runge-kutta methods without order reduction

Digital Object Identifier (DOI) 10.1007/s002110100332Numer. Math. (2002) 91: 577–603 Numerische

Mathematik

Runge-Kutta methods without order reductionfor linear initial boundary value problems

Isaıas Alonso-Mallo

DepartamentodeMatematicaAplicadayComputacion,FacultaddeCiencias,UniversidaddeValladolid, c/Doctor Mergelina s/n, 47005 Valladolid, Spain; e-mail: [email protected]

Received July 10, 2000 / Revised version received March 13, 2001 /Published online October 17, 2001 –c© Springer-Verlag 2001

Summary. It is well-known the loss of accuracy when a Runge–Kuttamethod is used together with the method of lines for the full discretiza-tion of an initial boundary value problem. We show that this phenomenon,called order reduction, is caused by wrong boundary values in intermedi-ate stages. With a right choice, the order reduction can be avoided and theoptimal order of convergence in time is achieved. We prove this fact fortime discretizations of abstract initial boundary value problems based onimplicit Runge–Kutta methods. Moreover, we apply these results to the fulldiscretization of parabolic problems by means of Galerkin finite elementtechniques. We present some numerical examples in order to confirm thatthe optimal order is actually achieved.

Mathematics Subject Classification (1991): 65M20, 65M12, 65M60, 65J10

1. Introduction

When applied to stiff systems of ordinary differential equations, the Runge–Kuttamethods suffer from reduction of order [13], evenwhen the solution isregular. Since the spatial semidiscretization of a partial differential equationbecomes stiffer when the spatial discretization is refined, it is natural toobserve the order reduction phenomenon when an evolutionary problem inpartial differential equations is solvedbyusing themethodof lines approach.

Theorderobserved in theapplications isgovernedessentiallyby thestageorderq of the Runge–Kutta method rather than the classical orderp [6,17–19,21] and it depends on several factors. However, it is well-known that the

The author has obtained financial support from DGICYT PB95-705 and JCyL VA025/01

578 I. Alonso-Mallo

order reduction is avoided when we use the method of lines and the solutionof the partial differential equation satisfies certain boundary constraints [9,21,29]. The crucial point is that these constraints do not agree in generalwith the boundary relations satisfied when the solution is regular.

There are several techniques used to avoid the order reduction when aRunge–Kutta method is applied. It is possible to find Runge–Kutta methodswith high stage order [7,16], but this technique increases the number ofstages and therefore the computational cost. It is possible to use modifiedversions of the classical implementation of a Runge–Kutta method [11,14,15], but the computational cost is also increased in some cases. Moreover,some substantial modifications of the usual implementation of Runge–Kuttamethods are required.

Since it seems that the order reduction is caused by the boundary valuesof the solution, it is natural to avoid this phenomenon by using the boundaryvalues of the exact solution. The first advantage is that the boundary valuescan be computedwith a smaller computational cost thanmaking evaluationsin the whole domain. This idea has been succesfully used in [2,3] to definepolynomial and rational methods without order reduction. These methodsare strongly related to Runge–Kutta methods [9,31]. However, there areseveral inconveniences in [2,3]. First, the Runge–Kutta methods reduce to apolynomial or rationalmethod only for the linear casewith time independentoperator, whichmakes impossible the generalization to other problems, eventhe linear ones. Second, the standard software is not written for the rationalformulation of a Runge–Kutta method, but for the classical formulation.Finally, we remark that the hypotheses on the spatial discretization in [2,3]are not adequate for some standard spatial discretizations.

In this paper, we use the classical formulation of a Runge–Kutta methodand we obtain the classical order by modifying only the boundary valuesusually assigned to the intermediate stages. A similar technique is succes-fully used in the numerical experiments presented in [1,27] in the context ofnonlinear hyperbolic equations discretizedwith explicit Runge–Kuttameth-ods and finite difference spatial discretizations. Since in the present workwe use implicit Runge–Kutta methods for the time integration, the stagesare defined implicitly and their boundary values are not determined by theRunge–Kutta method. We show that it is possible to avoid the order reduc-tion with an adequate choice. For this, we use boundary values obtainedfrom a recurrence relation where we have taken the usual boundary valuesas starting values. We remark that these boundary values are given only interms of the data and therefore, they are calculated without knowledge ofthe exact solution.

We begin the study with the semidiscretization in time of an abstractlinear initial boundary value problem (IBVP) with implicit Runge–Kutta

Runge-Kutta methods without order reduction 579

methods. We prove consistency and convergence avoiding the order reduc-tion. Previous semidiscrete analysis is used in order to derive estimates forthe error of the full discretizations. Although some parts of this paper arevalid for the hyperbolic case, we only consider examples given by parabolicproblems with Galerkin finite element techniques for the spatial discretiza-tion.

We remark that the order reduction phenomenon is also present whenother time integration methods are used. For example, [22] analyzes thecase of Rosenbrock methods, showing a sharp bound for the order of con-vergence. In [4], we use a similar technique to avoid the order reduction forfull discretizations with Rosenbrock methods and spectral methods for thespatial discretization.

The organization of the paper is as follows. The notation and the Runge–Kutta methods used for the time discretizations are the principal issues ofSect. 2. In Sect. 3 we study the error of the time discretization. Section 4 isdevoted to the full discretization. Section 5 presents some examples whichshow the applicability of the previous theory. Some numerical experimentsconfirm that the order reduction is avoided.

2. Notation and preliminaries

LetX andY be two complex Banach spaces, andD(A) be a dense subspaceof X andA : D(A) ⊂ X → X, ∂ : D(A) ⊂ X → Y be a pair of linearoperators. We wish to study time discretizations of the well-posed abstractnon-homogeneous linear IBVP

x′(t) = Ax(t) + f(t), 0 ≤ t ≤ T,x(0) = u0 ∈ X,∂x(t) = g(t) ∈ Y, 0 ≤ t ≤ T,

(2.1)

with an implicit Runge–Kutta method. We use the framework developed in[3,5,26], which covers several parabolic and hyperbolic problems of practi-cal interest. We present in this section the main features of this framework.Let us start making the following assumptions on the linear operatorsA and∂,

(A1) The boundary operator∂ : D(A) ⊂ X → Y is onto.(A2) Ker(∂) is dense andA0 : D(A0) = Ker(∂) ⊂ X → X, the restric-

tion ofA to Ker(∂), is the infinitesimal generator of aC0-semigroupS(t)t≥0 in X.

(A3) If z is a complex number with(z) greater than the typeω ofS(t)t≥0, then the eigenvalue problem

(z −A)x = 0,∂x = v,

580 I. Alonso-Mallo

possesses, for eachv ∈ Y , a unique solutionx = K(z)v. Moreover,this solution satisfies

‖K(z)v‖ ≤ L ‖v‖ ,for some constantL > 0 independent ofz, for z in a half-plane of theform(z) ≥ ω0 > ω.

Equivalently, we can impose (A1–2) and (A4) below, instead of (A3).

(A4) The operator(A, ∂) : D(A) ⊂ X → X × Y is closed.

Since we are interested in approximations of high order, we suppose thatthe solution of (2.1) is regular.Weassume that there exists an integer numberr ≥ 1 such that

Ar−ju(j) ∈ C([0, T ], X), 0 ≤ j ≤ r.(2.2)

We remark that the assumption (2.2) implies that the time derivatives of thesolution are regular in space, but without to impose any restriction on theboundary values.

Theorem 3.1 in [3] shows that (2.2) is satisfied when the datau0, f andg are regular and the boundary values∂u0, ∂f(0) andg(0) satisfy certainnatural compatibility constraints. As a consequence, we obtain

Aju(t) = u(j)(t)−j−1∑i=0

Aj−i−1f (i)(t), 0 ≤ j ≤ r,(2.3)

and by applying the boundary operator,

∂Aju(t) = g(j)(t)−j−1∑i=0

∂Aj−i−1f (i)(t), 0 ≤ j ≤ r.(2.4)

Later, we will use these boundary values of the exact solution. Notice thatthe right-hand side of (2.4) is given only in terms of the data of (2.1).

For the time discretization of (2.1) we consider ans-stages Runge–Kuttamethod with Butcher array,

c AbT

(2.5)

whereb = [b1, . . . , bs]T, c = [c1, . . . , cs]T, and

A =

a11 . . . a1s...

...as1 . . . ass

.


Also, we denote1 = [1, . . . , 1]T ∈ Rs, cl = [cl1, . . . , c

ls]

T, for l ≥ 0, andby I thes-dimensional identity matrix.

We suppose that (2.5) has classical orderp. We remember [13] that theorder conditions

bTAjcm =1

(j +m+ 1) . . . (m+ 1), 0 ≤ j +m ≤ p− 1,(2.6)

are necessary and sufficient so that (2.5) has classical orderp when ap-plied to a linear non-homogeneous ordinary differential equation with timeindependent coefficients similar to (2.1).

The convergence results of this paper depend on some rational func-tions defined by using the previous parameters; the stability function of theRunge–Kutta method is

r(z) = 1 + zbT(I − zA)−11.

We suppose that the Runge–Kutta method given by (2.5) is A-stable, i.e.A is regular and, for(z) ≤ 0, the matrixI − zA is regular and|r(z)| ≤ 1(Notice that this definition is slightly stronger than the usual one).

Further, we consider the functions,

Rl,j(z) = zbT(I − zA)−1Aj(cl − lAcl−1), j ≥ 0, l ≥ 1.

The stage order of the Runge–Kutta method is denoted byq, thusq is thehighest positive integer number satisfying

C(q) : Acl−1 =1lcl 1 ≤ l ≤ q.

Notice thatRl,j(z) ≡ 0 for 1 ≤ l ≤ q.For n > 0 integer, we consider time stepsizesk > 0 and we take

tn+1 = tn + k < T . In what follows, we seek an approximationun to theexact solutionu(tn) of (2.1) by applying the implicit Runge–Kutta methodgiven by (2.5). A direct implementation of this method give rise to thefollowing equations for the intermediate stages,

(I ⊗ I − A ⊗ kA)Un = (1 ⊗ I)un + k(A ⊗ I)Fn+c,(2.7)

whereUn = [U1n, . . . , U

sn]T, the vector of intermediate stages, andFn+c =

[f(tn + c1k), . . . , f(tn + csk)]T.TheRunge–Kuttamethod does not give all the information needed, since

the boundary values of the intermediate stages are not determined. As aconsequence, the solution to equations (2.7) is not unique. So, we have tospecify those values. By denotingGn = [∂U1

n, . . . , ∂Usn]T, we obtain the

following equations for the stages,

(I ⊗ I − A ⊗ kA)Un = (1 ⊗ I)un + k(A ⊗ I)Fn+c,∂Un = Gn.

(2.8a)

582 I. Alonso-Mallo

Now, we use the second equation of the Runge–Kutta method to obtain,

un+1 = un + (bT ⊗ kA)Un + k(bT ⊗ I)Fn+c.(2.8b)

In order to see that (2.8a) has a unique solution, it suffices to consider theproblems

(I ⊗ I − A ⊗ kA)U0n = (1 ⊗ I)un + k(A ⊗ I)Fn+c,

∂U0n = 0,

,(2.9)

and(I ⊗ I − A ⊗ kA)U b

n = 0,∂U b

n = Gn.

(2.10)

We endow the spaceXs with the usual norm product. Then, the solv-ability of (2.9) and (2.10) comes from the following two lemmas [6].

Lemma 2.1 There exists a constantC > 0 such that the operator(I ⊗ I −A ⊗ kA0) : D(As

0) ⊂ Xs → Xs is boundedly invertible and

‖(I ⊗ I − A ⊗ kA0)−1‖ ≤ C,

for k > 0 small enough.

Lemma 2.2 For k > 0 small enough andW = [W 1, . . . ,W s]T ∈ Y s, theproblem,

(I ⊗ I − A ⊗ kA)V = 0,∂V = W.

possesses a unique solutionV := K((kA)−1)W ∈ Xs. Moreover, thereexists a constantC > 0 such that,

‖K((kA)−1)W‖ ≤ C‖W‖.By using Lemma 2.1, we deduce that functions(bT ⊗kA0)(I ⊗I−A⊗

kA0)−1, r(kA0) andRl,j(kA0) are well-defined and bounded fork > 0.We also deduce the following result.

Lemma 2.3 The valuesUn andun defined in (2.8a) and (2.8b) satisfy

Un = (I ⊗ I − A ⊗ kA0)−1((1 ⊗ I)un + k(A ⊗ I)Fn+c)+K((kA)−1)Gn,(2.11)

and

un+1 = un − (bTA−1 ⊗ I)((1 ⊗ I)un + k(A ⊗ I)Fn+c)+(bTA−1 ⊗ I)(I ⊗ I − A ⊗ kA0)−1

×((1 ⊗ I)un + k(A ⊗ I)Fn+c)+(bTA−1 ⊗ I)K((kA)−1)Gn + k(bT ⊗ I)Fn+c.(2.12)


Proof. The expression (2.11) is straightforward by using the notation intro-duced in Lemmas 2.1 and 2.2. Now, to prove (2.12), we can write

un+1 = un + (bT ⊗ kA)Un + k(bT ⊗ I)Fn+c

= un + (bT ⊗ kA0)(I ⊗ I − A ⊗ kA0)−1

×((1 ⊗ I)un + k(A ⊗ I)Fn+c)+(bT ⊗ kA)K((kA)−1)Gn + k(bT ⊗ I)Fn+c,

= un + (bTA−1 ⊗ I)(A ⊗ kA0)(I ⊗ I − A ⊗ kA0)−1

×((1 ⊗ I)un + k(A ⊗ I)Fn+c)+(bTA−1 ⊗ I)(A ⊗ kA)K((kA)−1)Gn + k(bT ⊗ I)Fn+c

= un − (bTA−1 ⊗ I)((1 ⊗ I)un + k(A ⊗ I)Fn+c)+(bTA−1 ⊗ I)(I ⊗ I − A ⊗ kA0)−1

×((1 ⊗ I)u(tn) + k(A ⊗ I)Fn+c)+(bTA−1 ⊗ I)K((kA)−1)Gn + k(bT ⊗ I)Fn+c.

3. Error analysis of the semidiscrete problem

This section is devoted to the study the behaviour of local and global errorsof the semidiscrete method defined by (2.8a) and (2.8b). We denote byUn = [U1

n, . . . , Usn]T thes-dimensional vector that satisfies

(I ⊗ I − A ⊗ kA)Un = (1 ⊗ I)u(tn) + k(A ⊗ I)Fn+c,∂Un = Gn,

(3.1a)

and we next define

un+1 = u(tn) + (bT ⊗ kA)Un + k(bT ⊗ I)Fn+c.(3.1b)

With the previous notation, the semidiscrete local truncation error intn, isdefined by

ρn = u(tn) − un, 1 ≤ n ≤ N.(3.2)

Obviously, these local truncationerrors dependon theboundary valuesofthe intermediate stagesGn. As the stagesUn are approximations ofUn+c :=[u(tn +c1k), . . . , u(tn +csk)]T, a natural choice isGn = Gn+c := [g(tn +c1k), . . . , g(tn + csk)]T. As we show later, this usual choice is the origin oftheorder reduction. Inorder toavoid thisphenomenon,wedefine recursively,

U[0]n = Un+c,

U[j+1]n = (1 ⊗ I)u(tn) + (A ⊗ kA)U [j]

n

+k(A ⊗ I)Fn+c, j ≥ 0,

(3.3)

584 I. Alonso-Mallo

for each1 ≤ n ≤ N .By using a recursive argument, we deduce

U [j]n =

j−1∑l=0

(Al1 ⊗ klAl)u(tn) + (Aj ⊗ kjAj)Un+c

+kj∑

l=1

(Al ⊗ kl−1Al−1)Fn+c.(3.4)

Now, we define

G[j]n = ∂U [j]

n

=j−1∑l=0

(Al1 ⊗ kl∂Al)u(tn) + (Aj ⊗ kj∂Aj)Un+c

+kj∑

l=1

(Al ⊗ kl−1∂Al−1)Fn+c,(3.5)

for j ≥ 0, and1 ≤ n ≤ N . When these boundary values are used tocompute the intermediate stages, the corresponding solutions of (3.1a) and

(3.1b) are denoted byU[j]n andu[j]

n . In Sect. 4, we also need the followingboundary values

H [j]n = ∂AU [j]

n , j ≥ 0, 1 ≤ n ≤ N.(3.6)

Notice that the expressions (3.4) and (2.3) allow us to state (3.5) and (3.6)by using only the data of (2.1).

Theorem 3.1 Letu be the solution of (2.1) satisfying (2.2) forr = p + 1.We use the boundary valuesG[j]

n , 0 ≤ j ≤ p − q, in (3.1a). Then the localerrorsρ[j]

n = u(tn) − u[j]n , 1 ≤ n ≤ N , 0 ≤ j ≤ p− q, satisfy

‖ρ[j]n ‖ ≤ Ckq(j)+1, for k > 0,(3.7)

whereq(j) = min p, q + j, and constantC depends only on the deriva-tives ofu, the Runge–Kutta method and the operatorA0.

Proof. First, we takej = 0 and we useGn = G[0]n to get

(I ⊗ I − A ⊗ kA)U [0]n = (1 ⊗ I)u(tn) + k(A ⊗ I)Fn+c,

∂U[0]n = ∂U

[0]n ,

(3.8)

and we can write

(I ⊗ I − A ⊗ kA)U [0]n = (1 ⊗ I)u(tn) + k(A ⊗ I)Fn+c + δ[0]n ,(3.9)


where, by expanding into Taylor series,δ[0]n is given by

δ[0]n =p∑

l=q+1

kl

l!(cl − lAcl−1)u(l)(tn) +O(kp+1).

Denoting∆[0]n = U

[0]n − U [0]

n , and subtracting (3.8) from (3.9),

(I ⊗ I − A ⊗ kA)∆[0]n = δ

[0]n ,

∂∆[0]n = 0.

Therefore, by Lemma 2.1, we have

∆[0]n = (I ⊗ I − A ⊗ kA0)−1δ[0]n ,

for k > 0. On the other hand, since the Runge–Kutta method has orderp,

u(tn) + (bT ⊗ kA)U [0]n + k(bT ⊗ I)Fn+c

= u(tn) + (bT ⊗ kA)Un+c + k(bT ⊗ I)Fn+c

= u(tn) + k(bT ⊗ I)(AUn+c + Fn+c)= u(tn) + k(bT ⊗ I)U ′

n+c

= u(tn) +p−1∑l=0

kl+1

l!bTclu(l+1)(tn) +O(kp+1)

= u(tn) +p−1∑l=0

kl+1

(l + 1)!u(l+1)(tn) +O(kp+1)

= u(tn+1) +O(kp+1),(3.10)

where we have used the order conditionsbTcl = 1/(l+1), l = 0, . . . , p−1.Subtracting (3.1b) from (3.10), we get

ρ[0]n+1 = (bT ⊗ kA0)(I ⊗ I − A ⊗ kA0)−1δ[0]n

=p∑

l=q+1

kl

l!Rl,0(kA0)u(l)(tn) +O(kp+1) = O(kq(0)+1).

Let us take1 ≤ j ≤ p− q. Then we have(I ⊗ I − A ⊗ kA)U [j]

n = (1 ⊗ I)u(tn) + k(A ⊗ I)Fn+c,

∂U[j]n = G

[j]n ,

and

(I ⊗ I − A ⊗ kA)U [j]n

= (1 ⊗ I)u(tn) + k(A ⊗ I)Fn+c + δ[j]n ,(3.11)

586 I. Alonso-Mallo

whence, by subtracting (3.3) from (3.11),

δ[j]n = U [j]n − U [j+1]

n

= (A ⊗ kA)(U [j−1]n − U [j]

n )

= (A ⊗ kA)j(U [0]n − U [1]

n )

= (A ⊗ kA)jδ[0]n

=p∑

l=q+1

kl+j

l!Aj(cl − Acl−1)Aju(l)(tn) +O(kp+j+1).(3.12)

We define

u[j]n = u(tn) + (bT ⊗ kA)U [j]

n + k(bT ⊗ I)Fn+c.

We deduce from (3.10) thatu[0]n = u(tn+1) +O(kp+1). Moreover, we have

u[j+1]n = u(tn) + (bT ⊗ kA)U [j]

n + k(bT ⊗ I)Fn+c

−(bT ⊗ kA)(U [j]n − U [j+1]

n )

= u(tn) + (bT ⊗ kA)U [j]n + k(bT ⊗ I)Fn+c − (bT ⊗ kA)δ[j]n

= u(tn) + (bT ⊗ kA)U [j]n + k(bT ⊗ I)Fn+c

−p∑

l=q+1

kl+j+1

l!bTAj(cl − Acl−1)Aj+1u(l)(tn) +O(kp+j+1)

= u[j]n +O(kp+j+1),

where we have used the order conditions (2.6). By using a recursive argu-ment, we obtain

u[j]n = u(tn+1) +O(kp+1), 0 ≤ j ≤ p− q.(3.13)

Now, we denote∆[j]n = U

[j]n − U [j]

n . Fork > 0, we have

∆[j]n = (I ⊗ I − A ⊗ kA0)−1δ[j]n .

We subtract (3.1b) from (3.13), and we get,

ρ[j]n+1 = (bT ⊗ kA0)(I ⊗ I − A ⊗ kA0)−1δ[j]n +O(kp+1)

=p∑

l=q+1

kl+j

l!Rl,j(kA0)u(l)(tn) +O(kp+1)

= O(kq(j)+1).(3.14)


We now prove some additional results which will be useful in the fol-lowing section.

Lemma 3.2 With the notation and the hypotheses of Theorem 3.1, we have

∂ρ[j]n = O(kq(j)+1),(3.15)

for 1 ≤ n ≤ N and1 < j < p− q.Proof. From (3.14), we obtain

ρ[j]n+1 = (bT ⊗ kA0)(I ⊗ I − A ⊗ kA0)−1δ[j]n +O(kp+1)

= (bTA−1 ⊗ I)(A ⊗ kA0 − I ⊗ I + I ⊗ I)×(I ⊗ I − A ⊗ kA0)−1δ[j+1]

n +O(kp+1)

= −(bTA−1 ⊗ I)δ[j]n

+(bTA−1 ⊗ I)(I ⊗ I − A ⊗ kA0)−1δ[j]n +O(kp+1).

Therefore, we deduce that

∂ρ[j]n+1 = −∂(bTA−1 ⊗ I)δ[j]n +O(kp+1),

and we obtain the result from (3.12).

Lemma 3.3 With the notation and the hypotheses of Theorem 3.1, we have

Aρ[j]n = O(kq(j))

for 1 ≤ n ≤ N and1 ≤ j ≤ p− q.Proof. From (3.14), we obtain

Aρ[j]n+1 = (bT ⊗ kA)A0(I ⊗ I − A ⊗ kA0)−1δ[j]n +O(kp+1)

= (bT ⊗ kA)(k−1A−1 ⊗ I)(A ⊗ kA0 − I ⊗ I + I ⊗ I)×(I ⊗ I − A ⊗ kA0)−1δ[j+1]

n +O(kp+1)

= −(bTA−1 ⊗A)δ[j]n

+(bTA−1 ⊗ kA0)(I ⊗ I − A ⊗ kA0)−1k−1δ[j]n +O(kp+1).

and we obtain the result from (3.12).

Let us see two examples of Runge–Kutta methods used for the timediscretization of (2.1). First, let us consider the one stage Gauss method.The corresponding equations can be written as(

1 − kA

2

)Un = un +

k

2Fn+1/2,

∂Un = Gn,

588 I. Alonso-Mallo

and

un+1 = un +kA

2Un +

k

2Fn+1/2.

This method has classical orderp = 2 and stage orderq = 1. Our resultsprove order 1 whenGn = G

[0]n = g(tn+1/2) and order 2 whenGn = G

[1]n =

g(tn) + (k/2)g′(tn+1/2). Moreover, the values (3.6) are given byH [0]n =

g′(tn+1/2) − ∂f(tn+1/2) andH[1]n = g′(tn) − ∂f(tn) + k

2 (g′′(tn+1/2) −∂f ′(tn+1/2)).

Second, let us consider the two stagesSDIRKmethodwith Butcher array

γ γ 01 − γ 1 − 2γ γ

12

12

for γ = (3 +√

3)/6, which is A-stable. The equations of this method aregiven by

(1 − kγA)U1n = un + kγf(tn+γ),

(1 − kγA)U2n = un + k(1 − 2γ)AU1

n

+k(1 − 2γ)f(tn+γ) + kγf(tn+(1−γ)),∂U1

n = G1n,

∂U2n = G2

n,

and

un+1 = un +kA

2(U1

n + U2n) +

k

2(f(tn+γ) + f(tn+(1−γ))).

This method has classical orderp = 3 and stage orderq = 1. In this case,our results prove order 1 when

Gn = G[0]n =

[g(tn+γ)

g(tn+(1−γ))

],

order 2 when

Gn = G[1]n =

[g(tn)g(tn)

]+ k

[γ 0

(1 − 2γ) γ

] [g′(tn+γ)

g′(tn+(1−γ))

],

and order 3 when

Gn = G[2]n

=[g(tn)g(tn)

]+ k

[γ(g′(tn) − ∂f(tn))

(1 − γ)(g′(tn) − ∂f(tn))

]+k2

[γ2 0

2γ(1 − 2γ) γ2

] [g′′(tn+γ) − ∂f(tn+γ)

g′′(tn+(1−2γ) − ∂f(tn+(1−γ))

]+k[

γ 0(1 − 2γ) γ

] [∂f(tn+γ)

∂f(tn+(1−γ))

].


Now, the values (3.6) are given by

H [0]n =

[g′(tn+γ) − ∂f(tn+γ)

g′(tn+(1−γ)) − ∂f(tn+(1−γ))

],

H [1]n =

[g′(tn) − ∂f(tn)g′(tn) − ∂f(tn)

]+k[

γ 0(1 − 2γ) γ

] [g′′(tn+γ) − ∂f ′(tn+γ)

g′′(tn+(1−γ)) − ∂f ′(tn+(1−γ))

],

H [2]n =

[g′(tn) − ∂f(tn)g′(tn) − ∂f(tn)

]+ k

[γ(g′′(tn) − ∂f ′(tn) − ∂Af(tn))

(1 − γ)(g′′(tn) − ∂f ′(tn) − ∂Af(tn)))

]+k2

[γ2 0

2γ(1 − 2γ) γ2

]×[

g′′′(tn+γ) − ∂f ′′(tn+γ) − ∂Af ′(tn+γ)g′′′(tn+(1−γ)) − ∂f ′′(tn+(1−γ)) − ∂Af ′(tn+(1−γ))

]+k[

γ 0(1 − 2γ) γ

] [∂Af(tn+γ)

∂Af(tn+(1−γ))

].

We end this section studying the convergence with optimal orderp. Forthis, we need an additional hypothesis on stability. We suppose that variabletime stepsizeskn > 0 are used, withtn+1 = tn +kn < T . Then, we assumethe following stability condition,

“there exists a constantC = C(T ) such that

||n∏

m=1

r(kmA0)|| ≤ C(T ),(3.16)

for 0 ≤ n ≤ N . ”

For the case of A-stable Runge–Kutta methods, it is well-known thatcondition (3.16) is nearly satisfied for constant time stepsizes. The bound

||rn(knA0)|| ≤ C(T )N1/2, 0 ≤ n ≤ N,(3.17)

is proved in [8]. Bound (3.17) is the best possible for the general case and ahalf order can be lost. However, (3.17) may be improved for some particularmethods. This happens for instance for the RadauIIA methods ofs stages,whose stability functions fulfill the sufficient conditions in [8]. Moreover,there are two cases for which the termN1/2 can be dropped even for variabletime stepsizes, whenX is a Hilbert space andA0 is a dissipative operator[13,20] and whenA0 is the infinitesimal generator of an analytic semigroupand the Runge–Kutta method (2.5) is assumed to beA(β)-stable [23,24].

We define the semidiscrete global error intn as

e[j]n = u(tn) − u[j]n , 1 ≤ n ≤ N, 0 ≤ j ≤ p− q.(3.18)

590 I. Alonso-Mallo

Theorem 3.4 Assume that the hypotheses of Theorem 3.1 and the stabilitycondition (3.16) are satisfied. Then the global errors (3.18) satisfy

‖e[j]n ‖ ≤ C(T )n∑

m=1

kq(j)+1m−1 ,(3.19)

for 1 ≤ n ≤ N , 0 ≤ j ≤ p − q, where constantC depends only on thederivatives ofu, the Runge–Kutta method and the operatorA0.

Proof. For simplicity, we drop the dependence on0 ≤ j ≤ p − q in thenotation. Forn ≥ 1, Un andUn share the same boundary values. Thus,subtracting (2.8a) and (3.1a), we obtain

(I ⊗ I − A ⊗ knA)(Un − Un) = (1 ⊗ I)en,∂(Un − Un) = 0,

and subtracting (2.8b) and (3.1b)

un+1 − un+1 = en + (bT ⊗ knA)(Un − Un)= en + (bT ⊗ knA0)(I ⊗ I − A ⊗ knA0)−1(1 ⊗ I)en= r(knA0)en.

This equality allows us to obtain the recurrence relation for the globalerror

en+1 = u(tn+1) − un+1 + un+1 − un+1 = ρn+1 + r(knA0)en,

and by induction, supposinge0 = 0,

en =n∑

m=1

n−1∏l=j

r(klA0)ρm.

Finally, from the stability condition (3.16) and Theorem 3.1 we obtain(3.19).

Theorems 3.1 and 3.4 are sufficient in order to prove that the orderreduction can be avoided when the boundary valuesGn = ∂U

[j]n , 1 ≤

j ≤ p − q, are used. However, we note that the estimate for the localerrors obtained in Theorem 3.1 is not optimal [6,21]. Therefore, the orderof convergence obtained in the applications may be greater than the orderobtained in Theorems 3.1 and 3.4.

For instance, suppose that the IBVP can be written as the abstract initialvalue problem

x′(t) = A0x(t) + f(t), 0 ≤ t ≤ T,x(0) = u0 ∈ X,

(3.20)


beingA0 : D(A0) → X an infinitesimal generator of aC0-semigroup.This case agrees with the problem (2.1) ifg ≡ 0, i.e. the boundary operatorapplied to the solution vanishes. Equivalently, the solutionu belongs to asubspaceD(Aµ

0 ),µ ≥ 1 (the numberµmay be fractional). In this case, [21]proves the estimate

en = O(kmin(p,q+1+µ)),

for the global errors when the time stepsizek is constant andA0 is a self-adjoint operator in a Hilbert space setting. With the techniques in [22], it ispossible to obtain the same result whenA0 is the infinitesimal generator ofan analytic semigroup,µ ≥ 1, the stepsizek is constant, andr(∞) = 1. Allthese hypotheses are weakened in [6].

As a consequence, the one stage Gauss method applied to the abstractinitial value problem (3.20) (with solution vanishing on the boundary) hasorder 2 and therefore, the order reduction is not present. Analogously, theSDIRK method has order2 when it is applied to (3.20). Moreover, an ex-tra order of convergence is achieved for constant stepsizes and the orderreduction in the global errors is avoided in this case.

4. Full discretization

In this section we shall study the application of the above results whendiscretization also takes place with regard to the spaceX. As we will seein the examples of Sect. 5, the framework studied here includes problemsdiscretized in space by Galerkin finite element techniques.

Let us denote byh ∈ (0, h0] the parameter of the spatial discretization.LetXh be a family of finite dimensional spaces, approximatingX and wesuppose thatXh = Xh,0 ⊕Xh,b. The norm inXh is denoted by‖·‖h. Letus take a projection operatorLh : X → Xh,0 such thatLhx is thebestapproximation inXh,0 to x ∈ X. We also suppose that there exists anotheroperatorQh : Y → Xh,b and we then definePh := (Lh − LhQh∂) :D(A) → Xh,0.

For the approximation of the operatorA : D(A) ⊂ X → X, we assumethat we are given the operatorsAh : Xh → Xh,0 in such a way thatAh,0 :Xh,0 → Xh,0, the restrictions ofAh to the subspaceXh,0, are approximationofA0 : D(A0) ⊂ D(A) → X (Remind that elements ofD(A0) are regularin space and vanish in the boundary. However, it is possible to consideran elementu ∈ X regular in space but with no zero boundary value, i.e.u ∈ D(A)). Therefore, whenxh = xh,0 + xh,b ∈ Xh,0 ⊕Xh,b = Xh, wehaveAhxh = Ah,0xh,0 +Ahxh,b.

592 I. Alonso-Mallo

We pose the following semidiscrete problem. Finduh(t) ∈ Xh,0 suchthat

u′h(t) + LhQhg

′(t) = Ah(uh(t) + LhQhg(t)) + Lhf(t)uh(0) + LhQhg(0) = Lhu(0),

(4.1)

or, equivalently,

u′h(t) = Ah,0uh(t) +AhQhg(t) + LhQh(∂f(t) − g′(t)) + Phf(t)uh(0) = Phu(0),

which results from the discretization in space of (2.1).

The subsequent analysis is carried out under the following hypotheses,which are very close to those in [9].

(H1) The operatorsAh,0 are invertible and generate uniformly boundedC0-semigroupsetAh,0 onXh satisfying

||etAh,0 ||h ≤ M,(4.2)

whereM ≥ 1 is a constant (notice that (4.2) implies the invertibility of(εI−Ah,0) for all ε > 0, i.e. the operatorsAh,0 are nearly invertible).

(H2) For eachx ∈ X,

‖Lhu‖h ≤ C‖u‖,(4.3)

and for eachv ∈ Y ,‖Qhv‖h ≤ γh‖v‖.(4.4)

(H3) We define the elliptic projectionRh : D(A) → Xh,0 as follows. Ifu ∈ D(A), thenRhu satisfy

Ah(Rhu+Qh∂u) = LhAu,

or, equivalently

Rhu = A−1h,0(LhAu−AhQh∂u)

(In applications to Galerkin finite element methods,Rhu is the dis-cretized solution of the elliptic problem with exact solutionu).Now, we suppose that, foru ∈ D(A),

‖Rhu+ LhQh∂u‖h ≤ αh ‖u‖ + βh ‖Au‖ ,(4.5)

whereαh is bounded andβh is small withh. We also assume thatthere exists a subspaceZ of X, such that,(a) foru ∈ Z, c ∈ C with (z) > 0 andk > 0,

(cI − kA0)−1u ∈ Z, K(c/k)∂u ∈ Z,(4.6)


(b) for u ∈ Z ∩D(A),

‖Lhu−Rhu− LhQh∂u‖h

= ‖Phu−Rhu‖h ≤ εh ‖u‖Z ,(4.7)

whereεh is small withh.

Remark 4.1In order to prove the estimate( 4.5), it is possible to obtain

‖Phu−Rhu‖h ≤ βh ‖Au‖ ,which is very similar to (4.7) (withZ = D(A) andεh = βh), and we have

‖Rhu+ LhQh∂u‖h ≤ ‖Rhu− Phu‖h + ‖Lhu‖h

≤ βh ‖Au‖ + C ‖u‖ .Wepose the following full discretization for thenumerical approximation

of (2.1),

Uh,n = (1 ⊗ Ih)uh,n + (A ⊗ kAh,0)Uh,n + (A ⊗ kAhQh)Gn

−(A ⊗ kLhQh)Hn + k(A ⊗ Ih)PhFn+c,(4.8a)

uh,n+1 = uh,n + (bT ⊗ kAh,0)Uh,n + (bT ⊗ kAhQh)Gn

−(bT ⊗ kLhQh)Hn + k(bT ⊗ Ih)PhFn+c,(4.8b)

whereGn = G[p−q]n andHn = H

[p−q]n . We remember that, with these

boundary values, the order reduction is completely avoided in Sect. 3. Themethod (4.8a), (4.8b) is obtained by applying the standard formulation of aRunge-Kutta method given by (2.5), with modified boundary values, to theordinary differential system (4.1).

We denote byUh,n = [U1h,n, . . . , U

sh,n]T thes-dimensional vector sat-

isfying

Uh,n = (1 ⊗ Ih)Rhu(tn) + (A ⊗ kAh,0)Uh,n + (A ⊗ kAhQh)Gn

−(A ⊗ kLhQh)Hn + k(A ⊗ Ih)RhFn+c,(4.9)

and the valueuh,n+1 as

uh,n+1 = Rhu(tn) + (bT ⊗ kAh,0)Uh,n + (bT ⊗ kAhQh)Gn

−(bT ⊗ kLhQh)Hn + k(bT ⊗ Ih)RhFn+c.(4.10)

Now, we define the full local truncation error intn as

ρh,n = Rhu(tn) − uh,n, 1 ≤ n ≤ N.(4.11)

594 I. Alonso-Mallo

Theorem 4.2 Assume that the hypotheses of Theorem 3.1 as well as (H1)–(H3) are satisfied. Moreover, we suppose the solutionu and the source termof (2.1) satisfyAju ∈ C([0, T ], Z), 0 ≤ j ≤ p − q, Ajf ∈ C([0, T ], Z),0 ≤ j ≤ p − q − 1, for 0 ≤ t ≤ T . Then the method (4.8a), (4.8b) isconsistent and the estimate of the full local truncation error

ρh,n+1 = Rhu(tn+1) − uh,n+1

= O(αhkp+1 + βhk

p + γhkp+1 + kεh),(4.12)

holds for0 ≤ n ≤ N − 1.

Proof. We have

Rhu(tn+1) − uh,n+1 = (Rhu(tn+1) −Rhun+1) + (Rhun+1 − uh,n+1)),

and we wish to estimate the right hand side.From (H1), (H3) and Lemmas 3.2 and 3.3,

‖Rhu(tn+1) −Rhun+1‖h

= ‖Rhρn+1‖h

≤ ‖Rhρn+1 + LhQh∂ρn+1‖h + ‖LhQh∂ρn+1‖h

≤ αh‖ρn+1‖ + βh‖Aρn+1‖ + Cγh‖∂ρn+1‖= O(αhk

p+1 + βhkp + γhk

p+1).

By using (4.6) and Lemma 2.3, we deduce that the valuesUn andun

defined in (3.1a) and (3.1b) belong to the subspaceZ. By applying theoperatorRh to (3.1a), we have

RhUn = (1 ⊗ Ih)Rhu(tn) + (A ⊗ kRhA)Un + k(A ⊗ Ih)RhFn+c

= (1 ⊗ Ih)Rhu(tn) + (A ⊗ k(Ah,0Rh +AhQh∂ − LhQh∂A))Un

+k(A ⊗ Ih)RhFn+c + (A ⊗ k(RhA− PhA))Un

= (1 ⊗ Ih)Rhu(tn) + (A ⊗ k(Ah,0Rh +AhQh∂ − LhQh∂A))Un

+k(A ⊗ Ih)RhFn+c + (A ⊗ k((Rh − Ph)A))Un.(4.13)

From (H1) and (H3), we can write

RhUn − Uh,n = (I ⊗ Ih − A ⊗ kAh,0)−1(A ⊗ k((Rh − Ph)A)Un)= O(kεh).

Now, we apply the operatorRh to (3.1b) to get

Rhun+1 = Rhu(tn) + (bT ⊗ k(Ah,0Rh +AhQh∂ − LhQh∂A))Un

+k(bT ⊗ Ih)RhFn+c + (bT ⊗ k((Rh − Ph)A)Un).(4.14)


Therefore, by subtracting (4.10) from (4.14)

Rhun+1 − uh,n+1 = (bT ⊗ kAh,0)(RhUn − Uh,n)

+(bT ⊗ k((Rh − Ph)A)Un

= O(kεh),(4.15)

concluding the proof. As in the semidiscrete case, we need a stability condition to obtain con-

vergence. From now on, we assume that we use variable time stepsizes andthe following condition holds:

“there exists a constantC = C(T ) such that

‖n∏

m=1

r(kmAh,0)‖h ≤ C(T ),(4.16)

for eachh > 0 with tn+1 = kn + tn < T and0 ≤ n ≤ N .”

Notice that from hypothesis (H1), we have the following stability con-dition [8],

‖rn(kAh,0)‖h ≤ Kn1/2,

for eachh, k andn with fixed t = nk, andK being a constant independentof n, k andh .

Theorem 4.3 Assume that the hypotheses of Theorem 3.1 as well as (H1)–(H3) are satisfied. Moreover, we suppose the solutionu and the source termof (2.1) satisfyAju ∈ C([0, T ], Z), 0 ≤ j ≤ p − q, Ajf ∈ C([0, T ], Z),0 ≤ j ≤ p − q − 1, for 0 ≤ t ≤ T . Then the method (4.8a), (4.8b) isconvergent and the estimate of the full global error

eh,n+1 = Lhu(tn+1) − uh,n+1 − LhQh∂u(tn+1)= Phu(tn+1) − uh,n+1(4.17)

= O(αh

n+1∑m=1

kp+1m−1 + βh

n+1∑m=1

kpm−1 + γh

n+1∑m=1

kp+1m−1 + εh),(4.18)

holds for0 ≤ n ≤ N − 1.

Proof. Since

‖Lhu(tn+1) − uh,n+1 − LhQh∂u(tn+1)‖h

= ‖Phu(tn+1) − uh,n+1‖h

≤ ‖Phu(tn+1) −Rhu(tn+1)‖h + ‖Rhu(tn+1) − uh,n+1‖h,

596 I. Alonso-Mallo

and, from (H3),

‖Phu(tn+1) −Rhu(tn+1)‖h ≤ εh ‖u(tn+1)‖Z = O(εh),

it suffices to estimate‖Rhu(tn+1) − uh,n+1‖h.Then, we have

Rhu(tn+1) − uh,n+1 = (Rhu(tn+1) − uh,n+1) + (uh,n+1 − uh,n+1),

and we wish to estimate the right hand side.From Theorem 4.2,

Rhu(tn+1) − uh,n+1 = O(αhkp+1n + βhk

pn + γhk

p+1n + knεh).

Subtracting (4.8a) from (4.9),

Uh,n − Uh,n = (I ⊗ Ih − A ⊗ knAh,0)−1((1 ⊗ Ih)(Rhu(tn) − uh,n))

+(I ⊗ Ih − A ⊗ knAh,0)−1

×(kn(A ⊗ Ih)(Rh − Ph)Fn+c),

and (4.8b) from (4.10),

uh,n+1 − uh,n+1 = Rhu(tn) − uh,n + (bT ⊗ knAh,0)(Uh,n − Uh,n)

+kn(bT ⊗ Ih)(Rh − Ph)Fn+c

= r(knAh,0)(Rhu(tn) − uh,n)

+(bT ⊗ knAh,0)(I ⊗ Ih − A ⊗ knAh,0)−1

×(kn(A ⊗ Ih)(Rh − Ph)Fn+c)+kn(bT ⊗ Ih)(Rh − Ph)Fn+c.

Thus, we have proved the expression

Rhu(tn+1) − uh,n+1 = r(knAh,0)(Rhu(tn) − uh,n) + Fh,n

where

Fh,n = O(αhkp+1n + βhk

pn + γhk

p+1n + knεh).

The proof ends by using a standard recurrence argument.


5. Examples and numerical results

We show the feasibility of the previous theory for the discretization ofparabolic evolution problems. The spatial semidiscretization is achievedwith Galerkin finite element methods. We use here the notation introducedin [28]. The references [10,12,30–32] are also of interest. We consider aparabolic problem with non-homogeneous Dirichlet boundary conditions,but we remark that it is possible to consider other boundary conditions withslight modifications.

Let us suppose thatΩ is a bounded domain inRd (d = 2, 3) with aLipschitz continuous boundary∂Ω. We consider the second order linearoperator

Lw := −d∑

i,j=1

Di(aijDjw) +d∑

i=1

(Di(biw) + ciDiw) + a0w,

whereDi = ∂∂xj

. We suppose that the coefficientsaij(x), bi(x), ci(x) and

a0(x) are real smooth functions onΩ.L is elliptic, i.e. there exists a constantα0 > 0 such that

d∑i,j=1

aijξiξj ≥ α0|ξ|2

for all ξ ∈ Rd, a.e. onΩ.

The operatorL is associated to the bilinear form

a(w, v) :=∫

Ω

d∑i,j=1

aijDjwDiv −d∑

i=1

(biwDiv − civDiw) + a0wv

.ForV = H1

0 (Ω), the bilinear forma(·, ·) is well defined and continuousin V × V , i.e.

|a(u, v)| ≤ C||u||V ||v||V , u, v ∈ V.Moreover, under suitable conditions,a(·, ·) is coercive (see [28], p.164), i.e.

a(u, u) ≥ α||u||2V .Then the variational problem

“find u ∈ V such that

a(u, v) = (f, v), v ∈ V, ”(5.1)

is uniquely solvable forf ∈ L2(Ω).

598 I. Alonso-Mallo

Under the previous assumptions, a smooth solution of (5.1) is also thesolution to the homogeneous Dirichlet problem

Lu = f,u|∂Ω = 0.

In order to consider the non-homogeneous Dirichlet problem, we take

a boundary datumg ∈ H1/2(∂Ω). We extendg in wholeΩ to a functiong ∈ H1(Ω). Now, the variational problem is

“find u ∈ V such that

a(u+ g, v) = (f, v), v ∈ V. ”(5.2)

As it is well-known (see Remark 7.1.4 in [28]), it is possible to con-sider other formulations of the non-homogeneous Dirichlet problem. Theseformulations are based in approximations by mixed methods and are notconsidered here.

TakeX = L2(Ω), Y = H3/2(∂Ω) and denoteD(A) = H2(Ω) ⊂ X.Consider the operators acting onD(A),

Au = −Lu, ∂u = u|∂Ω .

On the other hand, we consider the operatorA0 = A|ker(∂). Then,A0 is the

infinitesimal generator of a holomorphic semigroupetA0 inX. Therefore,with the notation above, the IBVP

ut = −Lu+ f, onΩ × [0, T ],u|t=0 = u0, onΩ,u = g, on∂Ω × [0, T ],

(5.3)

can be fitted into the theory of abstract IBVPs developed in [3,5,26].Finite elements are used for the semidiscretization of the weak formula-

tion of (5.3),

“find u ∈ L2(0, T ;V ) ∩ C([0, T ], L2(Ω)) such that

d

dt(u(t) + g(t), v) + a(u(t) + g(t), v) = (f, v), v ∈ V.”

whereu(0)+ g(0) = u0, g(t) is a suitable extension of the boundary datum

g(t) in wholeΩ andf ∈ L2(Ω × (0, T )). Assuming that the boundarydatumg is regular, we follow the remark 6.2.2 in [28].

Wesuppose thatTh is apartitionofΩh =⋃

T∈ThT , a suitable subdomain

ofΩ. We take a family of finite dimensional spacesVhh>0 ⊂ X with theinherited norm andmade up of finite elements.We suppose that the partition


Ωh =⋃

T∈ThT and the finite elements satisfy the assumptions in Theorem

6.2.1 in [28].Let us takeXh = Vh and we writeXh = Xh,0 ⊕Xh,b whereXh,0 and

Xh,b collect the internal and boundary nodes, respectively. We denote byQh∂u ∈ Xh,b to the interpolation operator in the boundary ofu ∈ D(A).Therefore, in (4.4), the factorγh depends on the approximation of domainΩand boundary condition (see for example Sect. 4.4 in [30]). LetLh : X →Xh,0 be the orthogonal projection defined by

(Lhu, χ) = (u, χ), u ∈ X = L2(Ω), χ ∈ Xh,0,

which satisfies (4.3).We define the operatorsAh : Xh → Xh,0, through the relation

(Ahuh, χ) = a(uh, χ), uh ∈ Xh, χ ∈ Xh,0,

and, as in Sect. 4,Ah,0 is the restriction ofAh toXh,0. Then, the operatorsAh,0 are invertible and generateC0-semigroups onXh satisfying (H1) (see[12], Sect. 6 and7).Wealso introduce theRitz (or elliptic) projection definedby

(Ah(Rhu+Qh∂u), χ) = (Au, χ) = (LhAu, χ),

for u ∈ D(A) andχ ∈ Xh,0.We assume that the solutionu of the variational problem (5.2) is in the

spaceZ = Hr(Ω). SinceD(As) = H2s, s > 0, we deduce that (4.6) issatisfied. Moreover, with suitable hypotheses on the finite elements used(see e.g. Remark 6.2.2 in [28]), we can obtain (4.7), the second part of (H3),with εh = O(hr−1).

Now,we obtain (4.5) by usingRemark 4.1. By takingr = 2 and since thefunction(A, ∂) : D(A) = H2(Ω) → L2(Ω)×H3/2(Ω) is an isomorphism,we obtain (4.5), the first part of (H3), withαh = O(1) andβh = O(h). Theestimate on the global error obtained in Theorem 4.3 is

O

((γh + 1)

n+1∑m=1

kp+1m−1 + h

n+1∑m=1

kpm−1 + hr−1

),

and with additional hypotheses (Theorem 6.2.2 in [28]), it is possible toprove the estimate

O

((γh + 1)

n+1∑m=1

kp+1m−1 + h2

n+1∑m=1

kpm−1 + hr

).

The mild conditionh2 ≤ Ckn, C constant, allows us to obtain

O

((γh + 1)

n+1∑m=1

kp+1m−1 + hr

).

600 I. Alonso-Mallo

10−2

10−1

100

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

timestep

loca

l err

or

Fig. 1. Comparison of efficiency. Local errors

Remark 5.1Since the semidiscretization in time in Sect. 3 holds for a gen-eral Banach space framework, it is also possible to consider the maximumnorm. See for example [25].

5.1. Numerical experiments

We present some numerical experiments for the one-dimensional parabolicexample

ut(x, t) = uxx(x, t) + f(x, t), 0 ≤ x ≤ 1, 0 ≤ t ≤ 1,u(x, 0) = u0(x), 0 ≤ x ≤ 1,u(0, t) = g0(t), 0 ≤ t ≤ 1,u(1, t) = g1(t), 0 ≤ t ≤ 1.

(5.4)

This example is suitable for the studyof theorder of convergencebecause theboundary is very simple and certain sources of error, as a curved boundary,are not present. Therefore, it is possible to eliminate the spatial error andto find the order of convergence in time. This problem can be fitted intothe theory of abstract IBVPs developed in [26] takingX = L2(0, 1), Y =C

2 and defining, foru ∈ D(A) = H2(0, 1), Au = −uxx and ∂u =[u(0), u(1)].

The space discretization is carried out as follows [30]. ForJ a positiveinteger, we denoteh = 1/J , we introduce the uniform gridxj = jh,1 ≤ j ≤ J in (0, 1), and we takeXh the space of finite elements which are


10−2

10−1

100

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

glob

al e

rror

timestep

Fig. 2. Comparison of efficiency. Global errors

quadratic in each subinterval[xj , xj+1], j = 0, . . . , J−1, with theL2-norminherited fromX. The operatorsPh,Qh, andAh are defined in an analogousway to the previous example. We deduce thatγh = O(h1/2) in (4.4). Letus takeZ = D(A) = H2(0, 1) together with the usual norm. Ifu ∈ Z, theconditions (4.5) and (4.7) are obtained withαh = O(1), βh = O(h3) andεh = O(h3).

For the time discretization of (5.4), we use the two stage SDIRKmethodstudied in Sect. 3.

The numerical experiment is carried out for the datau0(x) = x2 −2x + 0.75, g0(t) = 0.75 exp(t), g1(t) = −0.25 exp(t) and f(x, t) =exp(t)(x2 − 2x− 1.25), which correspond to the solutionu(x, t) = exp(t)(x2 − 2x+ 0.75). We take the very small stepsizeh = 1/4096 in order toannihilate the error due to the space discretization (notice that this strategymay be disastrous when the Runge–Kutta method is explicit because we donot have unconditional stability).

Figure 1 displays the local errors in front of the time stepsizes. The lineswith circles, crosses and asterisks correspond, respectively, to the use of theboundary valuesG[0]

n andH [0]n ,G[1]

n andH [1]n ,G[2]

n andH [2]n . The numerical

order of consistency observed is roughly 2, 3 and 5. Therefore the orderreduction is totally avoided in the third case. We also remark the drasticdecrease of the sizes of the local errors.

We have also computed the global errors int = 1. The results are dis-played in Fig. 2 with the same plot symbols as Fig. 1. The numerical orderof convergence is 2, 2 and 4 respectively and the order reduction is also

602 I. Alonso-Mallo

totally avoided with the last modification of the boundary values. We alsonote that the order displayed for the first case is 2 because the time stepsizek is constant [6,13,21].

References

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runge-kutta methods without order reduction

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abstract ibvps developed

single step methods

homogeneous dirichlet problem

implicit rungekutta methods

order reduction phenomenon

banach spaces

initial boundary