and

∂fk(t, v)

∂vk

� −ρ ∀k, t, v

}.

We proved (see [12, Lemma 8]) thatF∗∞(ρ) ⊂ F∗(ρ) and one can prove similarly thatf ∈ F∗∞,g(ρ)

implies that the circle condition holds forf in ‖ · ‖∞,g with constantρ.In the literature an other type of positive problem sets is considered as well (see, e.g., [3,10,14])

has no concern with dissipativity by definition. Namely, for anyα ∈ R we define

Pα := {f | f (t, v) + αv � 0 for all t, v � 0

}.

It is clear by (2) thatPα ⊂ P for arbitraryα.The relation between thePα sets and the other ones introduced above can be summarized

following lemma.

Lemma 2. Letα,ρ > 0. ThenF∗(ρ) ⊂ Pα iff α � 2ρ.

Proof. Let f ∈ F∗(ρ) be arbitrary. Then for anyk index,t ∈ R andv � 0 we havefk(t, v) − fk(t,w) �−2ρvk (see [12, Lemma 3]) withw := v − vkek whereek is thekth coordinate unit vector. Moreovefk(t,w) � 0 sincef ∈ P andw � 0,wk = 0. Thereforefk(t, v) � −2ρvk , which provesF∗(ρ) ⊂ P2ρ .Further,f (t, v) := −2ρv belongs toF∗(ρ) and clearlyf ∈ Pα implies 2ρ � α, which completes theproof of the lemma. �2.3. The scheme functions and the positivity radius of RK schemes

For the construction of some positivity thresholds w.r.t. the positive function classes introducedwe found (see [12]) that the so-called scheme functions play an important role. These are rationtions that appear in the formulas when RK(A,b) is applied to linear IVPs with forcing terms. Asmatter of fact these scheme functions are important also for the positivity analysis of general noproblems.

Thus, we call the functionsKA,JA,Kb, Jb the scheme functions of the scheme(A,b) which are de-fined atX diagonals × s matrices for whichI − XA (and thereforeI − AX) is invertible and

KA(X) = (I − AX)−1e, JA(X) = A(I − XA)−1,

Kb(X) = 1+ bTX(I − AX)−1e, Jb(X) = bT(I − XA)−1.

Here and in the followingI (d) denotes thed-by-d identity matrix for alld positive integer,I withoutsuperscript meansI (s) ande = (1, . . . ,1)T ∈ R

s .We introduced also the positivity radius,R(A,b) of the scheme by

R(A,b) = sup{r � 0 | K ,J ,K ,J � 0 on[−rI,0I ]}.
A A b b


meth-elyequal[16].

es in-e also

rem 7],

t

using

lds the

Notice that we make use of the matrix interval[−rI,0I ] which is the set ofX matrices with−rI �X � 0I , i.e., diagonal matrices with diagonal entries from[−r,0]. Observe that theith component ofKA and theith row of JA can be written byaT

i := (ai1, . . . , ais) = eTi A, the ith row of A, in the form

eTi KA(X) = eT

i (I + AX(I − AX)−1)e = 1 + aTi X(I − AX)−1e andeT

i JA(X) = aTi (I − XA)−1 which

simplify to Kai(X) andJai

(X), respectively, if for anyw ∈ Rs we writeKw(X) = 1+wTX(I −AX)−1e,

Jw(X) = wT(I − XA)−1, in accordance with the definition ofKb,Jb.We shall see below that the positivity radius characterizes the irreducible and non-confluent RK

ods that are positive with some thresholdH > 0 on the positive problems introduced above, namR(A,b) > 0 is equivalent with this. We remark that the positivity radius is proved (see [12]) to beto the absolute monotonicity radius of the scheme, which is introduced and studied thoroughly in

2.4. Positivity thresholds

We proved the following lower bounds for the strict positivity threshold on the problem classtroduced above. Note that for the definition of irreducibility of RK methods consult, e.g., [11], seSection 3.

Proposition 3. Letρ > 0 and(A,b) be the scheme of an irreducible RK method withR(A,b) > 0. Thenwe have the following results:

(1) RK(A,b) is positive onF∗ with thresholdH = ∞ wheneverR(A,b) = ∞.(2) RK(A,b) is positive onF∗(ρ) with thresholdH = R(A,b)

2ρ.

(3) RK(A,b) is positive onF∗∞(ρ) with thresholdH = R(A,b)

ρ.

(4) RK(A,b) is positive onF∗∞,w(ρ) with thresholdH = R(A,b)

ρ.

Proof. For the proofs of the statements (1)–(3) see [12, Theorem 6, parts (a) and (c) and Theorespectively. The proof of the last statement is essentially the same as that of the statement (3).�Theorem 4. Let (A,b) be the scheme of an RK method and∅ �= F ⊂ Pα with anα > 0. Suppose thaRK(A,b) is well-defined on(1) with anyf ∈ F , t0 ∈ R, u0 ∈ R

n and step sizes not larger thanHdef =Hdef((A,b),F). Then

H := min

{R(A,b)

α,Hdef

}is a positivity step size threshold ofRK(A,b) w.r.t.F .

Proof. Observe thatf ∈ Pα implies thatf can be written in the formf (t, v) = D(t, v)v + p(t, v) withD(t, v) := −αI (n), p(t, v) := f (t, v) + αv and herep(t, v) � 0 for all t andv � 0 by the definitionof Pα.

In order to prove the statement of the theorem we apply the proof of [12, Theorem 1] directly,the new definition ofD andp. This may be done since the required properties ofD andp in the proofof Theorem 1 in [12] are fulfilled, which are presented just in the preceding paragraph. This yieproof of the theorem. �


a given

we knowesericaln

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dnd the.7]).

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ion

trict foration.ility of

d with

For a comparison of the results of Proposition 3 and Theorem 4, suppose that we have to solvepositive IVP of form (1) with a fixed RK method of scheme(A,b) with 0 < R(A,b) < ∞ and we areinterested in the largest step size a priori that guarantees positive numerical solutions. Supposethe circle condition holds forf in someLp-norm, f ∈ F∗(ρ). Then, Proposition 3, part (2) ensurthath � R(A,b)/(2ρ) is sufficient to that the method be well-defined on the problem and the numapproximations be positive. Due to Lemma 2, thish � R(A,b)/(2ρ) condition cannot be improved igeneral by applying Theorem 4 only, without knowing more onf .

However, for the given functionf , we may find anα smaller than 2ρ with f ∈ Pα (usuallyf ∈ Pα

can be checked simpler thanf ∈ F∗∞(ρ)) and if in some other ways we can ensure the well-definiteof the method with step sizes not larger thanR(A,b)/α, then Theorem 4 ensures positivity with largthreshold.

To see a typical example of this situation consider (1) and RK(A,b) with

f (t, v) = Lv, L =(−1 η

0 −1

), and A =

(0 0

1/2 1/2

), b = (1/2,1/2)T,

respectively. Supposeη � 0, i.e., the problem is positive. Simple calculations show thatf is dissipativein ‖ · ‖2 iff η ∈ [0,2] and the circle condition holds forf in ‖ · ‖2 with a constantρ < ∞ iff η ∈ [0,2) andin the latter case the smallest constant equals 1/(2 − η). Moreover,R(A,b) = 2. Hence Proposition 3part (2) ensures positivity under conditionh � H = 2 − η, wheneverη ∈ [0,2) and gives no result iη = 2. Observe thatH → 0 whenη → 2−.

On the other hand, for allη ∈ [0,2] we havef ∈ P1 and, in addition, the method is well-definewith all positive step sizes because of the dissipativity of the problem in an inner product norm acoercivity property of theA coefficient matrix of the method (for more details see [11, Theorem 14Hence Theorem 4 ensures positivity of the numerical method under conditionh � 2. A direct calculationof the method applied to the linear problem shows thath � 2 is the sufficient and necessary conditionpositivity, for all η � 0.

For the completeness of this study we consider the case ofη > 2. Then the problem under considetion is not dissipative in‖ ·‖∞ (sinceη > 1) hence in this case part (3) of Proposition 3 cannot be appHowever, we can see that−L is an M-matrix (see [1]) hence it is dissipative in‖ · ‖∞,g with anyg > 0for which Lg � 0 holds and in this norm even the circle condition holds withρ = −mini Lii . In oursituation this applies withg = (η,1)T andρ = 1, hence part (4) of Proposition 3 results in the conditof positivity h � 2, which is the sufficient and necessary condition for positivity for allη � 0.

3. Strict positivity thresholds

In this section we shall prove, as is announced in [12], that the thresholds in Proposition 3 are sirreducible and non-confluent methods. First we discuss shortly the role of irreducibility in our situNotice that the irreducibility condition was used in Proposition 3 only to guarantee unique solvabthe equations (cf. [16]), not directly in connection with positivity.

Now, consider the following two RK methods withs = 2 and schemeA = ( 1 0−1 1

), b = (1 0)T and

A = ( 1 0−1 2

), b = (1/2 1/2)T, respectively.

As is easily seen these methods are equivalent to the implicit Euler method (the RK methoschemeA = b = (1)), i.e., they are well-defined iff the implicit Euler method is such andu ,u , . . . are
1 2


ive onidered

ond-, in thee prover than

t

n

n

e also

ion

the same for these three methods when applied to the same IVP. Thus these methods are positF∗with H = ∞ (because the implicit Euler method is of this property, see [12]), although both consmethods withs = 2 have positivity radius equal to−∞ (sinceJA(X)21 < 0 for all X � 0 diagonalmatrices).

Thus the formulas forH cannot provide in general the strict positivity threshold on the corresping problem sets. However, observe that the RK methods in the previous example are reduciblefirst one in DJ-sense and in the second one in HS-sense (for the definitions see, e.g., [11]). Wbelow in Theorem 8 that assuming DJ-irreducibility and non-confluency, which is a bit strongeHS-irreducibility, the positivity thresholds in Theorem 3 are strict.

Recall that the RK method of scheme(A,b) is said to be non-confluent if all of the entries ofc aredifferent. Further, the reducibility in DJ-sense of thes-stage RK method of scheme(A,b) means thathere exists a non-trivial partition of{1, . . . , s}, {S1, S2} such thatbj = 0 andaij = 0 for all i ∈ S1, j ∈ S2.In our analysis below we need the following equivalent form of DJ-reducibility.

Lemma 5. The RK method ofs stages with scheme(A,b, c) is DJ-reducible if and only if the functioX → Jb(X) has at least one identically zero component.

Proof. It is easily seen that if the method is DJ-reducible and the partition{S1, S2} is taken by thedefinition above thenJb(X)j = bT(I −XA)−1ej ≡ 0 wheneverj ∈ S2. To prove the converse implicatioof the assertion assume thatJb(X)j ≡ 0. Then 0= Jb(X)Xej = bTX(I − AX)−1ej for all X diagonalmatrices. From this relation DJ-reducibility of the method follows by application a result of [6] (se[11, Lemma 15.7]). �

For finding strict positivity thresholds we make use of the test problem sets defined for arbitraryα > 0as

Ltest :={fL,g := L(t)v + g(t) | L : R → R

2×2, g : R → R2 cont.,L21 ≡ 0

},

L∗(α) := {fL,g ∈ Ltest | g � 0, L22 ∈ [−α,0], L11 ∈ [−α/2,0], −L11 � L12 � 0

}.

Lemma 6. For arbitrary α > 0 we have∅ �= L∗(α) ⊂ F∗ ∩F∗(α/2) ∩F∗∞(α).

Proof. Let α > 0 andf = fL,g ∈ L∗(α) be arbitrary. It is enough to prove that the circle conditholds forf in ‖ · ‖∞ with constantα/2. Taking into account (5) we have to prove‖(α

2I (2) + L)w‖∞ �α2‖w‖∞ wherew = v − v is arbitrary. By the definition ofL∗(α) we have‖(α

2I (2) +L)w‖∞ = max{(α2 +

L11)|w1| + L12|w2|, |α2 + L22||w2|} � max{(α

2 + L11 + L12), |α2 + L22|}‖w‖∞ � α

2‖w‖∞. �Lemma 7. Let fL,g ∈ Ltest, t0, u0 be arbitrary and applyRK(A,b) to IVP (1) with step sizeh > 0.Then the first step is uniquely defined iffI − XkA is regular forXk := hdiagi(Lkk(t0 + cih)) (k = 1,2).Moreover, in this case

u1 =(

Kb(X1) Jb(X1)Y1KA(X2)

0 Kb(X2)

)u0 +

(Jb(X1) Jb(X1)Y1JA(X2)

0 Jb(X2)

)γ, (6)

whereY := diag (L (t + c h)) ∈ Rs×s , γ := h(g (t + c h), . . . ,g (t + c h))T ∈ R

s , γ := (γ T, γ T)T.
1 i 12 0 i k k 0 1 k 0 s 1 2


e

ne

the

hod.

on

Proof. For shortness we introduce the following notations:

y := (yT

1 , . . . , yTs

)T, L = diagi

(hL(t0 + cih)

), g = (

hg(t0 + c1h)T, . . . , hg(t0 + csh)T)T

.

Further, the Kronecker product of matrices is denoted by⊗. In connection with the Kronecker product wdenote byπ = π2,s the 2s-by-2s permutation matrix that maps all the 2s-vectors of form(xT

1 , . . . , xTs )T

to (x1,1, . . . , xs,1, x1,2, . . . , xs,2)T. It is straightforward to verify the following assertions: for allv ∈ R

s

andS ∈ Rs×s,N ∈ R

2×2 we have(vT ⊗ I (2)

)πT = I (2) ⊗ vT, π

(v ⊗ I (2)

) = I (2) ⊗ v, π(S ⊗ N)πT = N ⊗ S.

Applying the introduced notations (3), (4) can be written in the form

u1 = u0 + (bT ⊗ I (2)

)(Ly + g), (7)

y = (e ⊗ I (2)

)u0 + (

A ⊗ I (2))(Ly + g). (8)

It is clear that (8) has a unique solution iffI ⊗I (2) −(A⊗I (2))L invertible. Making use of the permutatiomatrix π introduced above we may write with the matricesX1,X2, Y1 defined in the declaration of thlemma

I ⊗ I (2) − (A ⊗ I (2)

)L = πTπ

(I ⊗ I (2) − (

A ⊗ I (2))L

)πTπ = πT

(I − AX1 −AY1

0 I − AX2

)π

and this is invertible iffI − AX1 and I − AX2 are invertible, which shows the first statement oflemma.

Assuming the solvability of (8) and expressingy from (8) and inserting into (7) we obtain

u1 = (I (2) + (

bT ⊗ I (2))L

(I ⊗ I (2) − (

A ⊗ I (2))L

)−1(e ⊗ I (2)

))u0

+ (bT ⊗ I (2)

)(I ⊗ I (2) − L

(A ⊗ I (2)

))−1g. (9)

Manipulating with the matrixπ on (9) we get

u1 = (I (2) + (

bT ⊗ I (2))πTπLπTπ

(I ⊗ I (2) − (

A ⊗ I (2))L

)−1πTπ

(e ⊗ I (2)

))u0

+ (bT ⊗ I (2)

)πTπ

(I ⊗ I (2) − L

(A ⊗ I (2)

))−1πTπg

={I (2) +

(bT 00 bT

)(X1 Y1

0 X2

)(I − AX1 −AY1

0 I − AX2

)−1 (e 00 e

)}u0

+(

bT 00 bT

)(I − X1A −Y1A

0 I − X2A

)−1

γ

which results in (6) directly. �Theorem 8. Let H,α > 0 and (A,b) the scheme of a DJ-irreducible and non-confluent RK metThen, ifRK(A,b) is positive onL∗(α) with step sizeH , we haveH � R(A,b)

α.

Consequently, the thresholds appearing in Proposition3 are strict, whenever the method is in additinon-confluent.


lsn

on

t

fment is

-

Proof. Let Z ∈ [−HαI,0I ] be an arbitrary fixed diagonal matrix. It is enough to prove thatI − ZA isregular and

Kw(Z) � 0, Jw(Z) � 0, ∀w ∈ {b, a1, . . . , as}.Indeed, these imply that the scheme functions are non-negative on[−HαI,0I ], henceR(A,b) � Hα,which is to be proved.

We may assume that the stages are labeled such thatc1 < c2 < · · · < cs hold. Further, leth := H andt0 := 0 be fixed.

In the proof we say that the functionλ :R → R fits a diagonals-by-s matrix X if hλ(cih) = Xii

(∀i ∈ {1, . . . , s}), λ is linear betweencih andci+1h (∀i ∈ {1, . . . , s − 1}), and constant on the interva(−∞, c1h] and[csh,∞). It is clear that for eachX diagonals-by-s matrix there exists uniquely a functiothat fits it.

First we show thatI −ZA is invertible. For this letλ fit Z, L(t) = ( 0 00 λ(t)

)andg(t) = (0, g2(t))

T withan arbitraryg2 � 0 continuous function. ThusfL,g ∈ L∗(α) and according to the positivity assumptiof the method (8) has a unique solution. Now this equation reads, after manipulating withπ as above,

πT

(I 00 I − AZ

)πy = e ⊗ u0 + A ⊗ I (2)g. (10)

Since (10) has a unique solution for allu0 and, say,g = 0, I − AZ and thereforeI − ZA are invertible,which was to be proved.

Inserting the solution of (10) into (7) we obtainu1 = ( u0,1Kb(Z)u0,2+Jb(Z)γ2

). Since hereu0, γ2 � 0 are

arbitrary, we getKb(Z), Jb(Z) � 0. This implies thatKb,Jb are non-negative on[−HαI,0I ].Now let i ∈ {1, . . . , s} be arbitrary andw = ai := (ai1, . . . , ais)

T, theith row ofA. We shall prove thaKw(Z) � 0 andJw(Z) � 0 hold true which will complete the proof of the theorem.

For this letL22 fit Z and chooseX1 ∈ [−H α2I,0I ) such thatJb(X1)ei �= 0. Such anX1 does exist

due to Lemma 5. (Notice that each component ofJb(X) is a rational function ofs variables hence ithey are not identically zero then they are zero only on a set of zero measure hence its compledense near 0I .) Applying the results above thatKb andJb are non-negative on[−HαI,0I ] we obtainthatξ := Jb(X1)ei > 0.

Chooseη > 0 such thatη + X1,ii < 0 and letY1 := η eieTi . Further, letL11 andL12 fit X1 andY1,

respectively, andL21 ≡ 0. ThenfL,g ∈ L∗(α) is fulfilled with arbitrary continuousg � 0. Indeed,fL,g ∈Ltest, furtherL22(t) ∈ [−α,0], L11(t) ∈ [−α/2,0] and−L11(t) � L12(t) � 0 hold for t ∈ (−∞, c1h] ∪{c1h, . . . , csh} ∪ [csh,∞) by the construction ofX1, Y1 and the assumption onZ; betweencih andci+1h

this follows from the nature of linear interpolation.In this way we have

u1 =(

Kb(X1) ηξKw(Z)

0 Kb(Z)

)u0 +

(Jb(X1) ηξJw(Z)

0 Jb(Z)

)γ.

Sinceξη > 0 andu1 � 0 for all u0 � 0 andγ � 0,

Kw(Z) � 0, Jw(Z) � 0,

which was to be proved.�Remark 9. From this theorem we can see that the propertyR(A,b) > 0 really characterizes the irreducible and non-confluent RK methods which are positive, say, onF∗ (ρ). But if R(A,b) > 0 holds for
∞


in [16]cit

ining

licitods weods

er

of the

prob-,

old

ortion

sly, weinitial

e heat

the sec-m

RK(A,b), it has many negative consequences on the order of the method. Namely, it is provedthatR(A,b) > 0 implies—among others—A � 0, moreover, the classical order of explicit and implimethods cannot be greater than 4, or 6, respectively, and even the stage order is at most 2.

Remark 10. Theorem 8 shows that for an irreducible and non-confluent RK method RK(A,b) thepositivity step size thresholdR(A,b)/α cannot be increased concerning positive problem sets contaL∗(α). In [14] a positivity step size threshold w.r.t.L∗(α) is given in the formC(A,b)/α whereC(A,b)

comes from the coefficients when RK(A,b) is expressed as linear combination of steps with the expand implicit Euler method. By Theorem 8 we conclude that for irreducible and non-confluent methhaveC(A,b) � R(A,b). This result has a contribution to total variation diminishing (TVD) RK methas well because the step size condition ensuring TVD property ish � C(A,b)/α whereα is a measureof the problem, see [17]. Thus, if, for RK(A,b) we do not haveR(A,b) > 0 then the results in the pap[17] do not ensure discrete TVD property. Note that the resultC(A,b) � R(A,b) is also shown in [8],with the help of a non-linear, non-continuous construction. There, in [8] an extensive examinationTVD step size thresholds is presented and the role ofR(A,b) for RK methods with TVD property isdiscussed thoroughly.

4. Step size thresholds of RK methods w.r.t. a discrete positively invariant positive cone

4.1. Motivation

In the previous section we proved strict positivity thresholds for RK methods on several positivelem sets. This gives us, for instance, that considering the positive problem setF∗∞(ρ) and an irreduciblenon-confluent RK(A,b) method, the largest step size which guarantees a priorium � 0 (m = 1,2, . . .) forall u0 � 0 andf ∈ F∗∞(ρ), equalsR(A,b)/ρ if R(A,b) > 0 and 0 otherwise. Concerning the threshwe can see that it is often very small if at all positive. For example,R(A,b) = −∞ for many meth-ods of practical interest (since already one negative coefficient inA implies this). Even ifR(A,b) ispositive, it is a constant typically not greater than 3. Further,ρ is rather large for many problems. Fexample,ρ ∝ 1/(x) andρ ∝ 1/(x)2 for problems corresponding to semi-discretization of advecor respectively diffusion problems wherex is a typical mesh size of the space discretization.

However, it is a computational experience that for manyparticular u0 � 0 the strict threshold givea too pessimistic a priori prescription to guarantee positivity at numerical time integration. Namecan see from actual computations that for diffusion problems starting from a quite smooth positivevector the numerical solution remains positive with much larger step sizes thanR(A,b)/ρ, and, after fewexceptional steps it becomes and thereafter remains positive.

To illustrate the ideas above we take the following simple test example.Consider the initial-boundary value problem for one-dimensional parabolic problem called th

equation

ut = uxx, u(t,0) = u(t,1) = 0, u(0, x) = given, ∀t > 0, x ∈ [0,1].Semi-discretization with finite differences based on the second order standard approximation ofond derivative with central schemes on an equidistant mesh ofn internal points gives the linear proble(1) with

f (t, v) = Lv, L = 1/(x)2 tridiag(1,−2,1) ∈ Rn×n, x := 1/(n + 1). (11)


s

sitive

initialhandlescould

yed inRadau

es withectorsher someinitial

ition toena

ions and

Table 1Time-step sizes ofh = a10−k (a, k ∈ {0,1, . . . ,10}) when u1 � 0 for both x ∈{0.01,0.001} (second and third line) and strict positivity thresholds (last line)

u0 θ = 0.6 Lobatto IIIC,s = 2 Radau IIA,s = 2

δ1/40 h � 2(x)2 h � 0.1 –x(1− x) h � 0.3 ∀h � 0 h � 0.3

theory (∀u0) h � 1.25(x)2 h = 0 h = 0

We considered two testproblems with the following initial values:

• the projection ofδ1/40, Dirac’s delta function concentrated atx = 1/40; henceu0,i = 1/x wheneveri = �1/(40x)� andu0,i = 0 otherwise;

• the projection ofx(1− x), i.e.,u0,i = ix (1− ix) for all i.

As numerical methods we took theθ -method withθ = 0.6, Lobatto IIIC method withs = 2 and RadauIIA method with s = 2 (see [11]). We applied these RK methods to the testproblem withn ∈ {99,999}and took one time-step with step sizeh = a 10−k , a, k ∈ {0,1, . . . ,10} and investigated which time-stepproduceu1 � 0 for bothn; the results are given in the second and third line of Table 1.

Observe thatf ∈ F∗∞(ρ) with ρ = 2/(x)2. SinceR(A,b) equals 1/(1 − θ) for the θ methods and−∞ for the other two methods, we could expect the largest step size yieldingu1 � 0 as is written in thelast line of Table 1 (cf. Theorem 8). We note that the theory of Bolley and Crouzeix for linear poproblems (see [3]) gives exactly the same theoretical thresholds.

Comparing the theoretical thresholds with the results of the experiments with the “non-smooth”vector coming from the delta function we can see a good accordance. However, this theory thatall u0 � 0 at the same time gives no relevant result to the second initial value, the “smooth” one: wenot expect from this theory that positivity will be preserved with so large time-steps that are displathe third line of Table 1 (for other numerical experiments demonstrating this nice property of theIIA method see also [5]).

In the rest of this section we try to give a precise mathematical explanation of these experiencpositive and smooth initial vector. Roughly speaking, we shall specify subsets of positive initial vand step size restrictions for RK methods such that theum approximations remain in this set during ttime-integration and are proved, therefore, positive. We shall do this for some linear and also fonon-linear problems. Finally, the explanation of the actual numerical experiment with the “smooth”vector can be found in Remarks 18 and 19.

4.2. Definitions

We consider here IVPs of type (1) and use the definitions and notations as previously. In addthese, we writeV = R

n, V + = [0,∞)n and introduce some further definitions to treat the phenommentioned in the preceding subsection. We remark that for a deeper discussion of these definitnotions see, e.g., [18,20].


ern of

hat

ay

)ults

ten-

.

t

Definition 11. Let ∅ �= S ⊂ V andf be given. We callS positively invariant w.r.t. (1) iff∀t0∀u0 ∈ S∀t � t0 we haveU(t) ∈ S . If F is a set of right side functions then we callS positively invariant w.r.t.Fif S is positively invariant for all IVP (1) withf ∈F .

Notice that the word “positively” means the direction in time axis only and has no direct conc“positivity” of the previous sections.

Definition 12. Let ∅ �= S,S0 ⊂ V . We say thatS attractsS0 w.r.t. (1) iff ∀t0∀u0 ∈ S0 ∃t1 � t0 such thatU(t) ∈ S for all t � t1.

Now we formulate the discrete analogue of these definitions.

Definition 13. Let an IVP be given by (1),S ⊂ V and(A,b) the scheme of an RK method. We say tS is discrete positively invariant w.r.t. (1) under RK(A,b) with step size thresholdH if for all t0, u0 ∈ Sandhm ∈ [0,H ] (m = 1,2, . . .) we haveum ∈ S .

Definition 14. Let an IVP be given by (1),S,S0 ⊂ V and(A,b) the scheme of an RK method. We sthatS attracts discretelyS0 w.r.t. (1) under RK(A,b) with step size thresholdH iff for all t0, u0 ∈ S0 andhm ∈ [0,H ] (m = 1,2, . . .) there existsm0 positive integer such that for allm � m0 we haveum ∈ S .

We remark that using these definitions, the formerly used terms “IVP (1) is positive” or “RK(A,b)

is positive on (1) with step size thresholdH ” can be formulated as “V + is positively invariant w.r.t.IVP (1)” and “V + is discrete positively invariant w.r.t. (1) under RK(A,b) with step size thresholdH ”,respectively.

Our aim can be formulated now as follows: for a given problem setF and method RK(A,b) findS = S(F) ⊂ V + andH = H(S, (A,b),F) > 0 such thatS is discrete positively invariant w.r.t. IVP (1with all f ∈ F and strict step size thresholdH . (Continuing the line of the previous paragraph, the resof Theorem 8 give formulas forH(V +, (A,b),F) for certainF , for exampleH(V +, (A,b),F∗∞(ρ)) =R(A,b)

ρfor irreducible and non-confluent methods whereR(A,b) > 0.) Then, if H(S, (A,b),F) is

much larger thanH(V +, (A,b),F) and the givenu0 � 0 happens to belong toS , the requiremenhm � H(S, (A,b),F) (for all m) is a more relevant a priori prescription on the time-step sizes tosureum � 0 for all m than that one containing the globalH(V +, (A,b),F).

We shall see below that some cones form the appropriate positively invariant sets in our theory

Definition 15. Let V be a linear space,C ⊂ V . Then we callC a cone if∀x, y ∈ C, ∀α,β ∈ [0,∞) wehaveαx + βy ∈ C.

We remark thatV + is a cone inV .

4.3. Construction of a positive coneC

In this subsection we construct a positive coneC, i.e., a cone withC ⊂ V +. We shall see below thathisC will serve as a positively invariant set for certain IVPs.


e

in

ave

with

led by

Lemma 16. Suppose that{sk} is a fixed basis ofV such thats1 > 0 (i.e., each component ofs1 is strictlypositive) and letσk be the smallest positive number such thatσks1 ± sk � 0 (k = 1, . . . , n). Let us define‖ · ‖P :V → [0,∞) as‖v‖P := ∑

k σk|ηk| wheneverv = ∑k ηksk.

Then‖ · ‖P is a norm onV andC := {v = ∑k ηksk ∈ V | ‖v‖P � 2η1} is a positive cone, i.e., a con

with C ⊂ V +.

Proof. The lemma is a consequence of the more general results of [13, Lemma 6].�4.4. Positively invariant cones for linear problems

To formulate our main results with the linear problems we use the following notations: ifL ∈ Rn×n

then fL :Rn → Rn with fL(v) := Lv. Further, if r is a one-variable rational function thenMr is the

monotonicity radius of the functionr defined by

Mr := sup{µ � 0 | r is strictly increasing on[−µ,0] andr(−µ) �

∣∣r(ξ)∣∣ ∀ξ ∈ (−∞,−µ)

}. (12)

Theorem 17. Suppose that{sk} is a given basis ofV , s1 > 0, � � 0 and letC be the cone constructedLemma16. Consider theL ∈ R

n×n matrices with the property

Lsk = λksk, 0� λ1 � λk ∀k, andλ1 � −� (13)

and let us defineF�({sk}) := {fL | L is of type(13)}. Then we have the following results:

(1) C is positively invariant w.r.t.F�({sk}).(2) C is discrete positively invariant w.r.t.F�({sk}) underRK(A,b) with step size thresholdH = Mr

+�

wherer is the stability function of the method: r(x) := 1 + bT(I − xA)−1xe (i.e., r(x) = Kb(xI))andξ+ := max{ξ,0} (for arbitrary ξ ∈ [−∞,∞]).

(3) If in additionλk < λ1 for all k �= 1 and{sk} is orthogonal in thel2 inner product,C attractsV + w.r.t.F�({sk}). Moreover,C discretely attractsV + w.r.t.F�({sk}) underRK(A,b) wheneverhm � Mr

+�

forall m.

Proof. Suppose thatMr > 0. Then, ifh � Mr/� � Mr/(−λ1), we have for the eigenvalues ofr(hL) that|r(hλk)| � r(hλ1). Therefore, ifu0 = ∑

k ηksk ∈ C, i.e.,∑

k σk|ηk| � 2η1 thenu1 = ∑k r(hλk)ηksk and

‖u1‖P = ∑k σk|r(hλk)||ηk| � r(hλ1)

∑k σk|ηk| = r(hλ1)‖u0‖P � r(hλ1)2η1 which showsu1 ∈ C. This

proves statement (2) of the theorem.Statement (1) can be proved in the same way, justr has to be replaced by exp and in this case we h

|r(hλk)| � r(hλ1) for all h � 0.To prove statement (3) of the theorem first observe, thatv ∈ V + impliesη1 = 〈v, s1〉/〈s1, s1〉 > 0 and

then our statement follows from [13, Theorem 7].�Remark 18. Suppose that−L is an M-matrix andL + ρI (n) � 0, i.e.,fL ∈ F∗∞,w(ρ). Then we knowfrom [3] that H(V +, (A,b), {fL}) = R+

r /ρ whereRr is the absolute monotonicity radius ofr . SinceRr = −∞ for many methods, for example for the considered Lobatto IIIC and Radau IIA methodss = 2 (see, e.g., [11]), this theory does not guarantee positivity for any step sizes. But, ifL has a completeeigenvector system, e.g., it is in addition symmetric then the conditions of Theorem 17 are fulfil


u

st 2.

an

eorem.

mooth

ome

lemivity for

thatappliest paper.

in

the Perron–Frobenius theory (see, e.g., [1]), hence we can guaranteeum � 0 (or, what is sharper,um ∈ C)for all m wheneverMr > 0, hm � Mr/(−λ1) for all m andu0 ∈ C do hold. Further, usuallyMr > 0 is amuch weaker condition thanRr > 0. For example,Mr = +∞ andMr ≈ 2.26 for the Lobatto and Radamethods under consideration, respectively. Hence these methods are positive on{fL} wheneveru0 ∈ C.For completeness we mention that for theθ -methods withθ > 0.5 we haveMr = 2θ−1

2θ(1−θ).

In [3] it is proved also thatRr = ∞ cannot be achieved with an RK method of classical order at leaHence 1 is the order barrier for unconditional positive methods (i.e., whenH = ∞ is the strict thresholdfor positivity) onV +. However, for smaller cones likeC second order and unconditional positivity chappen, see the Lobatto IIIC method withs = 2.

Remark 19. For an application of Theorem 17 we have to check whether the conditions of the thare fulfilled. Here below we give some examples for this case. First considerL as is defined in (11)(This is a typical example of the case when−L is a symmetric M-matrix.) Here, for alln, � = 10 appliesandsk are the vectors withsk,i := √

2sin(kπix) for all k = 1, . . . , n, i = 1, . . . , n. Then{sk} fulfills theconditions of Theorem 17 and it can be seen thatσk � k. Hence, if

∑k k|ηk| � 2η1 thenu0 = ∑

k ηksk ∈ C.As an example of applications of this we can see that the projection of the functionx(1 − x) belongsto C. This explains the good positivity behaviour of our methods on the linear problem with the sinitial vector discussed in Section 4.1.

Remark 20. Let nowL = M−1K from the finite element semi-discretization of the heat equation or slinear integral equation by piecewise linear elements (M is the mass matrix,K is the stiffness matrix). Weknow that the sign structure of the elements ofL is in general of checkerboard pattern, hence the probis not positive (see, e.g., [19,21]). Hence, for the time stepping schemes we cannot expect positsmall step sizes for allu0 and in fact this has been proved (see [7] in connection with theθ -methods).Since the underlying PDE problem is positive we want to have positive solutions. One can proveL

has a positive eigenvector corresponding to the largest eigenvalue (cf. [21]) hence Theorem 17here. We shall give a more detailed analysis of positivity of finite element methods in a subsequen

4.5. Positively invariant cones for quasilinear problems

We have the following results as generalizations of the previous subsection.

Theorem 21. Suppose that{sk} is a given base ofV , s1 > 0, � � 0 and letC be the cone constructedLemma16. Further, suppose thatL is of type(13), g = g(t, v) a continuous function andγ � 0 such thatγ v + g(t, v) ∈ C for all t andv ∈ C. We assume thatγ = 0 if g(t, v) does not depend onv.

ThenC is discrete positively invariant w.r.t.{fL,g} under RK(A,b) with step size thresholdM(A,b)

�+γ

wherefL,g(t, v) := Lv + g(t, v) andM(A,b) is the monotonicity radius of the method:

M(A,b) := min{Mr | r(x) is a component ofKA(xI), JA(xI),Kb(xI), Jb(xI )

}.

Moreover, ifg depends only ont thenM(A,b) has to be replaced by

Mlin(A,b) := min{Mr | r(x) is a component ofKb(xI), Jb(xI )

}.

Proof. The main idea of the proof is to considerf (t, v) = (L−γ I (n))v + (γ v +g(t, v)) =: Lv +p(t, v)

and write themth step of the method in the form (cf. [12])


isusing

givena

err-m.tarting

i-ant

n earlier

.

979.Numer.

um = Kb

(hL

)um−1 + Jb

(hL

)P, (14)

y = KA

(hL

)um−1 + JA

(hL

)P, (15)

wherey := (yT1 , . . . , yT

s )T, P := (hp(tm−1 + c1h,y1)T, . . . , hp(tm−1 + csh, ys)

T)T. Notice thatKb(hL) in(14) is to be understood as the rational function with formulax → Kb(xI) evaluated athL; the samedefinition is valid for the other three matrix coefficients in (14), (15).

Observe that the cone constructed by the eigenvectors ofL according to Lemma 16 is againC andλk(hL) = λk(hL) − hγ � −h(� + γ ). Moreover,p(t, v) ∈ C for all t andv ∈ C, thusP ∈ Cs if y ∈ Cs .

First we prove the statement of the theorem in the linear case with forcing term, i.e., wheng = g(t).In this caseP ∈ Cs always holds, independently ofy. Now supposeum−1 ∈ C andh � Mlin(A,b)/�; thenapplying statement (2) of Theorem 17 to the components ofKb,Jb andum−1,P playing the role ofr andu0, respectively and exploiting the cone property ofC, we can see from (14) thatum ∈ C.

The proof of the statement with the non-linear case is more complicated because then forP ∈ Cs weneedy ∈ Cs hence (15) has to be taken into account as well. To provey ∈ Cs the same technique asgiven in [12], Theorem 1 applies here, starting from (14), (15) instead of (9.a), (9.b) in loc. cit. andagain statement (2) of Theorem 17 to the subproblems as in the previous paragraph.�Example 22. One example to a non-linear problem which fulfills the conditions of the theorem isby L as in (11) andg(v) = Γ (‖v‖2

2)v whereΓ a real function with|Γ | � γ . This problem representsnon-local diffusion–reaction equation (cf. [20, Example 6.3]).

Remark 23. Notice that the very favourable situationMlin(A,b) = ∞ can happen to second or highmethods, e.g., to the Lobatto IIIC method withs = 2, which results in unconditional positivity presevation (i.e.,H = +∞) starting from the coneC for linear problems with time dependent forcing terWe remark that unconditional positivity with second or higher order RK methods is impossible, sfrom arbitrary positive initial vector (see [3]).

As to the thresholdM(A,b)/(� + γ ) to ensure discrete positive invariance ofC for quasilinear prob-lems, in many applications it is much greater than the strict positivity threshold of formR(A,b)/ρ,despite the relationM(A,b) � R(A,b). For instance, the denominator� + γ is constant for the semdiscretized 1D diffusion–reaction problems (here� is approximately the one-sided Lipschitz constw.r.t. ‖ · ‖2 of the operator corresponding to the diffusion), whileρ involves 2/(x)2, which is propor-tional to the Lipschitz constant of the diffusion part.

Acknowledgements

The author thanks the unknown referees for their valuable comments on the presentation of aversion of this paper.

The work of this paper was supported in part by CBC PHARE award No. 2002/000-317-02-20

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