 # A Relaxation/Finite Difference discretization of a 2D Semilinear Heat Equation over a rectangular domain

We consider an initial and Dirichlet boundary value problem for a semilinear, two dimensional heat equation over a rectangular domain. The problem is discretized in time by a version of the Relaxation Scheme proposed by C. Besse (C. R. Acad. Sci. Paris Sér. I, vol. 326 (1998)) for the nonlinear Schrödinger equation and in space by a standard second order finite difference method. The proposed method is unconditionally well-posed and its convergence is established by proving an optimal second order error estimate allowing a mild mesh condition to hold.

## Authors

##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

## 1. Introduction

### 1.1. Formulation of the problem

Let , with and , , and be the solution of the following initial and boundary value problem:

 (1.1) ut=Δu+g(u)+f\rm on  Q, (1.2) u(t,x)=0∀(t,x)∈[0,T]×∂D, (1.3) u(0,x)=u0(x)∀x∈D,

where with , and with

 (1.4) u0|∂D=0.

### 1.2. Formulation of the numerical method

Let be the set of all positive integers. For a given , we define a uniform partition of the time interval with time-step , nodes for , and intermediate nodes for . Also, for given , we consider a uniform partition of with mesh-width and nodes for , and a uniform partition of with mesh-width and nodes for . Also, we set , , , and introduce the discrete space

 X∘H:={V=(Vi,j)(i,j)∈I∈R(J1+2)×(J2+2):Vi,j=0∀(i,j)∈∂I},

and a discrete Laplacian operator by

 (ΔHV)i,j:=Vi−1,j−2Vi,j+Vi+1,jh21+Vi,j−1−2Vi,j+Vi,j+1h22,∀(i,j)∈I∘∀V∈X∘H.

In addition, we introduce an operator , which, for given , is defined by for all . Finally, for any , we define by for all .

The Relaxation Finite Difference (RFD) method uses a standard finite difference scheme for space discretization along with a variant of the Relaxation Scheme for time stepping (cf. ).

Step 1: First, set

 (1.5) U0:=I∘H[u0]∈X∘H

and find such that

 (1.6) U12−U0(τ/2)=ΔHU12+g(u0)+I∘H[f(t12,⋅)].

Step 2: Set

 (1.7) Φ12:=g(U12)∈X∘H

and find such that

 (1.8) U1−U0τ=ΔH(U1+U02)+Φ12+I∘H[f(t1,⋅)+f(t0,⋅)2].

Step 3: For , first set

 (1.9) Φn+12:=2g(Un)−Φn−12∈X∘H

and then find such that

 (1.10) Un+1−Unτ=ΔH(Un+1+Un2)+Φn+12+I∘H[f(tn+1,⋅)+f(tn,⋅)2].

### 1.3. References and main results

The Relaxation Scheme (RS) has been introduced by C. Besse  as a linearly implicit, conservative, time stepping method for the approximation of the solution to the nonlinear Schrödinger equation. Convergence results has been obtained in  and  for the (RS) time-discrete approximations of the Cauchy problem for the nonlinear Schrödinger equation, which can not be concluded in the fully-discrete case, because on the one hand they are valid for small final time , and on the other hand are based on the derivation of a priori bounds (with ) for the time discrete approximations. Recently, in , the (RS) joined with a finite difference scheme is proposed for the approximation of the solution to a semilinear heat equation in the 1D case, and the corresponding error analysis arrived at optimal second order error estimates. Here, we investigate the extension of the results obtained in  in the 2D case. We would like to stress that a fylly-discrete version of the (RS) applied on a parabolic problem can be analyzed by using energy estimates, which, however, are not efficient in the case of the nonlinear Schrödinger equation (see ).

To develop an error analysis for the (RFD) method, we introduce the modified scheme (see Section 3.2) that follows from the (RFD) method after mollifying properly the terms with nonlinear structure (cf. , , ). For the approximations obtained from the modified scheme, we provide an optimal, second order error estimate in the discrete norm at the nodes and in the discrete norm at the intermediate nodes (see Theorem 3.1). After applying an inverse inequality (see (2.3)) and imposing a proper mesh condition (see (3.31)), the latter convergence result implies that the discrete norm of the modified approximations is uniformly bounded, and thus they are the same with those derived by the (RFD) method and hence inherit their convergence properties o (see Theorem 3.2), i.e. that there exist constant , independent of , , , such that

 max0≤n≤N[∣∣Φn+12−I∘H[g(u(tn+12,⋅))]∣∣0,H+∣∣Un−I∘H[u(tn,⋅)]∣∣1,H]≤C(τ2+h21+h22),

where is a discrete norm and is a discrete norm.

## 2. Preliminaries

### 2.1. Discrete relations

We provide with the discrete inner product given by

 (V,Z)0,H:=h1h2∑(i,j)∈I∘Vi,jZi,j∀V,Z∈X∘H

and we shall denote by its induced norm, i.e. for . Also, we equip with a discrete -norm defined by for , and with a discrete -type norm given by

 |V|1,H:=[h1h2J2∑j=1J1∑i=0∣∣∣Vi+1,j−Vi,jh1∣∣∣2+h1h2J1∑i=1J2∑j=0∣∣∣Vi,j+1−Vi,jh2∣∣∣2]1/2∀V∈X∘H.

In the convergence analysis of the method, we will make use of the following, easy to verify, relation

 (2.1) (ΔHV,V)0,H=−|V|21,H∀V∈X∘H,

of the discrete Poincaré-Friedrishs inequality

 (2.2) ∥V∥0,H≤12min{a2−a1,b2−b1}|V|1,H∀V∈X∘H,

of the inverse inequality

###### Lemma 2.1.

For it holds that

 (2.3) |V|∞,H≤L√h1+h2|V|1,H

with .

###### Proof.

Let and such that . Since , we conclude (see ) that

 (2.4) |Vi0,j0|2≤(a2−a1)h1J1∑i=0∣∣∣Vi+1,j0−Vi,j0h1∣∣∣2≤a2−a1h2h1h2J2∑j=1J1∑i=0∣∣∣Vi+1,j−Vi,jh1∣∣∣2

and

 (2.5) |Vi0,j0|2≤(b2−b1)h2J2∑j=0∣∣∣Vi0,j+1−Vi0,jh2∣∣∣2≤b2−b1h1h1h2J1∑i=1J2∑j=0∣∣∣Vi,j+1−Vi,jh2∣∣∣2.

From (2.4) and (2.5), we conclude that

 |V|2∞,H≤(b2−b1)(a2−a1)(a2−a1)h1+(b2−b1)h2|V|21,H

which, easily, yields (2.3). ∎

and of the following Lipschitz-type inequality:

###### Lemma 2.2.

Let with . Then, for , it holds that

 (2.6) ∥g(va)−g(vb)−g(za)+g(zb)∥0,H≤supR|g′||za−zb|∞,H∥vb−zb∥0,H+(supR|g′|+supR|g′′||za−zb|∞,H)∥va−vb−za+zb∥0,H.
###### Proof.

It is similar to the proof of (2.9) in Lemma 2.3 in . ∎

### 2.2. Consistency Errors

To simplify the notation, we set for , and for .

#### 2.2.1. Consistency error in time

For , let be defined by

 (2.7) g(un+12)+g(un−12)2=g(un)+rn.

Then, setting and expanding, by using the Taylor formula around , we obtain:

 rn=τ22I∘H[∫120[(12−s)ζtt(tn+sτ,⋅)+sζtt(tn−12+sτ,⋅)]ds],n=1,…,N−1,

which, easily, yields

 (2.8) τmax1≤n≤N−1|rn|∞,H+max2≤n≤N−1|rn−rn−1|∞,H≤CIτ3maxQ(|ζtt|+|ζttt|).

Let be defined by

 (2.9) u12−u0(τ/2)=I∘H[uxx(t12,⋅)]+g(u0)+I∘H[f(t12,⋅)]+r14

and, for , let be specified by

 (2.10) un+1−unτ=I∘H[uxx(tn+1,⋅)+uxx(tn,⋅)2]+g(un+12)+I∘H[f(tn+1,⋅)+f(tn,⋅)2]+rn+12.

Using (1.1), from (2.9) and (2.10), we obtain

 (2.11) r14=r14A+r14B,\rm andrn+12:=rn+12A+rn+12B,n=0,…,N−1,

where

 r14A=u12−u0(τ/2)−I∘H[ut(t12,⋅)],r14B:=g(u12)−g(u0)

and

 rn+12A:=un+1−unτ−I∘H[ut(tn+12,⋅)]−I∘H[ut(tn+1,⋅)+ut(tn,⋅)2−ut(tn+12,⋅)],rn+12B:=g(un+1)+g(un)2−g(un+12).

Applying the Taylor formula we obtain

 (2.12) r14A=−2τI∘H[∫120sutt(sτ,⋅)ds],r14B=I∘H[∫t1t0ζt(s,⋅)ds]

and

 (2.13) rn+12A=τ22I∘H[∫120[s2uttt(tn+sτ,⋅)+(12−s)2uttt(tn+12+sτ,⋅)]ds]−τ22I∘H[∫120[suttt(tn+τs,⋅)+(12−s)uttt(tn+12+τs,⋅)]ds],rn+12B=τ22I∘H[∫120[sζtt(tn+τs,⋅)+(12−s)ζtt(tn+12+τs,⋅)]ds],n=0,…,N−1.

Thus, from (2.11), (2.13) and (2.12), we arrive at

 (2.14) τ|r14|∞,H+max0≤n≤N−1|rn+12|∞,H≤CIIτ2maxQ(|ζt|+|ζtt|+|utt|+|uttt|).

#### 2.2.2. Consistency error in space

Also, let be defined by

 (2.15) u1−u0(τ/2)=ΔHu1+g(u0)+I∘H[f(t12,⋅)]+s14

and, for , let be given by

 (2.16)

Subtracting (2.15) from (2.9) and (2.16) from (2.10), we obtain

 (2.17) r14−s14=I∘H[uxx(t12,⋅)]−ΔHu12,rn+12−sn+12=I∘H[uxx(tn+1,⋅)+uxx(tn,⋅)2]−ΔH(un+1+un2),n=0,…,N−1.

For , we use of the Taylor formula with respect to the space variables around to get

 I∘H[uxx(t,⋅)]−ΔH(IH[u(t,⋅)])i,j=h216∫10(1−z)3∂4x1u(t,x1,i+h1z,x2,j)dz+h216∫10z3∂4x1u(t,x1,i−1+h1z,x2,j)dz+h226∫10(1−z)3∂4x2u(t,x1,i,x2,j+h2z)dz+h226∫10z3∂4x2u(t,x1,i,x2,j−1+h2z)dz∀(i,j)∈I∘,

which, along with (2.17), yields

 (2.18) |s14−r14|∞,H+max0≤n≤N−1|sn+12−rn+12|∞,H≤CIII(h21+h22)maxQ(|∂4x1u|+|∂4x2u|).

## 3. Convergence Analysis

### 3.1. A mollifier

For , let (cf. , , 

) be an odd fuction defined by

 (3.1)

where is the unique polynomial of that satisfies the following conditions:

 pδ(δ)=δ,p′δ(δ)=1,p′′δ(δ)=p′′′δ(δ)=0,pδ(2δ)=2δ,p′δ(2δ)=p′′δ(2δ)=p′′′δ(2δ)=0.

### 3.2. The modified scheme

For given , the modified version of the (RFD) method (cf. , , ), derives approximations of the solution as follows:

Step M1: First, we set

 (3.2) V0δ:=U0andV12δ:=U12.

Step M2: Set

 (3.3) Φ12δ:=g(nδ(V12δ))

and find such that

 (3.4) V1δ−V0δτ=ΔH(V1δ+V0δ2)+Φ12δ+I∘H[f(t1,⋅)+f(t0,⋅)2].

Step M3: For , first define by

 (3.5) Φn+12δ:=2g(nδ(Vnδ))−Φn−12δ

and, then, find such that

 (3.6) Vn+1δ−Vnδτ=ΔH(Vn+1δ+Vnδ2)+Φn+12δ+I∘H[f(tn+1,⋅)+f(tn,⋅)2].

### 3.3. Convergence of the modified scheme

###### Theorem 3.1.

Let and . Then, there exist positive constants , and , independent of and , such that: if , then

 (3.7) |u12−V12δ⋆|1,H≤CBCV,2δ⋆(τ32+τ12h21+τ12h22)

and

 (3.8) max0≤m≤N−1∥g(um+12)−Φm+12δ⋆∥0,H+max0≤m≤N|um−Vmδ⋆|1,H≤CBCV,3δ⋆(τ2+h21+h22)
###### Proof.

To simplify the notation, we set for , and for . In the sequel, we will use the symbol to denote a generic constant that is independent of , and , and may changes value from one line to the other. Also, we will use the symbol to denote a generic constant that depends on but is independent of , , and may changes value from one line to the other.

Since , after subtracting (1.6) and (1.8) from (2.15) and (2.16) (with ), respectively, we obtain

 (3.9) e12=τ2ΔHe12+τ2s14, (3.10) e1=τ2ΔHe1+τθ0+τs12.

Taking the inner product of (3.9) with , and then using (2.1), the Cauchy-Schwarz inequality, (2.14), (2.18) and the arithmetic mean inequality, we get

 ∥e12∥20,H+τ2|e12|21,H=τ2(s14,e12)0,H≤τ2(∥s14−r14∥0,H+∥r14∥0,