# Convergence in the maximum norm of ADI-type methods for parabolic problems

Results on unconditional convergence in the Maximum norm for ADI-type methods, such as the Douglas method, applied to the time integration of semilinear parabolic problems are quite difficult to get, mainly when the number of space dimensions m is greater than two. Such a result is obtained here under quite general conditions on the PDE problem in case that time-independent Dirichlet boundary conditions are imposed. To get these bounds, a theorem that guarantees, in some sense, power-boundeness of the stability function independently of both the space and time resolutions is proved.

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06/01/2020

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## 1 Introduction

The present article considers the numerical solution of ODE systems

 ˙U=DU+g(t),U(0)=U0,t∈[0,t∗],D:=m∑j=1Dj,g(t):=m∑j=1gj(t), (1)

stemming from the spatial discretization by using finite differences (or finite volumes) of semilinear parabolic PDEs with constant diffusion coefficients and an initial condition and Dirichlet Boundary Conditions (BCs) of the form

 ∂tu(t,→x)=m∑j=1βj∂xjxju(t,→x)+c(t,→x),t∈[0,t∗],→x=(x1,…,xm)⊤∈I:=(0,1)m,u(0,→x)=u0(→x),→x∈∂I;u(t,→x)=h(t),t∈(0,t∗],→x∈∂I. (2)

Here, is a source term and we assume that its discretization is entirely included in , so that , consist only of contributions from the boundary conditions in the

-direction. In particular, we shall be concerned with time-independent Dirichlet boundary conditions, in which case the vectors

are constant.

To prove convergence in the maximum norm for many numerical methods of splitting type applied to (1), it is customary to get uniform bounds for , where is the time stepsize and is a rational mapping acting on the matrices , . Typically, we have , with when second order central differences are considered in the spatial discretization of (2). Here, we denote the spacing , where is the number of equidistant grid-points on the -direction, and stands for the Kronecker product of matrices. It should be observed that the matrices pairwise commute. Such methods of splitting type when applied to (1) typically produce a recursion for the global errors , , of the form , where stands for the numerical solution at , denotes the local error and is the stability matrix associated to the numerical integrator. For ADI-type integrators the stability matrix depends on (see, e.g., (Hun-Ver03, , Sec. II.2.3)). A relevant example is

 R(τD1,…,τDm)=I+Π(θ)−1τD,D=m∑j=1Dj, (3)

where , which has the associated stability function of complex variables

 R(z1,z2,…,zm)=1+zQ,z:=m∑j=1zj,Q:=m∏j=1(1−θzj). (4)

For the choice , this is the stability matrix of the Peaceman-Rachford method (when ), also the one of the Douglas scheme (Douglas62 , Hun-98a (Hun-Ver03, , p. 373)) and the one of the one-stage AMF-W-method GHH-sinum20 . Furthermore, the stability matrix of the so-called Hundsdorfer–Verwer scheme (Hun-Ver03, , Section IV.5.2), which is a -stage W-method of order in general, and of order for , is given by

 R(τD1,…,τDm)=I+2Π(θ)−1τD−Π(θ)−2τD+12(Π(θ)−1τD)2.

In this case the stability function is given by (with and defined in (4)). The power boundedness in the maximum norm of some -stability functions was already considered in GHH-BIT21 . However, for the power bound there obtained is not uniform, since it allows a logarithmic growth in terms of or GHH-BIT21 , i.e.,

 ∥R(τD1,…,τDm)n∥∞≤Cmmin{(ln(n+1))m,(ln(N+1))m},τ>0,nτ≤t∗,N=max{Nj,j=1,…,m}.

With this power bound, convergence results in the maximum norm of size , when the local errors are of size can be obtained. However, with power bounds of the stability matrix as the one in (5), it can be shown convergence of size in case of time independent BCs in (2). In Section 2, we prove a result related to the power boundedness for rational functions. This result is applied in Section 3 to show unconditional convergence in the maximum norm for some ADI-methods. In Section 4, numerical experiments are included to illustrate the orders of convergence regarding the PDE solution for some relevant ADI-type methods.

## 2 Bounds in the maximum norm for rational functions

We look for bounds in the maximum norm of the form

 ∥D−1R(τD1,τD2,…,τDm)n∥∞≤C<∞,tn=nτ,n=0,1,2,…,τ>0,D=m∑j=1Dj,Dj=βj(Im⊗⋯⊗Ij+1⊗Lj⊗Ij−1⊗⋯⊗I1),Lj=1Δx2j\rm tridiag(1,−2,1)Nj,Ij\rm is the identity matrix of dimension Nj,Δxj=1Nj+1, (5)

where is a rational function (or a mapping when acting on the matrices ) of complex variables that is -stable, i.e.

 |R(z1,…,zm)|≤1,∀zj∈W(α):={z∈C:|arg(−z)|≤α}∪{0},\rm for% some 0<α≤π/4. (6)

Of course, if (6) holds true for some then it also holds for .

###### Theorem 1

If is a rational function that satisfies (6), then there exists a constant only depending on and such that (5) holds for .

The proof of this theorem is given below and makes use of the following two lemmas.

###### Lemma 1

For any matrix (5) it holds that

 ∥L−1j∥∞≤18, (7)
 ∥(zIj−τLj)−1∥∞≤sec(θ/2)|z|,∀τ>0\rm and z=|z|eiθ,|θ|<π, (8)
 ∥(zIj−τLj)−1∥∞≤18τ−|z|,∀|z|<8τ, (9)

and

 ∥(zIj−τLj)−1∥∞≤1|z|−4τ/Δx2j,∀|z|>4τ/Δx2j>0. (10)

Proof. The formula in (7) is well known in the literature (see, e.g., (Mattheij, , formula (4.10)) or (lar-thom, , p. 43-45)). The formula (8) is an immediate consequence of Lemma 4.1 in GHH-BIT21 (see also (farago02ste, , formula (5))), with and , since

 ∥(zIj−τLj)−1∥∞=μ−1∥(μ−1zIj−Tj)−1∥∞≤μ−1sec(arg(μ−1z)/2)μ−1|z|=sec(θ/2)|z|.

To show (9), by considering , we have that

 ∥(zIj−τLj)−1∥∞=∥(τLj)−1(Ij−zτ−1L−1j)−1∥∞≤τ−1∥L−1j∥∞1−|z|τ−1∥L−1j∥∞≤τ−18−11−|z|τ−18−1=18τ−|z|.

To show (10), for it holds

 ∥(zIj−τLj)−1∥∞=∥z−1(Ij−z−1τLj)−1∥∞≤|z|−11−|z|−1τ∥Lj∥∞≤1|z|−4τ/Δx2j.

###### Lemma 2

Assume that for any positive integer we have that

 zj=−rjeiθj,rj>0,−α≤θj≤α,(j=1,2,…,m),0≤α≤π/4. (11)

Then

 |m∑j=1zj|≥ ⎷m∑j=1|zj|2≥1√mm∑j=1|zj|≥√mm∏j=1|zj|1/m. (12)

Proof. The last two inequalities in (12

) follow from the fact that for positive numbers the Quadratic Mean is greater or equal than the Arithmetic Mean and this is greater or equal than the Geometric Mean. To show the first inequality, we observe that for complex numbers

satisfying (11) it holds that Hence, fulfils and it has an angle with the negative axis. In particular, takes the form (11). Then, adding a new complex number (11) and using the same argument we deduce that fulfils and it has an angle with the negative axis. The application of the induction principle concludes the proof.

Proof of Theorem 1. We define and use below the following notation for the Kronecker product of matrices Consider the positively oriented boundary of the open domain , which is symmetric with respect to the negative real axis in the complex plane,

 γ1:={−reiθ;−α≤θ≤α},γ3:={−r∗eiθ;−α≤θ≤α},γ2:={−ρe−iα,r≤ρ≤r∗}∪{−(r+r∗−ρ)eiα,r≤ρ≤r∗},r:=4^βτ,r∗:=8βτ/(Δx)2,Δx:=min{Δx1,…,Δxm}<1,\rm since each Δxj=1/(Nj+1)<1. (13)

Observe that Let us define the rational function (and the associated mapping when acting on matrices)

 ϕ(z1,z2,…,zm)=(z1+z2+…+zm)−1R(z1,z2,…,zm)n. (14)

Taking into account that is analytic if , from the Cauchy’s integral formula applied on each variable we get the following formula by using iterated integrals

 ϕ(z∗1,z∗2,…,z∗m)=1(2πi)m∮Γ⋯∮Γϕ(z1,z2,…,zm)m∏j=1dzjzj−z∗j,∀(z∗1,z∗2,…,z∗m)∈Ωm. (15)

By considering the mapping acting on the matrices we deduce that

 ϕ(τD1,…,τDm)=1(2πi)m∮Γ⋯∮Γϕ(z1,z2,…,zm)[⊗(zjIj−τβjLj)−1dzj]mj=1. (16)

Observe that the eigenvalues of each matrix

are , , where

 −r∗=−8βτ(Δx)2<λ(j)Nj<λ(j)1=−4βjτ(sin(π/2⋅Δxj)Δxj)2<−4βjτ≤−r.

Hence, the spectrum of falls in .

At this point we should notice the identity From here, taking the maximum norm and using that for two matrices and , we get that

 ∥D−1R(τD1,τD2,…,τDm)n∥∞≤A:=τ(2π)m∮Γ⋯∮Γ|ϕ(z1,z2,…,zm)|m∏j=1∥(zjIj−τβjLj)−1∥∞|dzj|. (17)

Next we bound when . We distinguish three cases.

1. , then , and . From (9) it follows that

 ∥(zjIj−τβjLj)−1∥∞≤18τβj−r≤18τ^β−r=1r

and we deduce that

 zj∈γ1⟹∥(zjIj−τβjLj)−1∥∞|dzj|≤rrdθj=dθj. (18)
2. , then , and . From (8) it follows that

 ∥(zjIj−τβjLj)−1∥∞≤sec((π−α)/2)|zj|=1ρjsin(α/2)

and then

 zj∈γ2⟹∥(zjIj−τβjLj)−1∥∞|dzj|≤1ρjsin(α/2)dρj. (19)
3. , then , and . From (10) it follows that

 ∥(zjIj−τβjLj)−1∥∞≤1|zj|−4τβj/Δx2j≤1r∗−r∗/2=2r∗

and we get that

 zj∈γ3⟹∥(zjIj−τβjLj)−1∥∞|dzj|≤2dθj. (20)

From the A()-stability of we deduce that

 |ϕ(z1,z2,…,zm)|≤1|z1+…+zm|,∀(z1,…,zm)∈(¯Ω)m. (21)

From (12) in Lemma 2, we have that Consequently, this together with (21) yields

 |ϕ(z1,z2,…,zm)|≤m−1/2m∏j=1|zj|−1/m,∀(z1,…,zm)∈(¯Ω)m. (22)

Now, by considering (17) and (22) we deduce that

 \bf A≤\bf B:=τ√m(2π)m∑1≤i1,i2,…,im≤3∫γi1⋯∫γimm∏j=1(∥(zjIj−τβjLj)−1∥∞|zj|−1/m|dzj|). (23)

Taking account that all these iterated integrals can be transformed into products of integrals in one variable, we get

 \bf B≤τ√m(2π)m∑m1+m2+m3=mmj≥0m!m1!m2!m3!(A1)m1(A2)m2(A3)m3,\rm where, using (???), (???) and (???) respectively,A1:=∫α−αr−1/mdθ=2αr−1/m,A2:=∫r∗rρ−1−1/msin(α/2)dρ=msin(α/2)(r−1/m−(r∗)−1/m)

Then, we have for ,

 τ(A1)m1(A2)m2(A3)m3≤τCm1,m2,m3r−(m1+m2)/m(r∗)−m3/m,

with

 Cm1,m2,m3:=(2α)m1(msin(α/2))m2(4α)m3,

and

 r−(m1+m2)/m(r∗)−m3/m=r−1(rr∗)m3/m≤14^βτ(12)m3/m.

Hence, each term is bounded since

 τ(A1)m1(A2)m2(A3)m3≤^β−1Cm1,m2,m34⋅2m3/m.

This concludes the proof.

## 3 Convergence in the uniform norm of some ADI-type methods

The first goal of this section is to show unconditional convergence of order two in the maximum norm for semilinear parabolic problems with constant diffusion coefficients (and a time dependent source term) and time-independent Dirichlet boundary conditions (1)-(2), when the one-step AMF-W-method (henceforth denoted as AMF-W1) in GHH-sinum20 ; gonzalez18amf is considered with the parameter choice

 K(0)1 =τDUn+τg(tn), (25) (I−θτDj)K(j)1 =K(j−1)1+θτ2˙gj(tn),j=1,…,m, Un+1 =Un+K(m)1,

where stands for the derivative of a function regarding . The following discussion can be applied in similar terms to the Douglas method (Hun-Ver03, , p. 373). We use the same notations as in GHH-sinum20 . The global error at the time step is denoted as in (GHH-sinum20, , formula (2.3)) by where is the solution of the numerical method and is at the same time the exact solution of the (1) and the exact solution of the PDE on the set of discrete points of the spatial mesh-grid . Observe that we will not consider in our analysis the truncation errors introduced in the spatial discretization of the PDE, since when using central differences we get a stable space discretization and the truncated spatial errors do not play any important role in the analysis of global errors (space truncation errors plus time integration errors) as it can be seen e.g. in (Hun-Ver03, , Chapt. IV). It should be remarked that the discretization of the source term is entirely included in (GHH-sinum20, , Sect. 1). Besides, the terms

 φi(t):=DiU(t)+gi(t),t∈[0,t∗],(i=1,…,m),\rm satisfym∑i=1φi(t)=˙U(t) (26)

and they are smooth (i.e. they have bounded first and second derivatives independently of the spatial resolution), since is a smooth function and we have (below for and )

 φi(t):=DiU(t)+gi(t)=βi∂xixiu(t,→xG)+δi1⋅c(t,→xG)+O((Δxi)2∂xixixixiu(t,→xG)).

Additionally, when time independent boundary conditions are assumed in the PDE problem (2), we have and (GHH-sinum20, , Sect. 1 and 4),

 Di1Di2…Dir˙φj(t)=βi1βi2⋯βir∂2r+2˙u(t,→xG)∂x2i1∂x2i2⋯∂x2ir∂x2j+O((Δxi1)2+⋯+(Δxir)2+(Δxj)2),i1
###### Theorem 2

Assume that the exact solution of the discretized problem (1) satisfies the following uniform bounds

 ∥U(j)(t)∥∞≤C,∥g(i)(t)∥∞≤C,i=0,…,3,j=0,…,4,t∈[0,t∗],

that and that (27) holds. Then, the global errors, with , for the AMF-W-method (25) with fulfill where the constant only depends on and .

Proof. According to (GHH-sinum20, , formula (2.11)) the global errors of (25) follow the recursion

 En=(RnD−1)DE0+n−1∑j=0Rn−1−jSj,n=1,2,…,n∗=t∗/τ, (28)

where the matix is given by (3) and the discretization local errors are given by (GHH-sinum20, , formula (2.10))

 Sn=Π(θ)−1(τ˙U(tn)+θτ2˙G(tn))−(U(tn+τ)−U(tn)), (29)

with , and the convention (see (GHH-sinum20, , formula (2.4))). We also make use of other expression for the global errors (see (GHH-sinum20, , formula (4.11))), obtained by partial summation in (28),

 En=(I−Rn)(I−R)−1S0+n−2∑j=0(I−Rn−1−j)(I−R)−1(Sj+1−Sj),n=1,2,…, (30)

and of a simplified expression for the local errors given in (GHH-sinum20, , formula (4.7))

 Sn=S(1)n+S(2)n,S(1)n=τ22Π−1m∑i=1(Πi−Π)˙φi(tn),S(2)n=−τ32∫10(1−s)2...U(tn+sτ)ds,Π=(I−θτD1)⋯(I−θτDm),Πi:=(I−θτD1)⋯(I−θτDi−1),(i>1),Π1:=I. (31)

Since, from (26),