Positivity of Runge-Kutta and diagonally split Runge-Kutta methods

APPUED NUMERICAL

MATHEMATICS ELSEVIER Applied Numerical Mathematics 28 (1998) 309-326

Positivity of Runge-Kutta and diagonally split Runge-Kutta methods

Zolt&n Horvdth * Department of Mathematics, Szkhenyi Istvrin College, F! 0. Box 701, H-9007 Gy& Hungary

Received 1 September 1997; received in revised form 27 February 1998; accepted 2 April 1998

Abstract

In this paper we study positivity of general Runge-Kutta (RK) and diagonally split Runge-Kutta (DSRK) methods when applied to the numerical solution of positive initial value problems for ordinary differential equations. Here we mean by positivity that the nonnegativity of the components of the initial vector is preserved. First we state and prove a theorem that gives conditions under which a general RK or DSRK method is positive on arbitrary positive problem set. Then we study problems which are simultaneously positive and dissipative. For such problems we give the maximal step size that-under a solvability assumption on the algebraic equations defining the method-guarantees positivity. We show how the step size threshold is governed by the radius of positivity, which is an inherent property of the scheme. This result ensures that we can construct DSRK methods which are unconditionally positive and have an order higher than 1. Note that such a method does not exist between the classical methods. Investigating the radius of positivity of RK methods further we can get rid of the additional solvability condition. In this way we can give a complete positivity analysis for RK methods. We calculate the positivity threshold for some methods, which are of practical interest. Finally we prove a theorem which generalizes the well-known result of Bolley and Crouzeix to nonlinear problems. 0 1998 Elsevier Science B.V. and IMACS. All rights reserved.

Keywords: Runge-Kutta methods; Diagonally split Runge-Kutta methods

1. Introduction

Consider the following initial value problem (IVP) for systems of ordinary differential equations (ODES)

U’(t) = f(t, u(t)), t 3 to, U(t0) = UO? (1)

where to E IR, n is a positive integer, uo E IP and f : IR x IR” + R” has the following property:

f is continuous and (1) has a unique uncontinuable solution for all to E R and u. E KY.

* E-mail: zhorvathQd7.szif.hu.

(2)

0168-9274/98/$19.00 0 1998 Elsevier Science B.V. and IMACS. All rights reserved. PII: SO 168-9274(98)00050-6

310 Z. Horvdth /Applied Numerical Mathematics 28 (1998) 309-326

We call the ODE in (1) as well the IVP (1) positive if f is of type (2) and U(t) 2 0 holds for all t > to whenever 10 E Iw and uo 3 0. (Throughout the paper 3 and > are meant entrywise for matrices or vectors.) We denote by P the set of functions f for which the corresponding ODE in (1) is positive.

Positive IVPs arise very often, for example when modeling chemical reactions or semidiscretizing partial differential equations of advection-diffusion type (see, e.g., [ 10,11,20]). In both cases the components of U denote concentrations or temperature. For other applications where positivity of IVPs are investigated we refer to [ 191. Note that positive problems arising from practical problems often are dissipative in a suitable norm.

Solving a positive IVP with nonnegative initial vector numerically, it is a natural demand that the resulting numerical approximations should be nonnegative. This is often necessary to assure a correct physical meaning of the approximations as is in the problems just cited above. Furthermore, a negative value may cause blow up of the numerical solution process (see [ 163 for a study of this possibility).

Motivated by these reasons it is important to analyze numerical methods from the point of view of positivity, the conservation of nonnegativity of the initial vector. For such an analysis we make use of the following definition.

Definition 1. Let there be given a one-step method, F c P and 0 < H < 00. We call the method positive on F with threshold H if the numerical approximations obtained by the method are uniquely defined and are nonnegative whenever the method is applied to the IVP (1) with any f E J=‘, to E Iw, ua 2 0 and with step size at most H. If this holds with H = 00 or some H E (0, w) then we call the method unconditionally or, respectively, conditionally positive on 3.

Concerning linear problems, the most important result in the literature has been obtained by Bolley and Crouzeix in [3] where conditional and unconditional positivity of methods (including the class of Runge- Kutta (RK) methods) is examined on some dissipative, positive and linear problem sets. It is proved there that unconditional positivity of the methods on the considered problems implies the order barrier of 1 for the methods. With regard to a class of positive nonlinear problems, Hundsdorfer investigated sufficient conditions for positivity for a class of explicit and diagonally implicit RK methods (see [lo, 1 l]), following an idea of Shu and Osher [ 171. However, the literature does not provide a thorough analysis for positivity of general RK methods. The primary aim of this paper is to examine positivity of RK methods under as general conditions as possible.

Studying methods that are unconditionally contractive in the maximum norm for linear problems, Spijker proved in [18] that a general class of methods, which includes the RK methods, is subject to an order barrier of 1 and he obtained, moreover, that the positivity and contractivity order barriers arise from the same property of the methods. To get unconditionally contractive methods of an order higher than 1, Bellen et al. introduced a generalization of RK methods, the family of diagonally split Runge- Kutta (DSRK) methods (see [l]). They and Bellen and Torelli [2] proved that a class of DSRK methods, namely the strongly ANf(O)-stable DSRK methods are unconditionally contractive in the maximum norm and they can have an order higher than 1.

Hence it seems worth to investigate DSRK methods from the point of view of positivity; for linear problems such an investigation is given in [8,9]. As one could expect the strongly ANf(O)-stable DSRK methods “break” the order barrier of [3]. Note that the strongly ANf (0)-stable DSRK methods that fulfill a slight additional condition are subject to an order barrier of 4, as is shown by in ‘t Hout in [ 131.

Z. Horvdth / Applied Numerical Mathematics 28 (1998) 309-326 311

In the rest of the paper we examine positivity of general RK and DSRK methods on general positive and particularly positive and dissipative problem sets, in the following organization.

In Section 2 we present a characterization of positive problems, introduce the diagonal split form of functions, which is of crucial importance in our analysis, and give the definition of the methods considered. In Section 3 we state and prove a theorem that gives conditions under which a general RK or DSRK method is positive on arbitrary positive problem set. This is formulated in terms of the scheme functions, some rational functions determined by the scheme of the method. Then, in Sections 4 and 5, we study problems which are simultaneously positive and dissipative. First we introduce the positivity radius of RK and DSRK methods, a real number computed only from the scheme of the method. Then we obtain our first main result, Theorem 2, in which-under an assumption on solvability of the algebraic equations that define the method-we give the maximal step size that guarantees positivity; this threshold is a simple expression of the positivity radius and a measure of dissipativity. This theorem ensures that we have unconditionally positive DSRK methods on positive and dissipative problems of an order higher than 1. In Section 5 we analyze further the results of the previous sections for RK methods. First we prove that the positivity radius equals the absolute monotonicity radius for RK methods; the latter one is introduced and completely investigated by Kmaijevanger in [14]. Employing and investigating further some results from [ 141 we can get rid of the additional solvability condition. In this way we can give a complete positivity analysis for RK methods. presented in Theorem 6. As an application of the theory developed in this paper we calculate the positivity threshold for some methods in Example 1 and demonstrate with an example (see Example 2) that violation of our conditions may lead to a complete loss of conservation of nonnegativity. Finally, with Theorem 7 we arrive at the third main result of this paper, a generalization of the well-known theorem of [3] to nonlinear problems.

2. Preliminaries: notations, definitions, basic results

Throughout the paper we use the following notations: 0 for Vi EIFxm (i=l,..., s), diag,Vi denotes the sn-by-sm blockdiagonal matrix whose blocks in

the main diagonal are VI, . . . , V, in this order; ?? e(“’ := (1, . . . , l)T E IKY, I@) is the n-by-n identity matrix; we omit the superindex denoting the

dimension of the vector and matrix if this does not do confusion: ?? we use subindices to denote the components of vectors and (vector valued) functions: if the name

of the vector or function involves a subindex then a back subindex denotes the component, i.e., vi-k denotes the kth component of Ui.

2.1. Characterization of positive problems

In this subsection we present a lemma that characterizes positive IVPs which constitutes a slight generalization of the corresponding lemma of [lo].

For the ease of presentation we introduce the following notation: for all n positive integer, k E { 1,

. . . ) n}andu~IR”,u~~ddenotes(~~ ,..., vk-i.a,uk+i ,..., U,)T E Iw”.

Lemma 1. Let f be of type (2). Then the corresponding IVP (1) is positive iJg

fk(t,u&O)>O foraZZkE{l,..., n}, t EIR, u E IR; := [O, w)n. (3)


Proof. The necessity of condition (3) is clear: if, on the contrary, f would be the right-hand side function

ofthepositiveIVP(l)withatER,kE{l,... , n} and V E !X; for which fk(t, i? & 0) < 0 holds then

for (1) with to = 7, ua = V -k 0 we would have uo 3 0, U,(to) = 0, UL(to) -C 0 hence Uk < 0 in a right neighbourhood of to, in contradiction with the positivity of (1).

In order to prove the sufficiency, suppose on the contrary that f is of type (2) and (3), moreover ua E rW; and to E R are such that for the solution of the corresponding IVP (1) there holds

tt = sup {? > to 1 U(t) 3 0 for all t E [to, t]} < T.

Here T denotes the endpoint of the interval on which the unique uncontinuable solution exists. Let t2 E (tl , T) be arbitrary and, for any E > 0, consider the problem

U’(t) = f(t, U(t)) + ce, t 3 to, U(t0) = 240 + Et?. (4)

Let U, be any uncontinuable solution of (4). Since f is continuous and U is the unique solution of (1) on [to, tz] we have, for sufficiently small E > 0, that ZJ, is defined on [to, t2] and lim,,o UE(t) = U(t) for all t E [to, t2] (see [5, p. 181). Hence we can choose an ~1 > 0, t3 E [to, t2] and k E { 1, . . . , n} such that U,,,,(t,) = 0 and U,,(t) > 0 for all t E [to, t3). This implies Uj,,,(t,) < 0.

On the other hand, UL,,k(t3) = fk(t3, U,, (tg)) + ~1 > ~1 > 0, which is a contradiction. ??

2.2. Diagonal split form offunctions

In connection with Lemma 1 we introduce the following form of functions, the diagonal split form, which turns out to be useful in analyzing numerical methods from the point of view of positivity.

Definition 2. Let f : IF! x IIt” --+ IP. The diagonal split form of f is defined as f (t, v) = D(t, u)v + p(t, v) with D(t, v) = diag(di (t, v), . . . , d,(t, v)), where for all k E { 1, . . . , n} and t E IR, v E IP

pk(t, v> := fk(t, 2, -& o),

~(fi(t.~)-.fx(t,uLo)), ifuk#&

afk if vk = 0 and av (t, v) exists , k

otherwise.

Remark 1. Referring diagonal split forms, Lemma 1 can be reformulated as follows: let f be a function of type (2) and f = Du + p be its diagonal split form; then IVP (1) is positive if and only if p(t, v) > 0 foralltEIWandvEIW1.

Remark 2. If f is linear, i.e. f (t, u) = L(t)v + g(t) with some square matrix valued function L, then D(t, u) = diag, L(t)ii, the main diagonal part of L(t). Further, in a few problems the diagonal split form of f arises in a natural way. For example, if modeling chemical reactions, DZJ and p are called the loss and the production terms, respectively (see, e.g., [20]).

We conclude this subsection with the following lemma, which gives a simple means to get bounds for the functions dk .

Z. Horvdth /Applied Numerical Mathematics 28 (1998) 309-326 313

Lemma 2. Let f : II% x IF’ + BY’ be a function whose kth component function, fk = fk(t, v) is differentiable with respect to vk. Then

afk afk inf -(t, w) 6 dk(t, v) < sup -(t, w) UJ&” auk w&@” auk

for all t E iI%, v E IV.

Proof. The statement of the lemma is a straightforward consequence of the mean value theorem. ??

2.3. RK and DSRK methods

Here we specify the numerical methods that we examine in the rest of the paper. These are the RK and the DSRK methods (see, e.g., [ 1,2,4,6,7]). Both are one-step methods for solving (1) numerically, hence it is enough to define the first step and the others are defined recursively.

Consider the IVP (1). The first step of an RK method of step size h t > 0 produces an approximation u1 toUatti =to+hl thatisdefinedas

u~=uo+h,Cbif(to+cihl,yi), i=l

(54

where { yi }f= 1, the set of stage values, are solutions of the system of algebraic equations

yi=L10+h,Caijf(t~+cjhl,yj), i=l,..., s. (5b) j=l

In the formulas (5a) and (5b) the positive integer s, the number of stages, and the real numbers bi, Ci, Uij (i, j = 1, . . . , s) are the parameters defining the method. As usual, we make use of the notation A = (aij) E Iw”““, b = (bi) E IP, c = (ci) E R”. We assume that c = Ae. Hence the RK method is determined by the arrays A and b. The pair (A, b) is called the scheme of the method, which, for shortness, is denoted by RK(A, b).

In order to present the definition of the DSRK methods (cf. [ 1,2]) we need the splitting function related to the function f : IF% x IR” + Iw” as

F:IWxIR”xIW”-+IW”, F,(t,u,v)=fk(t&~,) forallkE{l,..., n}.

Then, the first step of a DSRK method applied to the IVP (1) with step size hl > 0 produces u1 * U(t,), where

UI =uo+hICbiF(tO+Cih,,Xi.Yi). i=l

(64

Here {xi]s=, and (yi}f=t, the sets of stage values, are solutions of the system of algebraic equations

xi=uo+hlCaiiF(to+cjhl,xj,yj), i=l,..., s,

j=l (6b)

Yi=uo+hlCl’!ijF(to+cjhI,Xj,Yj), i=l,...,s. j=l

(6~)

In the formulas (6a)-(6c) the positive integer s, the number of stages, and the real numbers bi, Ci, Uij. Wij (i, j = 1, . . . , s) are the parameters defining the method. We make use of the notation


A = (ajj) E JRsxs, b = (bi) E Iw”, c = (ci) E Iw”, W = (Wij) E IRsx”. We assume that c = Ae = We. Hence the DSRK method is determined by the arrays A, b and W. The triple (A, b, W) is called the scheme of the method. Similarly as before, we denote the corresponding DSRK method by DSRK(A, b, W).

We say that RK(A, b) or DSRK(A, b, W) is uniquely defined for the IVP (1) with given step sizes if the systems of algebraic equations (5b) or respectively (6b)-(6c) have a unique solution.

Taking into account that F(t, u, v) = f(t, u) for all f, t, u we can see easily that DSRK(A, b, A) is exactly RK(A, b), including the existence and uniqueness of the solution of the corresponding system algebraic equations. Therefore the class of DSRK methods includes the class of RK methods.

In order to have a convenient formulation of (6a)-(6c) we use the diagonal split form of functions to simplify the splitting function F. For this aim, consider a given function f : IR x IL?” + Iw” and let f(t, v) = D(t, v)u + p(t, v) be its diagonal split form (see Definition 2). Then we can write

Fk(t, u, v) = fk(t, v k uk) = fj(t, u k uk) - fk(t, u -k 0) + fk(t, u k 0)

=d,(t,v~~k)~k+~k(t,2’) forallk~{l,...,n),

hence

F(t, U, V) = DF(t, U, V)U +p(t, v) with DF(~, U, v) := diagk(4(t, u k Q)). (7)

Further, let

x := (XT,. . . ) X,T)T, y := (YT,. . . , Y,‘>‘,

M :=diagi(h1DF(t0+cihl,Xi,yi)) EIR~“~~~,

N := (hph + clh~, ydT, . . . t h~p@o + c,h~, Y.T)~)~ E IF”

and

b:=b@Z’“‘, A := A @I I’“‘, W := w @I I@), e := e(‘) @ I(“).

Here @I denotes the Kronecker product of matrices (see, e.g., [ 151). Then using (7) in (6a)-(6c) we obtain a system equivalent to (6a)-(6c).

ui =uo+bT(Mx+N), (W x=euo+A(Mx+N), (8b) y = euo + W(Mx + N). (8~)

At the end of this section we introduce two additional notations: first, Z := I(‘) @ I’“) = Ion), and second, n = rY is the sn-by-sn permutation matrix that maps all the sn-vectors of the form <vT, * *. 9 $)’ to (u1.1, . ..f 21s.1, . . . . Vl,n,*.., u,,,)~. The matrix n plays a role in our analysis when we change the multipliers in a Kronecker product.

3. Positivity of DSRK methods on general problems

In this section we investigate conditions under which DSRK methods are positive on arbitrary positive problem sets. We emphasize that the results of this section are valid for RK methods as well, see Section 2.3.

Z. Horva’th /Applied Numerical Mathematics 28 (1998) 309-326 315

To formulate the task of this section more precisely, let F c P, H > 0 and (A, b, W) a scheme of a DSRK method be given. In this section we give a sufficient condition under which DSRK(A, b, W) is positive on F with threshold H.

We make use of the following condition which is connected with the solvability of the algebraic equations (6b)-(6c).

(S) There exists a unique solution xi, yj, i = 1, . . . , s, of (6b)-(6c), and the solution vectors are continuous functions of uo and h 1 whenever DSRK(A, b, W) is applied to IVP (1) with any f E F, ua E Iw”, to E Iw, h, E [0, H]. Moreover, the solvability and continuous dependence on parameters holds even if the vectors ua are replaced by arbitrary constant n-vectors ci and vi (ti = ni in case of A = W) in the ith equations of (6b) and (6~) (i = 1, . . . , s), respectively.

Note that condition (S) is automatically fulfilled when we apply explicit RK methods to IVPs with continuous right-hand side functions.

It turns out that four rational functions called the scheme functions play an important role in our analysis.

Definition 3. For (A, b, W), the scheme of a given DSRK method of s stages, let

2, = { Z E Iw”” 1 Z is diagonal and I”’ - ZA is nonsingular}.

The scheme functions of DSRK(A, b, W) are defined on 2~ as

K/,,(Z) = 1 + bT(Z - ZA)-‘Ze, Jb(Z) = bT(Z - ZA)-I,

Kw(Z) = (I + W(Z - ZA)-‘Z)e, Jw(Z) = W(Z - ZA)-‘.

In case of RK methods, i.e., when W = A, Kw and JW is referred as KA and JA, respectively. Note that KA(Z) simplifies to (I - AZ)-‘e.

Definition 4. Let (A, b, W) be a scheme of a DSRK method of s stages, and let 2 be a set of diagonal s-by-s matrices. We call the scheme positive on 2 if 2 c 2~ and all the scheme functions are nonnegative on 2.

Now we are in a position to formulate and prove a theorem that provide the announced result.

Theorem 1. Let F be a given positive problem set, (A, b, W) a scheme of a DSRK method of s stages, H > 0 and 2 be a set of diagonal s-by-s matrices. Assume that

(Al) (S) holdsfor (A, b, W), .F and H; (A2)foraZZfE.E toEIF& u~,...,v,EIW”, h~[O,H]andk~{l,...,n}wehavehdiagi(d~(to+

ci h, Vi)) E 2 where dk is given in Definition 2; (A3) the scheme is positive on 2.

Then DSRK(A, b, W) is positive on F with threshold H.

Proof. Let f E 3, u. E IQ, to E IR be arbitrary and apply DSRK(A, b, W) to the corresponding IVP (1) with an h = hl E (0, H]. In this way we get (8a)-(8c).

First we prove that I - MA and Z - AM are nonsingular. Indeed, by means of the matrix n introduced in Section 2.3, see also [8,9],

nTMn = diag,Zk with Zk = diag, (hdk (to + cih, yi C ~i,k)) E 2,

316 Z. Horvdth /Applied Numerical Mathematics 28 (I 998) 309-326

hence

Z - MA = nnT(Z - MA)nnT = 7t ((I@) @I I(‘)) - diag,Zk (I’“’ @ A))nT

= ndiag, (I(‘) - ZkA)nT,

which is nonsingular by (A2) and (A3). The nonsingularity of Z - AM follows now from the identity (I - AM)(Z + A(Z - MA)-‘M) = 1.

Then we have from (8b) that x = (I - AM)-’ (euo + AN). Substituting this equation into (8a) and (8~) we obtain

ur = (I’“’ + bT(Z - MA)-‘Me)uo + bT (I - MA)-‘N, Pa) y = (I + W(Z - MA)-‘M)euo + W(Z - MA)-‘N. (9b)

The coefficient matrices of ua and n/ in (9a) and (9b) can be simplified in the same manner as was done for Z - MA. For example,

W(Z - MA)-’ = mrTWnnT(Z - MA)-‘rrnT

=~(Z’“‘~~)((Z’“‘~Z’s’) -diag,Zk(Z’“‘@A))-lirT=rrdiag,(.I~(Zk))rrT.

Similarly,

I(“’ + bT(Z - MA)-‘Me = diagk(&,(Zk)),

bT(Z - MA)-’ = diagk(Jb(Zk))nT,

(I + W(Z - MA)-‘M)e = ndiag,(Kw(Zk)).

Thus the coefficients of uo and N in (9a) and (9b) are nonnegative, due to the positivity of the scheme on 2.

We know that y > 0 implies N 2 0. Hence, by (9a), it is enough to prove that y 2 0. By (Al) we may assume that uo > 0 and p(t, u) > 0 for all t and u > 0. Indeed, replacing f by f + se

and ua by ua + ce in (l), the resulting equations (6b)-(6c) remain the same but in the ith equations ua + (1 + hci)se appears instead of ~0.

We show that y > 0 for any h E [0, H]. This is true for h = 0. Suppose on the contrary that this is not so for some step size from (0, H]. Then, again by (Al), there exists % E (0, H] such that, with h = h, there hold y > 0 and yL = 0 with some index 1 E { 1,. . .,sn}. Hence N > 0, because of y 2 0 and the assumption on f. Now consider the Ith component of the left- and right-hand side in (9b). The component on the left equals 0 hence so does the component on the right. The right-hand side is a sum of two matrix-vector products where both matrices are nonnegative and both vectors are strictly positive. Thus the lth row of both matrices is identically 0. But, if the ~th row of W(Z - MA)-’ is zero then the off-diagonal elements of the Ith row of Z + W(Z - MA)-‘M are 0 and the diagonal element equals 1, consequently the &h row of the first matrix on the right-hand side of (9b) is not identically 0, which is a contradiction. This contradiction completes the proof of the theorem. •I

Remark 3. We have also proved that, under the conditions of Theorem 1, y > 0 holds as well. However, generally x > 0 does not hold. To ensure x > 0 we should have additionally, as can be seen from the proof of Theorem 1, (I - AM)-‘e 2 0 and (I - AM)-' A 2 0. These conditions mean that KA and JA are nonnegative on 2, RK(A, b), besides DSRK(A, b, W), is also positive on 2 which is in general a much stronger requirement, cf. Theorem 3, Remark 6 and Theorem 5.


4. Positivity of DSRK methods on dissipative and related problem sets

In this section we examine positivity of DSRK methods applied to IVPs that are positive and even dissipative in some L, norm. As we have mentioned in the introduction this class of problems is of great importance in many applications. Besides dissipative IVPs we shall consider some larger subclasses as well. We shall use dissipativity and other additional conditions that define problem classes to obtain a simple suitable set for 2 in Theorem 1, which is the main tool in our approach.

4.1. Dissipative problems and the positivity radius of DSRK methods

In this paper we call IVP (1) dissipative in a given norm 11 . 1) if f is of type (2) and for any two solutions of the differential equation of (I), say U and 6, the function t H (1 U(t) - i?(t) 11 is monotone decreasing. This property of the IVP is equivalent to

I(Z-z-r(f(t,~)-f(t,~))Il~~llZ-zll~ foralltEIF& z,Z~IW”andt>O, (10)

see [14, pp. 500-5011. Let D* denote the set of functions f for which the corresponding IVP (1) is dissipative in some L,-norm with 1 < p < 00 and let 3” = P n D*.

For describing subclasses of dissipative problems one often uses the so-called circle condition. We say that f, the right-hand side function of the IVP (1) fulfills the circle condition with constant p > 0 in the norm I] . II if it is of type (2) and there holds

Ilp(z-z)+(f(t,~-_f(t,z))I(~pllz-zll foralltER, z,ZEIWn (11)

(cf., [ 14, p. 5011). For any p > 0, we denote by D*(p) the set of functions f for which the circle condition holds with p in some L,-norm with 1 < p < 00. Further, let F*(p) = P n 23*(p).

Using these notations, the aim of this subsection is to apply Theorem 1 to the cases F = F* and 3==3.-*(p>.

In connection with assumption (A2), one of the assumptions of Theorem 1, we have the following result.

Lemma 3. Let f be afinction from Iw x II%” to IF?, t E Iw, u E W, p > 0, &-ther k E { 1, . . .,, n}, and let dk be thefunction dejined in Dejinition 2. Then we have

(a) dk(t, v) < 0 whenever f E D*; (b) -2p < dk(t, V) < 0 whenever f E D*(p).

Proof. Let e, := (0, . . . , 0, 1 , 0, . . . , O)T E R” denote the mth unit vector in IFY for m = 1, . . . , n, z = u, i = u + aek with an arbitrary a! # 0 real number and cp := (l/cz)(f(t, Z) - f(t, z)). It is enough to prove that (pk < 0 for the (a) statement of the lemma and -2p 6 (ok < 0 for the (b) one whenever f E ?3* and f E D*(p), respectively. Indeed, the statements follow from the corresponding inequality by inserting

cx = -vk whenever uk # 0, and letting a T 0 whenever uk = 0 and &fk (t, II & 0) exists. (a) Suppose f E jT)* and p E (1,001 are such that (10) holds with I] . 1) = II . Ilp. Insert here z = u,

t = u + (Yek. Hence we obtain

kkllp < lkk - =& for all t > 0. (12)

318 Z. Horvrith /Applied Numerical Mathematics 28 (1998) 309-326

First let p = 00. It follows from (12) that

l~lle~-r~ll~=maw{ll-s~~l,t~~~~~~~}=~l-~~~~=l-~~~ forsmallt>O,

which implies the desired inequality. Now let 1 < p < 00. Then we have from ( 12), employing the notation C : = CmZk (pm I P, that

1 G llek - wllpP = (1 - tqkok)P + ctp = I - tPpk + t 2 c1 - =hdP - 1 + rpqk

t2 + CtP as t J, 0,

from where we get

(ok < t (1 - r%)P - 1 + rP% + CtP-1 \

t= as t J 0,

hence I& < 0 follows by taking ‘5 $0. (b) Let f E 27*(p) be arbitrary and 1 < p 6 00 such that the circle condition holds for f with p in

11 . (I = (I a lip. Substituting z = u and Z = u + &?k into (11) we obtain

Further

thus (p + pkl < p, which was to be proved.

Remark 4. We can derive bounds for dk in the same way also in the case when a generalized circle condition holds for f. Namely, if

IJyG-z)+ (fO,3-.ff(t,z))(j <pII?-zll forallteR, z,?ER” (13)

holds with some y E lw and p > 0 then - y - p < dk (t, v) 6 - y + p for all t, v. Note that (13) permits right-hand side functions of nondissipative IVPs as well.

In virtue of Lemma 3 it is advantageous to introduce some further notations.

Definition 5. For any positive integer s and nonnegative real number r let 2* and 2*(r) be the sets

2* := {Z E BY’” 1 Z is diagonal and Zii < 0 for all i},

z*(r) := {z E Iw”“” 1 Z is diagonal and - r < Zii < 0 for all i} if r 3 0 is arbitrary.

Definition 6. Let (A, b, W) be a scheme of a DSRK method. The radius of positivity of the scheme, R(A, b, W) is defined as

R(A,b, W):=sup{r~E%) r > 0 and (A, b, W) is positive on 2*(r)}.

Concerning RK methods, we also use the shorter notation R(A, b) = R(A, b, A).

Remark 5. Observe that the positivity radius played a role in the analysis of DSRK(A, b, W) from the point of view of maximum norm contractivity. (We mean by contractivity the discrete version of the

2. Horvcith /Applied Numerical Mathematics 28 (1998) 309-326 319

dissipativity.) Namely, it can be easily seen that [-R(A) b, W), 0] equals the strong (or “semi”) AN,f(O)- stability region of the considered method (c.f. [1,2]). This implies (see lot. cit.) that DSRK(A, b, W) is unconditionally contractive in the maximum norm whenever R(A, b, W) = 00 and conditionally contractive with step size threshold R(A, b, W)/p whenever p > 0, 0 < R(A, b, W) < 00 and f, the right-hand side function of the IVP has the property

f is dissipative in (1 . (1 co, differentiable and g(r, u) 3 -p for all k E { 1, . . . , n), k

tER, VEIRY. (14)

Note that (14) is a sufficient condition for the circle condition to hold for f with p in the maximum norm, cf. Lemma 8.

4.2. Positivity of DSRK methods on positive and dissipative problems

The next theorem explains which role R (A, b, W) plays in the analysis of positivity of DSRK methods on 3* and 3*(p) . Recall that by nonconfluency of RK methods we mean that the corresponding ci values are distinct. Further, concerning DJ-reducibility consult, e.g., [6] or [7].

Theorem 2. Let (A, b, W) be a scheme of a DSRK method and p > 0. Then we have the following results.

(a) If condition (S) holds with 3 = P and H = 00 then DSRK(A, b, W) is unconditionally positive on 3* whenever R(A, b, W) = 00. Conversely, if DSRK(A, b, W) is unconditionally positive on 3* and RK(A, b) is nonconfluent and DJ-irreducible then R(A, b, W) = 00.

(b) Ifcondition (S) holds with 3 = 3*(p) and H = R(A, b, W)/(2p) then DSRK(A, b, W) ispositive on 3*(p) with threshold H. Conversely, if DSRK(A, b, W) is unconditionally positive on 3*(p) and RK(A, 6) is nonconfluent and DJ-irreducible then H < R(A, b, W)/(2p).

Proof. In order to prove the first statement of part (a) and (b), apply Theorem 1 to 3 = 3*, 2 = 2*, H = 00 by employing Lemma 3(a) and, respectively, to 3 = 3”(p), 2 = 2*( R(A, b, W)), H = R(A, b, W)/(2p) by employing Lemma 3(b).

The proof of the second statement of part (a) and (b) is given in [9]. ??

We give now a similar result using the positivity radius but, instead of the dissipativity and the circle condition, we employ an assumption on the smoothness of f.

Theorem 3. Let (A, b, W) be a scheme of a DSRK method. Then the following statements hold. (a) If condition (S) holds with

3= {

f&‘jf ‘dlff k 1s afk t erentiable with respect to vk and av (t, v) < 0 V t, v k 1

and H = 00 then DSRK(A, b, W) is unconditionally positive on 3 whenever R(A, b, W) = 00. Conversely, ifDSRK(A , b, W) is unconditionally positive on 3 and RK(A , b) is nonconfluent and DJ-irreducible then R(A, b, W) = 00.

320 Z. Horvrith /Applied Numerical Mathematics 28 (1998) 309-326

(b) Zfp > 0 and condition (S) holds with

f~P(f ‘df t’bl k 1s 1 eren la afk e with respect to uk and - p < av < 0 k

and H = R(A, b, W)/p then DSRK(A, b, W) is positive on .F with threshold H. Conversely, if DSRK( A, b, W) is unconditionally positive on .F and RK(A, b) is noncon.uent and DJ-irreducible then R(A, b, W) > pH.

Proof. The statements follow in the same way as those of Theorem 2 but here we use Lemma 2 instead of Lemma 3. 0

Remark 6. As a consequence of Theorems 2 and 3, R(A, b, W) characterizes the step size threshold of positivity for many positive problem sets. In [1,2], R(A, b, W) is displayed for a few DSRK methods. There, R(A, 6, W) = 00 holds for a third order and several second order methods. For example, DSRK(A, b, W) determined by the arrays

is of second order with R(A, b, W) = 00. Hence, by Theorem 2, these methods are unconditionally positive on F*-under the (S) condition-and are of higher order than 1, the order barrier for RK methods that are unconditionally positive on F* (cf. [3] and Theorem 5). However, DSRK methods of this type seem to be subject to an order barrier of 4 (see [ 131 where this conjecture is proved for a large class of relevant DSRK methods, in the context of unconditional contractivity).

As to the unpleasant (S) condition, we proved (see [8,9]) that it can be dropped from the conditions of Theorem 2 if we confine ourselves to linear problems that are dissipative in the maximum norm, since then it follows from the positivity conditions.

5. Positivity of RK methods on dissipative and related problem sets

We have already investigated positivity of RK methods as they are included in the class of DSRK methods. However, there is an opportunity for getting better results if we investigate the positivity radius of RK methods further, as we shall see below,

5. I. A study of the positivity radius

Our investigation is based on the fact that there is a strong relation between positivity and absolute monotonicity of RK schemes. The latter one is introduced in [ 141 where the radius of absolute monotonicity was thoroughly examined. We prove here that the two radii are equal (cf. also Remark 2.6 in [14]), which provides us with a convenient way for analyzing questions related to positivity of RK methods.

Recall the following definitions (cf. [ 141).

Definition 7. Let (A, b) be the scheme of an RK method. (A, b) is absolutely monotonic at a given e E IR if (A, b) is positive at 61. Further, the absolute monotonicity radius of (A, b) is defined as

R(A, b) := sup {r E It3 1 (A, b) is absolutely monotonic on [-r, 011.

2. Horvdth /Applied Numerical Mathematics 28 (1998) 309-326 321

Definition 8. Let $r : IR” + IR” be a rational function and z E IR”. We say that + is absolutely monotonic at z if it is defined at z, $r (z) > 0 and all of the partial derivatives of the component functions of 1,4 are nonnegative at z .

Notice that we sometimes identify the diagonal matrix Z E IR”” and the vector z := (Zr 1, . . . , Z,,)T E IR” for any Z when we are talking about absolute monotonicity of scheme functions.

To prove the promised result, we need the following lemmas.

Lemma 4. Let $f : R” + IF’ be a rational function, Z and Z two elements of D, the domain of q!r, such that Z < Z and assume that o, the line segment from Z to Z belongs to D. Then absolute monotonicity of $ at Z implies absolute monotonicity of $ at Z.

Proof. Without loss of generality we may assume m = 1. As D is open and B is compact, one can easily construct a finite sequence of open balls, say Br , . . . , B, with centers Cl, . . . , C,, respectively, such that Bj C D, Cl 6 *.. < C,, Cl = Z, C, = Z, Cj E o and Cj E Bj+l for all j E { 1,. . . , m} (let, say, B mfl := B,). We know that I,!I can be expanded into a Taylor series around Cr = Z and B1 belongs to the region where the series converges to $. Because of the absolute monotonicity of @ at Ct we know that all of the coefficients of the series are nonnegative. Further C2 E B1 and C2 - Ct 3 0. Hence, by differentiating the terms of the series we see that $r is absolutely monotonic at Cz. By induction we obtain the desired result. ??

Lemma 5. Let (A, b) be a scheme of an RK method of s stages and Z E IFV”” a diagonal matrix. Then the scheme is positive at Z if and only if the scheme functions are absolutely monotonic at Z.

Proof. The “if” part is trivial. Conversely, suppose that Kb(Z) 2 0, &(Z) 2 0, KA (Z) > 0 and J,J(Z) 3 0 and let Ei E If%““” with Ei,kt := 1 iff k = 1 = i and 0 otherwise. Then one easily verifies thatforalliE{l,...,s}thereholds

from

$KbCZ) = Jb(Z)EiKA(Z), 1

$Jb(z, = Jb(Z)EiJA(Z), i

$A(Z) = JA(Z)&KA(Z)~ I

$A(z) = JA(Z)EiJA(Z), I

which the absolute monotonicity of the scheme functions follows by induction. ??

Theorem 4. Let (A, b) be a scheme of an RK method. Then R(A, b) = R(A, b).

Proof. It is clear that R (A, b) < R(A , b) . For the converse relation let r > 0 be arbitrary such that the scheme is absolutely monotonic at -r, i.e., the scheme is positive at -rZ. We have to prove that the scheme is positive at each Z E 2*(r).

First we show that I - ZA is nonsingular for all Z E 2*(r). For this aim let Z E 2*(r) be arbitrary and C := Z + r-l 3 0. Since Z - ZA = (I - CA(Z + rA)-‘)(I + rA), it is sufficient to prove that Z - CA(Z + rA)-’ is nonsingular. For shortness, let Pt := CA(Z + rA)-’ and P2 := rA(Z + rA)-‘. Notice that P2 = JA(-rz) > 0, Z - PZ = (I + rA)-’ hence Z - P2 is nonsingular and (I - P2)e = KA(-rZ) 3 0, i.e., P2e < e. Finally, P2 - PI = -(l/r>ZPz > 0, i.e., PI < P2. From P2 3 0, P2e < e it


follows that the spectral radius of P2, spr( P2), is an eigenvalue of P2 (using the well-known Frobenius- Perron theorem) and is at most 1. But it is strictly less than 1 because I - P2 is nonsingular. Combining this with 0 < P, < P2 we get spr( PI) -C 1. Therefore I - PI is nonsingular what was to be proved.

Now let 2 E 2*(r) be arbitrary. We know from Lemma 5 that the scheme functions are absolutely monotonic at -rl. Further, we have just proved that the scheme functions are defined on the segment from -rZ to Z. Hence, by Lemma 4, the scheme functions are absolutely monotonic rather nonnegative at Z. This means that the scheme is positive at Z. •I

Now we may adopt some results from [14]. Note that by irreducibility of RK methods we mean that they are irreducible in both DJ and HS sense, for details see [6] or [7].

Theorem 5. Let (A, b) be an irreducible scheme of an RK method. Then we have the following results. 1.

2. 3. 4.

R(A, b) = 00 if and only if A is nonsingular and the following properties are fu&Yled: A-’ is an M-matrix, A-‘e > 0, bTA-’ > 0 and bTA-‘e < 1. If R(A, b) = 00 then the classical order of RK(A, b) is at most 1. R(A, b) > 0 ifand only if A 2 0, b > 0 and Vi,j (A2ij # 0 =+ Aij # 0). If R(A, b) > 0 then the classical order of RK(A, b) is at most 4 for explicit methods and at most 6 for implicit methods. Furthen the stage order of such a method cannot exceed 2 and if it equals 2 then A has a zero row.

We conclude this subsection with investigating the positivity radius of some well-known RK methods.

Example 1. Applying the results of Theorem 5 to the 0-methods (i.e., RK(A, b) with A = (8 ), b = ( 1)) we immediately obtain that R(A, b) = 00 iff 0 2 1 and R(A, b) > 0 for all 19 > 0. A simple calculation shows that R(A, b) = I/( 1 - 0). Hence the radius of positivity of the explicit Euler, the implicit midpoint rule and the implicit Euler method equals 1, 2, CQ, respectively.

For the implicit trapezoidal rule, i.e., RK(A, b) with

we have that A 2 0, b > 0 and A2 = iA, hence R(A, b) > 0. In fact, R(A, b) = 2. For the classical fourth order RK method we have A 3 0, b > 0 but A31 = 0 and A231 # 0, which leads

to R(A,b) =O. Most of the well-known families of RK methods violate A 2 0, one of the necessary conditions for

R(A, b) > 0. So do, e.g., the Lobatto IIIC methods for which then R(A, b) = 0.

5.2. Existence, uniqueness and continuity of the solution of algebraic equations

As we shall see from Lemma 7 below, positivity of the scheme implies that the algebraic equations arising in the RK steps have a unique solution whenever the problem is dissipative and the step size is


taken in accordance with the positivity threshold. Moreover, even condition (S) holds for such problems and step sizes.

Lemma6. Let QclI%“, G:IWN x Q -+ IWN be continuous on its domain, 1) . 1) a given norm in IWN and y a positive real number: Suppose that

(16)

Then the equation G(z, q) = 0 has a unique solution for all q E Q and the solution, z = z(q), depends continuously on q with respect to Q.

Proof. The unique solvability of the equation under consideration is clear by (16) (see, e.g., [12]). As to continuity, suppose on the contrary that there exist q*, qn E Q (n = 1,2, . . .) such that limq, = q* but limz(q,) # z(q*). We may assume, without loss of generality, that either there exists z* E RN with limz(q,) = z* # z(q*) or lim lIz( = 00.

If limz(q,) = z* with some z* E RN then by the continuity of G we have 0 = G(z(q,), qn) --+ G(z*, q*) hence z(q*) = z* = limz(q,), which is impossible. Thus lim Ilz(qn)jI = 00. Now we have by (16) that

l(G(z(qd, qn) - G(z(q*), %)I[ 3 vllz(q,) - z(q*)(( forall n.

The limit of the left- and right-hand side equals 0 and 00, respectively, which is a contradiction. ??

Lemma 7. Let (A, b) be the scheme of an irreducible RK-method of s stages and p > 0. Then (a) R(A, b) = CC implies that (S) holds for the scheme considered, .F = F*, and H = 00; (b) 0 < R(A, b) -C cc implies that(S) holdsfor the scheme at hand, F = F*(p), and H = R(A, b)/p.

Proof. Let to E IR, II . 11 = (( . Jlp with some 1 < p < 00, and either f be dissipative in I( . II or the circle condition hold for f in I] . II with constant p in statements (a) and (b), respectively. Further, let ei = I]i (i= l,... , s) and h = hl vary in EC” and [0, R(A, b)/p], respectively.

The perturbed system of (6b)-(6c) is equivalent to the equation G(y, q) = 0 where y = (y:, . . . , y,:)‘,

q = (VT, . . . , VT, h)T E Q = IL?“” x [O, R(A, b)/p] and

G(y, q) = y - (r& . . . , q;)* - A(hf (to + cih, y,lT,. . . , hf (to + c,h, Y.T)~)~.

Then, in the same way as is in Theorem 7.1 of [14], we have IjG(z, q) - G(Z, q)ll 3 y411z - Zll for allz,zEIWS”andqEQwithy,=IIAIA-‘Illco in part (a) and v4 = ]](I + phA) ](Z + ,ohA)-‘( ((oo in part (b). (Here the absolute value function for matrices is taken entrywise.) y4 is a positive constant in part (a) and v4 = y(h) is a positive continuous function of h on [0, R(A, b)/p] in part (b). (The positivity of v4 is a consequence of the fact that P I P-’ ( # 0 for nonsingular matrices.) Hence there exists a positive real constant y such that (16) holds with this y for G. Now we can apply Lemma 6 to Q, G, ]I . 1) and y , which results in the statements of the lemma. ??

324 Z. Horva’th /Applied Numerical Mathematics 28 (1998) 309-326

5.3. Positivity of RK methods on dissipative and positive problem sets

Combining the results of Theorem 2 and Lemma 7 we arrive at the following theorem.

Theorem 6. Let (A, b) be the scheme of an irreducible RK method and p > 0, H > 0. Then we have the following results:

(a) RK(A, b) is unconditionally positive on .F* whenever R(A, b) = 00. (b) R(A, b) = CC whenever RK(A, b) is nonconfluent and unconditionally positive on .F*. (c) RK(A, b) ispositive on F*(p) with threshold H = R(A, b)/(2p) whenever 0 -C R(A, b). (d) R(A, b) 3 2p H whenever RK(A, b) is nonconjuent and positive on F*(p) with threshold H.

Example 2. Here we demonstrate that lack of R(A, b) > 0 can lead to poor numerical results from the point of view of positivity when RK(A, b) is applied to numerical solution of some positive IVPs, even if the method has good property otherwise.

Consider the Lobatto IIIC method with s = 2, i.e., RK(A, b) with A and b given in (15). We know (see Example 1) that now R(A, b) = 0 so the method is not positive, but, for example, algebraically stable (see, e.g., [4,7]). Following the ideas of [9] we arrive at the IVP (1) with n = 2, to = 0, ug = (0, O)T, f(t, u) = L(t)u + g(t), where

s(t) = (0, t’)‘, L(t)= (-r ‘r) fort30

and L(t) = L(0) for t < 0. Clearly f E F*(l). Then applying RK(A, b) under consideration to this problem with arbitrary h > 0 step size we obtain

‘I= 4+(l+e&(l+h))2h’y (

(-1 +edh(l + h))h4 h3

>

T

’

hence u 1.1 < 0 for all h > 0. Therefore nonnegativity is not conserved, not even “for small step sizes”.

In certain cases that include important problems from the applications we can get a less restrictive threshold for (conditional) positivity-see Theorem 7 below. This theorem is a generalization to nonlinear problems of the result proved in [3] for linear problems.

We first prove an auxiliary result (cf. [ 11).

Lemma 8. Let p > 0 and f be of type (14). Then the circle condition holds for f with constant p in II . Iloo.

Proof. We have to prove that (11) holds for f and 1) . 11 = )I . [loo. Let z, Z E Iw”, t E R arbitrary. Then, by the mean value theorem, there exist WI, . . . , W, E IR” such that for the matrix @ with @ii = (afi/auj)(t, wi) there holds f (t, 7) - f (t, z) = @(I? - z). Since f is dissipative in the maximum norm hence p,(Q) < 0 (see [l]). Here pco stands for the logarithmic norm corresponding to the maximum norm (for details see, e.g., [6]). Furthermore, (14) implies that @ii 2 -p for all i. Thus we have

llPz+@IIW=I~~~n IP+@iil+Cl@ijI 1 =P+@cxa(@),</3,

j#i >

Z. Horvcith /Applied Numerical Mathematics 28 (1998) 309-326 325

hence

what was to be proved. ??

Theorem 7. Let (A, b) be the scheme of an irreducible RK method with R (A, b) > 0, p > 0, and

F = If E P I .f is sf type (14)).

Then RK(A, b) ispositive on F with threshold H = R(A, b)/p. Further, ifRK(A, b) is nonconfluent and positive on F considered with some H > 0 then H 6 R(A, b)/p.

Proof. We know from Lemma 8 that F c F* (p) . Hence, by Lemma 7 we get that (S) holds for the given RK(A, b), F and H. Applying Lemma 3 and Theorem 3 we obtain the desired result. For the proof of our second assertion we refer the reader to [9]. ??

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