garcia de galdeano

Some methods of order three for reaction -diffusion problems

2003

“garcia de galdeano”

PRE-PUBLICACIONES del seminario matematico

C. Clavero J. L. Gracia

n. 29

seminario matemático

garcía de galdeano Universidad de Zaragoza

Some methods of order three for reaction-diffusion problems1 C. Clavero Departamento de Matemática Aplicada, Universidad de Zaragoza C/ Mar´ıa de Luna 3, C.P. 50018 Zaragoza, Spain E-mail: [email protected]

J.L. Gracia Departamento de Matemática Aplicada, Universidad de Zaragoza C/ Ciudad Escolar, 44003 Teruel, Spain Abstract — In this work we prove the uniform convergence of high order of some numerical methods used to solve a type of linear reaction-diffusion singularly perturbed problems. Firstly, we will consider the case of 1D stationary problems; based on the central finite difference scheme, constructed on a piecewise uniform Shishkin mesh, a new method of HODIE type is defined having third order of ε-uniform convergence in the maximum norm. Using this HODIE scheme jointly with a two stage SDIRK method, we solve a 1D parabolic singularly perturbed problem. Finally, we consider a parabolic 2D problem of the same type; in this case, the HODIE scheme is combined with a fractional step Runge Kutta method. In both time dependent problems, the resulting methods have third order of uniform convergence with respect both time and spatial variables. We show some numerical experiments confirming the theoretical results. 2000 Mathematics Subject Classification: 65N12; 65N30; 65N06. Keywords: Reaction-diffusion problems, HODIE schemes, SDIRK method, fractional RK method, uniform convergence, high order.

1. Introduction Some phenomena in different areas of science, by example in chemical reactions, are models of singularly perturbed problems of reaction-diffusion type, for which the diffusion coefficient can be very small with respect to the reaction term. The presence of parabolic boundary layers makes necessary the use of robust numerical methods, which give effective solutions for any value of the diffusion parameter. In [12] it was proved that, in general, even for the simple case of nonlinear 1D problems, on uniform meshes it is not possible to construct robust methods; therefore, it seems necessary to use fitted mesh methods, i.e., methods defined on special meshes condensing the mesh points in the boundary layer regions. Many papers have been developed following this way using Shishkin or Bakhvalov meshes for both stationary or time dependent problems, see [4, 5, 9] and references therein. Nevertheless, only 1

This research has been supported by the project MCYT/FEDER Project BFM2001–2521

2 C. Clavero and J.L. Gracia in few cases the methods have order of uniform convergence bigger than one; for example, in [9] is constructed a finite difference scheme of fourth order used to solve a 1D problem; in [8, 10, 11] the authors analyze the case of 1D parabolic problems and they construct numerical methods having order 2 in space and order 2 and 3 in time; finally, in [7] the authors study the case of 2D singularly perturbed parabolic convection-diffusion problem and they developed a third order method in both space and time variables. In this work we give a systematic way to construct a third order uniformly convergent method, used to solve different problems of reaction-diffusion type, including 1D elliptic and 1D or 2D parabolic problems. The paper is structured as follows; in section 2 we study the stationary 1D problems and we construct the basic HODIE finite difference scheme having order three; in section 3 we consider the 1D parabolic problem and we prove that combining adequately the HODIE scheme with a SDIRK method, the resulting numerical scheme is also a third order ε-uniform convergent method; in section 4 we analyze the 2D parabolic problems and we see that the combination of the HODIE scheme together with a fractional Runge Kutta scheme, permit us to deduce again a third order ε-uniform convergent method; finally, in section 5 we show the results obtained in different examples, using the methods previously developed; in all cases the results confirm in practice the predicted order of uniform convergence. Henceforth, C denote any positive constant independent of the diffusion parameter ε and the discretization parameters N and ∆t.

2. The one dimensional problem In this section we consider the linear 1D singularly perturbed reaction-diffusion problem Lε u ≡ −εu + d(x)u = f (x),

0 < x < 1,

u(0) = u0 , u(1) = u1 ,

(1)

where 0 < ε 1, u0 and u1 are given constants, we suppose that d and f are sufficiently smooth functions and also that the reaction term is positive (d(x) β > 0). It is well known, see [6], that √ for ε sufficiently small it appears a boundary layer at x = 0 and x = 1, both of width O( ε). To study the ε-uniform convergence of a numerical method, it is convenient to know the asymptotic behaviour, with respect to ε, of the exact solution of (1) and its derivatives. In [6] it was proved that this exact solution can be written as u = v + w, where |v (k) (x)| C, with

|w(k) (x)| Cε−k/2 e(x, ε, β),

e(x, ε, β) = e−

√

√ βx/ ε

+ e−

√

√ β(1−x)/ ε

(2)

,

and also that these functions satisfy Lε v = f, Lε w = 0. To discretize (1) we will use a finite difference scheme constructed on the following Shishkin mesh (see [6]): let N = 4k be, where k is a positive integer; the piecewise uniform mesh is defined dividing [0,1] into three intervals [0, σ], [σ, 1 − σ], [1 − σ, 1], where σ is √ (3) σ = min {1/4, σ0 ε ln N }, with σ0 a constant to be fixed later, taking uniform meshes on each one of these intervals, with k + 1 points in [0, σ] and [1 − σ, 1], and 2k + 1 in [σ, 1 − σ] respectively. Then, the mesh

3

Some methods of order three for reaction-diffusion problems N

I ε = {xj ; j = 0, · · · , N } is given by  j = 0, . . . , N/4,  jh, xN/4 + (j − N/4)H, j = N/4 + 1, . . . , 3N/4, xj =  x3N/4 + (j − 3N/4)h, j = 3N/4 + 1, . . . , N,

(4)

with H = 2(1 − 2σ)/N and h = 4σ/N . When σ = 1/4 the mesh is uniform, N −1 is very small in comparison with the size of ε and √ a classical analysis could be made; so, in the rest of the paper we will assume that σ = σ0 ε ln N . Moreover, in the analysis we will suppose that H 2 ||d ε, which is the proper singularly perturbed case in practice. On this mesh, the central finite difference scheme is defined by LN,dc Zj ≡ −εD+ D− Zj + dj Zj = fj , ε

1 j N − 1,

Z0 = u(0), ZN = u(1),

where D+ Uj = (Uj+1 − Uj )/hj+1 and D− Uj = (Uj − Uj−1 )/hj . It is known (see [6]) that this scheme is of positive type, it satisfies a discrete maximum principle and consequently it is uniformly stable in the maximum norm. On the other hand, it is straightforward to prove that the local error, at the mesh point xj , satisfies dc τj,u = −ε(hj+1 − hj )u j /3 + 2ε(R3 (xj , xj+1 , u)/hj+1 + R3 (xj , xj−1 , u)/hj )/(hj + hj+1 ),

where Rn is the Taylor expansion remainder. Using appropriately Taylor expansion, it is not difficult to obtain  2 −2 2 if xj ∈ (0, σ) ∪ (1 − σ, 1),  CN σ0 ln N,√ dc −1 − βσ0 C(N ε + N √ ), if xj ∈ {σ, 1 − σ}, |τj,u |  C(N −2 ε + N − βσ0 ), if xj ∈ (σ, 1 − σ). gives the bound From these bounds, the ε-uniform stability of operator LN,dc ε √

|u(xj ) − Zj | C(max {N −2 σ02 ln2 N, N −1 ε} + N − βσ0 ), 0 j N. √ Since H 2 ||d||∞ ε, if βσ0 2 then the scheme is uniformly convergent of order almost 2 (due to the logarithmic factor). To improve this order, we only need to change the discretization in the mesh points located at (0, σ) and (1 − σ, 1). So, following [3] we propose the HODIE discretization Uj ≡ rj− Uj−1 + rjc Uj + rj+ Uj+1 = qj1 fj−1 + qj2 fj + qj3 fj+1 , LN,HO ε

if xj ∈ / [σ, 1 − σ],

where the coefficients are calculated by imposing that the space of polynomials of degree less or equal that 3 belong to the kernel of the local error and also that the normalization condition qj1 + qj2 + qj3 = 1 holds. Similarly to [3], it is easy to prove that these coefficients are given by qj3 = qj1 rj+

rjc = qj1 dj−1 + qj2 dj + qj3 dj+1 − rj− − rj+ , 2

= −ε/h +

qj3 dj+1 ,

rj−

2

= −ε/h +

qj1 dj−1 .

(5) (6)

The free parameter qj1 is chosen in order to achieve third order of uniform convergence. For any value of ε and xj ∈ (0, σ) ∪ (1 − σ, 1), the local error satisfies (4)

HO τj,u = uj h2 ε(qj1 − 1/12) + R5 (xj , xj−1 , u)rj− + R5 (xj , xj+1 , u)rj+ + εqj1 (R3 (xj , xj−1 , u )+ +R3 (xj , xj+1 , u ) + qj1 (dj−1 R5 (xj , xj−1 , u) + dj+1 R5 (xj , xj+1 , u)).

4 C. Clavero and J.L. Gracia We want remark that the term of order O(N −3 ) disappears due the symmetry that the HO = coefficients of the scheme have. Then, taking qj1 = 1/12, trivially it follows that τj,u −4 O(N ). So, the HODIE scheme can be written as N,HO Lε Uj ≡ rj− Uj−1 + rjc Uj + rj+ Uj+1 = qj1 fj−1 + qj2 fj + qj3 fj+1 , 0 < j < N, (7) U0 = u(0), UN = u(1), where

qj1 = 0, qj2 = 1, qj3 = 0, if xj ∈ [σ, 1 − σ], 1 2 3 qj = 1/12, qj = 5/12, qj = 1/12, if xj ∈ (0, σ) ∪ (1 − σ, 1).

Lemma 2.1. Let N N0 be, where N0 > 0 is a positive integer independent of ε such that (8) 4σ02 d∞ /3 < N02 / ln2 N0 . Then, there exists a constant c such that rj− + rjc + rj+ c > 0, rj− < 0, rj+ < 0,

1 j < N.

Consequently the method (7) is of positive type and, therefore, it satisfies a discrete maximum principle. Moreover, the operator is ε-uniform stable in the maximum norm. Proof. The proof immediately follows from (5), (6) and the value of qj1 = 1/12. To study the consistency of the scheme, in a similar way to the continuous problem, we decompose the numerical solution as Uj = Vj + Wj , where {Vj } and {Wj } are respectively the solutions of the discrete problems LN,HO Vj = qj1 fj−1 + qj2 fj + qj3 fj+1 , 1 j < N, V0 = v(0), VN = v(1), ε Wj = 0, 1 j < N, W0 = w(0), WN = w(1). LN,HO ε Theorem 2.1. Let N N0 be, u the solution of (1) and {Uj } the solution of (7). Then, there exists a positive constant C independent of ε and N , such that |u(xj ) − Uj | C(max {N −4 σ04 ln4 N, N −1 ε} + N − Since H 2 d ε, if

√

√

βσ0

),

0 j N.

(9)

βσ0 3 the scheme is third order uniformly convergent.

Proof. If xj ∈ [σ, 1 − σ], using that on this interval we approach the first and the second derivative using central differences, easily we obtain √ CN −1 ε, if xj ∈ {σ, 1 − σ}, HO HO − βσ0 | CN . |τj,v | |τ j,w CN −2 ε, if xj ∈ (σ, 1 − σ), √ In the other case, using that now h = 4σ0 N −1 ε ln N , from (2) we can deduce (6)

(6)

HO HO |τj,v | C(ε + h2 )h4 |vj | CN −4 σ04 ε3 ln4 N, |τj,w | C(ε + h2 )h4 |wj | CN −4 σ04 ln4 N.

Therefore, the proof trivially follows from these bounds and the uniform stability of the scheme.

Some methods of order three for reaction-diffusion problems

5

3. The one dimensional parabolic problem Now we consider parabolic reaction-diffusion problems of type ut + Lx,ε u = f (x, t), (x, t) ∈ Ix × It ≡ (0, 1) × (0, T ], u(x, 0) = u0 (x), x ∈ I x , u(0, t) = 0, u(1, t) = 0, t ∈ It ,

(10)

where the differential operator Lx,ε is defined by Lx,ε u ≡ −εuxx + d(x)u. We assume that f and d are sufficiently smooth functions with d(x) β > 0, and moreover we suppose that there are sufficient compatibility conditions between data. It is known (see [8]) that the solution of (10) has √ parabolic layers in the neighbourhoods of x = 0 and x = 1, with a thickness of order O( ε). To solve (10), firstly we consider a time semidiscretization to find an intermediate approximation un (x) to the solution u(x, t) at the time levels tn = n∆t (for simplicity we take a constant time step ∆t). Concretely, we use the third order SDIRK method with two stages defined by  0 u = u(x, 0) = u0 (x),      (I + a11 ∆tLx,ε )g 1,n+1 = un + a11 ∆tf (x, tn + c1 ∆t),       2       (I + a22 ∆tLx,ε )g 2,n+1 = un + ∆t a2k f (x, tn + ck ∆t) − a21 ∆tLx,ε g 1,n+1 , (11)  k=1    2       n+1 n   u = u + ∆t bk (f (x, tn + ck ∆t) − Lx,ε g k,n+1 ),     k=1

whose Butcher’s table is

√ √ (3 + √3)/6 (3 +√ 3)/6 − 3/3 (3 − 3)/6 1/2

√ (3 + 3)/6 1/2

To analyze the convergence of this method, we introduce the auxiliary function uñ+1 , which is the solution obtained after one step of the semidiscrete scheme, taking the exact value u(x, tn ) instead of un as the starting data, i.e.,  g 1,n+1 = u(x, tn ) + a11 ∆tf (x, tn + c1 ∆t), (I + a11 ∆tLx,ε )˜    2    2,n+1  (I + a22 ∆tLx,ε )˜ g = u(x, tn ) + ∆t a2k f (x, tn + ck ∆t) − a21 ∆tLx,ε g˜1,n+1 , (12) k=1  2    n+1  u ˜ = u(x, t ) + ∆t bk (f (x, tn + ck ∆t) − Lx,ε g˜k,n+1 ).  n  k=1

Here we will assume that the local error associated to this discretization satisfies u(x, tn+1 ) − uñ+1 (x) C(∆t)4 .

(13)

A proof of this bound and also of the uniform stability of the SDIRK method can be found in [2]. Based on these two properties in [2] it is also proved that the global error satisfies u(x, tn ) − un (x) C(∆t)3 ,

(14)

and therefore the time semidiscretization process is uniformly convergent of order three.

6 C. Clavero and J.L. Gracia To find a totally discrete method we now discretize the 1D problems given in (12) by using the HODIE scheme of section 2 on the same piecewise uniform Shishkin mesh. Then, we have                                         

 2   1,n+1  (G ˜ ˜ 1,n+1+a11 ∆tLN,HO )= qjl+1 (u(xj−1+l , tn )+a11 ∆tf (xj−1+l , tn+c1 ∆t)), 0 0, r˜jk,− < 0, r˜jk,+ < 0,

k = 1, 2, 1 j < N.

Consequently the finite difference operators defined in (15) are both of positive type, satisfy a discrete maximum principle and they are uniformly stable in the maximum norm. Proof. Using (16), (17), Lemma 2.1 and the definition of the mesh, we deduce r˜jk,± < 0, k = 1, 2, 1 j < N. Finally, from (16) and Lemma 2.1 we obtain r˜jk,− + r˜jk,c + r˜jk,+ c > 0. In order to prove the convergence of the spatial discretization, we must analyze the local error associated to the HODIE scheme. Then, at each stage of (15) it holds 1,n+1 1,n+1 1 τj,˜ ˜j1,− g˜j−1 + r˜j1,c g˜j1,n+1 + r˜j1,+ g˜j+1 − g 1,n+1 = r

2

1,n+1 1,n+1 qjl (˜ gj−1+l +a11 ∆tLx,ε g˜j−1+l ) = a11 ∆tτj,˜g1,n+1 ,

l=0 2 τj,˜ g 2,n+1

+a21 ∆t

=

2,n+1 r˜j2,− g˜j−1

2 l=0

+

r˜j2,c g˜j2,n+1

+

2,n+1 r˜j2,+ g˜j+1

−

2

2,n+1 2,n+1 qjl (˜ gj−1+l + a22 ∆tLx,ε g˜j−1+l )+

l=0 1,N ˜ 1,n+1 1,n+1 qjl+1 (Lx,ε g˜j−1+l −Lx,ε G g 2,n+1 +(a21 /a11 ) j−1+l ) = a22 ∆tτj,˜

2 l=0

˜ 1,n+1 − g˜1,n+1 ), qjl+1 (G j−1+l j−1+l


7

where τj,˜gk,n+1 , k = 1, 2, are the local errors associated to the 1D stationary problem (see section 2). Lemma 3.2. If N N1 , then |˜ un+1 (xj ) − U˜jn+1 | C∆t(max {N −4 σ04 ln4 N, N −1 ε} + N −

√ βσ0

0 j N.

),

(18)

Proof. From the above expressions of the local error and Theorem 2.1, we deduce ˜ 1,n+1 | C∆t(max {N −4 σ04 ln4 N, N −1 ε} + N − |˜ gj1,n+1 − G j

√ βσ0

0 j N,

),

(19)

From (12), (15) and (19), it follows 1,N 1,n+1 ˜ | C(max {N −4 σ04 ln4 N, N −1 ε} + N − |Lx,ε g˜j1,n+1 − Lx,ε G j

√ βσ0

),

0 j N.

(20)

Using now (12), (15) and (20), we deduce ˜ 2,n+1 | C∆t(max {N −4 σ04 ln4 N, N −1 ε} + N − |˜ gj2,n+1 − G j

√ βσ0

0 j N.

),

(21)

From (12), (15), (20) and (21) it follows that 2,N 2,n+1 ˜ | C(max {N −4 σ04 ln4 N, N −1 ε} + N − |Lx,ε g˜j2,n+1 − Lx,ε G j

√ βσ0

),

0 j N.

(22)

Finally, from (12), (15), (20) and (22) we deduce the required result. Using both time and spatial discretization, the totally discrete scheme can be written in the form                                                   

Uj0 = u(xj , 0) = u0 (xj ), 0 j N, Forn = 0, 1, . . . , T /∆t :  2     n  (G1,n+1+a ∆tLN,HO G1,n+1 )=  qjl+1 (Uj−1+l +a11 ∆tf (xj−1+l , tn+c1 ∆t)), 0 0, i = 1, 2. Following [4], we decompose the source term in the form f2 (x1 , x2 , t) = f (x1 , 0, t) + x2 (f (x1 , 1, t) − f (x1 , 0, t)), f1 (x1 , x2 , t) = f (x1 , x2 , t) − f2 (x1 , x2 , t). Now we will suppose that the local error associated to this discretization satisfies u(x, tn+1 ) − uñ+1 (x) C(∆t)4 ,

(27)

where uñ+1 is the solution of the following auxiliary problem  s  n(i)  s,n+1  = u(x, tn ) + ∆t asi (fn(i) (x, tn + ci ∆t) − Ln(i),ε g˜i,n+1 ),   g˜ i=1

6  n(i)  n+1  = u(x, tn ) + ∆t bi (fn(i) (x, tn + ci ∆t) − Ln(i),ε g˜i,n+1 ).   u˜

1 s 6, (28)

i=1

A proof of bound (27), of the uniform stability and the third order of uniform convergence of the additive RK scheme can be found again in [2]. Using this time semidiscretization, we have reduced the 2D time dependent problem to a set of 1D stationary problems, where the operators Li,ε , i = 1, 2 are a family of 1D differential operators depending on the variables x2 ∈ (0, 1) and x1 ∈ (0, 1) respectively. To discretize (28), we use the HODIE scheme given in section 2 on the Shishkin mesh defined by the tensorial product of the corresponding 1D Shishkin meshes (4), i.e., the mesh N N N is defined by Ωε = I 1,ε × I 2,ε , (now xk,i , Hk , hk , σk , σk,0 , indicates the xk –direction). Then,

10 C. Clavero and J.L. Gracia the scheme can be written as                                                                                         

For  0 < j < N, For s = 1, 3, 5    2    s,n+1 s,n+1 1 N  ˜ ˜  + ∆tass L1,ε Gi,j ) = qi1,l (u(x1,i−1+l , x2,j , tn ) + f1 (x1,i−1+l , x2,j , tn + cs ∆t)+ (Gi,j     l=0   s−1  n(k) ˜ k,n+1 )), 0 < i < N, +∆t ask (fn(k) (x1,i−1+l , x2,j , tn + ck ∆t) − Ln(k),ε G i−1+l,j   k=1   s−1    n(k) s,n+1 s,n+1  ˜ ˜ L1,ε Gi,j = (u(x1,i , x2,j , tn ) − Gi,j + ∆t ask (fn(k) (x1,i , x2,j , tn + ck ∆t)−      k=1   ˜ k,n+1 ))/(∆ta1ss ) + f1 (x1,i , x2,j , tn + cs ∆t), 0 i N. −Ln(k),ε G i,j For 0 < i < N,  For s = 2, 4, 6    2     ˜ s,n+1 ) = ˜ s,n+1 + ∆ta2ss LN G  qj2,l (u(x1,i , x2,j−1+l , tn ) + f2 (x1,i , x2,j−1+l , tn + cs ∆t)+ ( G 2,ε i,j  i,j    l=0   s−1  n(k) ˜ k,n+1 )), 0 < j < N, ask (fn(k) (x1,i , x2,j−1+l , tn + ck ∆t) − Ln(k),ε G +∆t i,j−1+l   k=1   s−1    n(k) s,n+1 s,n+1  ˜ ˜ G L = (u(x , x , t ) − G + ∆t ask (fn(k) (x1,i , x2,j , tn + ck ∆t)−  2,ε i,j 1,i 2,j n i,j     k=1   ˜ k,n+1 ))/(∆ta2ss ) + f2 (x1,i , x2,j , tn + cs ∆t), 0 j N, −Ln(k),ε G i,j 6 n(k) n+1 ˜ k,n+1 ), 0 < i, j < N, ˜ Ui,j = u(x1,i , x2,j , tn ) + ∆t bk (fn(k) (x1,i , x2,j , tn + ck ∆t) − Ln(k),ε G i,j k=1

(29)

˜ s,n+1 |∂Ω = 0, 1 s 6 and q k,l , k = 1, 2, 1 l 3, 0 < i < N , are the coefficients where G i,j i in the xk –direction of 1D problem considered in section 2. Writing the reaction term in the form dk (x) + 1/(∆takk ), and denoting ˜ s,n+1 + ∆ta1ss LN ˜ s,n+1 ) ≡ r˜1,− G ˜ s,n+1 + r˜1,c G ˜ s,n+1 + r˜1,+ G ˜ s,n+1 , (G 1,ε Gi,j i,j i i−1,j i i,j i i+1,j s,n+1 2,− ˜ s,n+1 2,c ˜ s,n+1 2,+ ˜ s,n+1 2 N ˜ s,n+1 ˜ + ∆tass L2,ε G ) ≡ r˜ G + r˜ G + r˜ G , (G i,j

i,j

j

i,j−1

j

i,j

j

i,j+1

we deduce that r˜jk, = ∆takss rj , where rj , = c, +, −, are given in section 2 (see (5) and (6)). Lemma 4.1. Let N N2 be, where N2 > 0 is a positive integer independent of ε such that 2 (dk ∞ + 1/(a11 ∆t))/3 < N22 / ln2 N2 . (30) max 4σk,0 1k2

Then, there exists a constant c such that r˜jk,− + r˜jk,c + r˜jk,+ c > 0, r˜jk,− < 0, r˜jk,+ < 0,

k = 1, 2, 1 j < N.

Consequently the finite difference operators defined in (29) are of positive type, satisfy a discrete maximum principle and they are uniformly stable in the maximum norm. Proof. The proof is completely analogous to this one of Lemma 3.1. Similarly to the 1D parabolic problem, next we analyze the local error of the spatial discretization. In this case, it is straightforward to see that the local error satisfies s−1 s ˜ k,n+1 } − g˜k,n+1 |h ∞ , 1 s 6, ∆t|τi,j,˜gs,n+1 | + {G |τi,j,˜ g s,n+1 | CA,Q k=1

11


where τi,j,˜gs,n+1 are the local errors associated to the stationary problem (see section 2), and the constant CA,Q only depends of the coefficients of the matrices A1 and A2 and the coefficients qik,l of the HODIE method. Lemma 4.2. If N N2 , then for 0 i, j N it holds n+1 4 4 | C∆t(max {N −4 (σ1,0 + σ2,0 ) ln4 N, N −1 ε} + N − |˜ un+1 (x1,i , x2,j ) − U˜i,j

√ βσ1,0

+N−

√ βσ2,0

). (31)

Proof. Analogously to the 1D parabolic problem, the proof is deduced from the bounds k,n+1 4 4 ˜ k,n+1 | C∆t(max {N −4 (σ1,0 |˜ gi,j −G + σ2,0 ) ln4 N, N −1 ε} + N − i,j

√ βσ1,0

+ N−

√ βσ2,0

),

and N k,n+1 4 4 ˜ k,n+1 | C(max {N −4 (σ1,0 |Ln(k),ε g˜i,j −Ln(k),ε G +σ2,0 ) ln4 N, N −1 ε}+N − i,j

√ βσ1,0

+N −

√ βσ2,0

),

for 1 k 6 and 0 i, j N . Resuming, to approximate the solution of (25), the totally discrete scheme is given by

                                                                                                  

0 = u (x , x ), 0 i, j N, Ui,j 0 1,i 2,j For n = 0, 1, . . . , T /∆t :  For   0 < j < N,    For s = 1, 3, 5       2      s,n+1 s,n+1  n 1 N    (Gi,j + ∆tass L1,ε Gi,j ) = qi1,l (Ui−1+l,j + f1 (x1,i−1+l , x2,j , tn + cs ∆t)+         l=0     s−1     n(k)   ask (fn(k) (x1,i−1+l , x2,j , tn + ck ∆t) − Ln(k),ε Gk,n+1 +∆t  i−1+l,j )), 0 < i < N,      k=1     s−1      n(k) s,n+1 s,n+1   n  L1,ε Gi,j = (Ui,j − Gi,j + ∆t ask (fn(k) (x1,i , x2,j , tn + ck ∆t)−          k=1     k,n+1  1 ) + f (x , x , t + c ∆t),  G ))/(∆ta 0 i N. −L 1 1,i 2,j n s n(k),ε i,j  ss    For 0 < i < N,  For s = 2, 4, 6      2       s,n+1 n 2 LN Gs,n+1 ) =     + ∆ta qj2,l (Ui,j−1+l + f2 (x1,i , x2,j−1+l , tn + cs ∆t)+ (G ss 2,ε i,j  i,j        l=0     s−1    n(k)   ask (fn(k) (x1,i , x2,j−1+l , tn + ck ∆t) − Ln(k),ε Gk,n+1 +∆t  i,j−1+l )), 0 < j < N,      k=1     s−1       n(k) s,n+1 n − Gs,n+1 + ∆t    L G = (U ask (fn(k) (x1,i , x2,j , tn + ck ∆t)−  2,ε i,j  i,j i,j        k=1     k,n+1 2 ) + f (x , x , t + c ∆t),  −L G ))/(∆ta 0 j N,  2 1,i 2,j n s n(k),ε ss i,j    6   n(k)  U n+1 = U n + ∆t  bk (fn(k) (x1,i , x2,j , tn + ck ∆t) − Ln(k),ε Gk,n+1 ), 0 < i, j < N,  i,j i,j i,j k=1

(32)

where Gs,n+1 |∂Ω = 0, 1 s 6. i,j Theorem 4.1. Let N N2 be, u the solution of (25) and {Ujn } the solution of (32). Then, there exists a positive constant C independent of ε, N and ∆t such that 4 4 u(x, tn )|h −{U n }∞ C(max {N −4 (σ1,0 + σ2,0 ) ln4 N, N −1 ε}+N −

√ βσ1,0

+N −

√ βσ2,0

+(∆t)3 ).

12 C. Clavero and J.L. Gracia Proof. The proof is similar to the proof given for the case of 1D parabolic problem (see section 3). We only need to use the bounds of the expression (27), the Lemma 4.2 and that it holds T i ∞ C, i = 0, · · · , n − 1, where T is the appropriate discrete transition operator. Remark 4.1. We see again that if the constants σk,0 , k = 1, 2 are taken adequately, the totally discrete method is third uniformly convergent in both time and spatial variables.

5. Numerical examples In this section we show the results obtained for some examples with the previous schemes. In all examples we consider a modification of the HODIE method developed previously in order to obtain third order of convergence even when H 2 ||d||∞ < ε and xj ∈ [σ, 1 − σ]. In this case we consider the discretization  c rj = qj1 dj−1 + qj2 dj + qj3 dj+1 − rj− − rj+ ,    +   rj = −2ε/(hj+1 (hj + hj+1 )) + qj3 dj+1 , rj− = −2ε/(hj (hj + hj+1 )) + qj1 dj−1 ,    qj3 = (h3j + h3j+1 )/(6hj+1 (hj + hj+1 )2 ) + hj (hj+1 − hj )/(3hj+1 (hj + hj+1 )),   1 qj = (hj − hj+1 )/(3hj ) + qj3 hj+1 /hj , qj2 = 1 − qj1 − qj3 , which is the natural extension to nonuniform meshes of the coefficients obtained in section 2. Firstly, we consider the problem −εu (x) + u(x) = − cos2 (πx) − 2επ 2 cos (2πx), x ∈ (0, 1), (33) u(0) = 0, u(1) = 0. whose exact solution is

√ √ (exp (−(1 − x)/ ε) + exp (−x/ ε) √ u(x) = − cos2 (πx). 1 + exp (−1/ ε)

N Then, we compute exactly the maximum nodal errors, eN ε = max |u(xj )−Uj |, the ε-uniform 0jN

N

error, e

= max eN ε , ε N 2N

N 2N and the numerical orders of convergence pN ε = log (eε /eε )/ log 2,

pN = log (e /e )/ log 2. Table 1 shows the results obtained with the HODIE scheme of section 2 taking σ0 = 3 for some values of ε = 2−4 , 2−6 , · · · , 2−30 . From it clearly we see the order of convergence according Theorem 2.1. Moreover, when the mesh is uniform, we have fourth order of convergence, due to the special choice of the coefficients qji . Table 2 shows the dependence of the order of convergence with respect to the value of the constant σ0 . From it we see that if σ0 is too small, we do not achieve the required order; however, if σ0 is larger, the error increases but the order is preserved. The second example is ut − εuxx + (1 + x2 )u/2 = t3 , (x, t) ∈ (0, 1) × (0, 1], (34) u(0, t) = 0, u(1, t) = 0, t ∈ (0, 1], u(x, 0) = 0, x ∈ [0, 1], for which the exact solution is unknown. In figure 1 we can find an approximation of the solution for ε = 1.e−8 and t = 0.5 and t = 1.


Table 1. Table of errors and orders of convergence for HODIE scheme ε/N ε = 2−4 ε = 2−6 ε = 2−8 ε = 2−10 ε = 2−12 ε = 2−14

ε = 2−30 eN pN

16 4.074E-5 4.008 5.456E-5 3.988 1.001E-3 4.391 8.320E-3 3.488 9.317E-3 2.217 9.317E-3 2.217 ... ... ... 9.317E-3 2.217 9.317E-3 2.217

32 2.533E-6 4.002 3.430E-6 3.997 4.768E-5 3.990 7.414E-4 3.966 2.003E-3 2.894 2.003E-3 2.894 ... ... ... 2.003E-3 2.894 2.003E-3 2.894

64 1.581E-7 4.000 2.147E-7 3.999 3.001E-6 3.997 4.745E-5 3.990 2.695E-4 3.056 2.695E-4 3.056 ... ... ... 2.695E-4 3.056 2.696E-4 3.056

128 9.878E-9 4.000 1.343E-8 4.000 1.879E-7 3.999 2.987E-6 3.997 3.241E-5 3.440 3.241E-5 3.217 ... ... ... 3.241E-5 3.217 3.241E-5 3.217

256 6.173E-10 4.000 8.398E-10 4.000 1.175E-8 4.000 1.870E-7 3.999 2.986E-6 3.997 3.487E-6 3.317 ... ... ... 3.487E-6 3.317 3.487E-6 3.317

512 3.858E-11 4.000 5.249E-11 4.000 7.344E-10 4.000 1.169E-8 4.000 1.870E-7 3.999 3.499E-7 3.391 ... ... ... 3.499E-7 3.391 3.499E-7 3.391

1024 2.411E-12 3.281E-12 4.590E-11 7.309E-10 1.169E-8 3.335E-8 ... ... ... 3.335E-8 3.335E-8

Table 2. Dependence of parameter σ0 for HODIE scheme ε = 2−24 σ0 = 1 σ0 = 2 σ0 = 4 σ0 = 5

32 3.100E-2 1.011 9.680E-4 2.011 5.356E-3 2.632 1.046E-2 2.384

64 1.538E-2 1.022 2.402E-4 2.022 8.638E-4 3.107 2.003E-3 3.064

128 7.571E-3 1.045 5.913E-5 2.045 1.003E-4 3.189 2.395E-4 3.174

256 3.669E-3 1.090 1.433E-5 2.090 1.100E-5 3.315 2.654E-5 3.300

512 1.723E-3 1.180 3.365E-6 2.180 1.105E-6 3.390 2.695E-6 3.391

0.25 t=0.5 t=1

0.2

0.1

0 −0.1

0

1

1.1

1024 7.607E-4 7.425E-7 1.054E-7 2.569E-7

13

14 C. Clavero and J.L. Gracia For each value of ε, we estimate the maximum errors by N,∆t 2N,∆t/2 N,∆t eε = max max |U˜ (x2j , tn ) − U (xj , tn )| , tn

j

where U˜ 2N,∆t/2 (xj , tn ) is the numerical solution on a mesh which contains the mesh points xj of I¯εN and also the midpoints xj+1/2 = (xj + xj+1 )/2, j = 0, 1, . . . , N − 1. Table 3 shows the results obtained using the HODIE scheme and the SDIRK method, taking again σ0 = 3 From it we deduce the uniform convergence of third order with respect both time and spatial variables. Table 3. Maximum errors and numerical orders for problem (34) ε/N ∆t 2−4 2−6 2−8 2−10 2−12 2−14

2−30 eN,∆t pN,δt

N = 64 ∆t = 0.1 1.369E-4 2.708 1.393E-4 2.710 1.298E-4 2.640 5.481E-5 2.140 8.174E-4 3.221 8.143E-4 3.222 ... ... ... 8.112E-4 3.223 8.174E-4 3.221

N=128 ∆t = 0.1/2 2.095E-5 2.627 2.128E-5 2.633 2.082E-5 2.611 1.243E-5 2.127 8.767E-5 3.476 8.727E-5 3.180 ... ... ... 8.687E-5 3.181 8.767E-5 3.187

N=256 ∆t = 0.1/22 3.392E-6 2.704 3.430E-6 2.708 3.408E-6 2.701 2.846E-6 2.534 7.881E-6 4.343 9.626E-6 3.312 ... ... ... 9.578E-6 3.312 9.626E-6 3.312

N=512 ∆t = 0.1/23 5.205E-7 2.808 5.249E-7 2.812 5.243E-7 2.809 4.914E-7 2.751 3.882E-7 2.955 9.694E-7 3.535 ... ... ... 9.644E-7 3.537 9.694E-7 3.535

N=1024 ∆t = 0.1/24 7.433E-8 7.476E-8 7.480E-8 7.299E-8 5.006E-8 8.360E-8 ... ... ... 8.311E-8 8.360E-8

To analyze the influence of the compatibility conditions, we consider the problem ut − εuxx + (1 + x2 )u/2 = t1/2 sin(πx), (x, t) ∈ (0, 1) × (0, 1], u(0, t) = 0, u(1, t) = 0, t ∈ (0, 1], u(x, 0) = 0, x ∈ [0, 1].

(35)

Table 4 shows the results for the scheme (23) for the same values of σ0 and ε. Now, the order of convergence is smaller because we do not have sufficiently compatibility conditions between the data of the problem. Finally, we solve the 2D parabolic problem  (x, y, t) ∈ (0, 1)2 × (0, 1], ut − ε∆u + (5 + x2 y 2 + sin (πx) sin (πy))u = f,    u(0, y, t) = u(1, y, t) = 0, y ∈ [0, 1], t ∈ [0, 1], (36) u(x, 0, t) = u(x, 1, t) = 0, x ∈ [0, 1], t ∈ [0, 1],    u(x, y, 0) = 0, x, y ∈ [0, 1], where f is such that the exact solution is u = ((t + 1)e−t − 1)(h1 (x) − 1)(h2 (y) − 1)(x(1 − x) + y(1 − y)), and √ ε

h1 (x) = (e−x/

√

√

√

+ e−(1−x)/ ε )/(1 + e−1/ ε ), h2 (y) = (e−2y/

ε

√

√

+ e−2(1−y)/ ε )/(1 + e−2/ ε ).


15

Table 4. Maximum errors and numerical orders for problem (35) ε/N ∆t

N = 64 ∆t = 0.1 1.062E-4 1.204 1.249E-4 1.359 1.290E-4 1.389 1.300E-4 1.396 1.302E-4 1.397 1.302E-4 1.397 ... ... ... 1.302E-4 1.397 1.302E-4 1.397

2−4 2−6 2−8 2−10 2−12 2−14

2−30 eN,∆t pn,∆t

N=128 ∆t = 0.1/2 4.613E-5 1.399 4.870E-5 1.447 4.927E-5 1.457 4.941E-5 1.459 4.944E-5 1.460 4.945E-5 1.460 ... ... ... 4.945E-5 1.460 4.945E-5 1.460

N=256 ∆t = 0.1/22 1.750E-5 1.461 1.786E-5 1.477 1.795E-5 1.478 1.797E-5 1.479 1.797E-5 1.479 1.797E-5 1.479 ... ... ... 1.797E-5 1.479 1.797E-5 1.479

N=512 ∆t = 0.1/23 6.356E-6 1.479 6.417E-6 1.483 6.440E-6 1.485 6.446E-6 1.486 6.448E-6 1.486 6.448E-6 1.486 ... ... ... 6.448E-6 1.486 6.448E-6 1.486

N=1024 ∆t = 0.1/24 2.280E-6 2.296E-6 2.300E-6 2.301E-6 2.302E-6 2.302E-6 ... ... ... 2.302E-6 2.302E-6

In figures 2 and 3 we see the solution for ε = 1.e−8 and t = 1 and a zoom of corner (1,1) respectively.

−2.43e−17 0

−0.01 −0.02 −0.03 −0.04 −0.05 −0.06 −0.07 −0.08

−0.1

−0.09 −0.1 −0.11

−0.132 0 0

−0.12 −0.13

1

Z

X

1

Y

Figure 2. Solution of problem (36) for ε = 1.e−8 and t = 1.

16 C. Clavero and J.L. Gracia

−2.43e−17 −0.01 −0.02

0

−0.03 −0.04 −0.05 −0.06 −0.07 −0.08 −0.09 −0.1

−0.1 1 0.9 −0.132 0.7486

0.8

0.8 0.9

1

Z

1.059

−0.11 −0.12 −0.13

0.7039

Y X

Figure 3. Solution of problem (36) for ε = 1.e−8 and t = 1, near the corner (1, 1)

We decompose the reaction term in the form d1 = 1 + x2 y 2 /2 + sin (πx) sin (πy)/2,

d2 = 4 + x2 y 2 /2 + sin (πx) sin (πy)/2.

Table 5 shows the results for σ1,0 = 3, σ2,0 = 1.5 and ε = 10−4 , 10−5 , . . . , 10−8 , which are in agreement with the order of convergence proved in Theorem 4.1. Table 5. Maximum errors and numerical orders for problem (36) ε/N ∆t 10−4 10−5 10−6 10−7 10−8 eN,∆t pN,∆t

N=32 ∆t = 0.5 2.614E-3 2.772 2.604E-3 2.773 2.601E-3 2.773 2.600E-3 2.773 2.599E-3 2.773 2.614E-3 2.772

N=64 ∆t = 0.5/2 3.826E-4 2.729 3.810E-4 2.725 3.805E-4 2.724 3.803E-4 2.723 3.803E-4 2.723 3.826E-4 2.729

N=128 ∆t = 0.5/22 5.772E-5 2.853 5.764E-5 2.851 5.761E-5 2.850 5.760E-5 2.850 5.760E-5 2.850 5.772E-5 2.853

N=256 ∆t = 0.5/23 7.988E-6 7.988E-6 7.988E-6 7.988E-6 7.988E-6 7.988E-6


17

References [1] B. Bujanda and J.C. Jorge, Third order fractional step methods for multidimensional evolutionary convection-diffusion problems, Finite Difference Methods: Theory and Applications, Nova Science, (1999), pp. 110–117. [2] C. Clavero, J.L. Gracia and J.C. Jorge, High order time methods for 1D and 2D parabolic reactiondiffusion singularly perturbed problems, in preparation. [3] C. Clavero, J.L. Gracia and F. Lisbona, High order methods on Shishkin meshes for singular perturbation problems of convection-diffusion type, Numerical Algorithms 22 (1999), pp. 73–97. [4] C. Clavero, J.C. Jorge, F. Lisbona and G.I. Shishkin, A fractional step method on a special mesh for the resolution of multidimensional evolutionary convection–diffusion problem, Appl. Num. Math. 27 (1998), pp. 211–231. [5] C. Clavero, J.C. Jorge, F. Lisbona and G.I. Shishkin, An alternating direction scheme on a nonuniform mesh for reaction–diffusion parabolic problem, IMA J. Numer. Anal., 20 (2000), pp. 263–280. [6] Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I., Robust computational techniques for boundary layers, Chapman & Hall (2000) [7] J.L. Gracia, J.C. Jorge, B. Bujanda and C. Clavero, High order uniformly convergent fractional step RK methods and a HODIE finite difference scheme for 2D evolutionary convection–diffusion problems, to appear in Journal of Computational Methods in Sciences and Enginnering. [8] P.W. Hemker, G.I. Shishkin and L.P. Shishkina, ε-uniform schemes with high-order time-accuracy for parabolic singular perurbation problems, IMA J. Num. Anal. 20 (2000), pp. 99–121. [9] D. Herceg, Uniform fourth order difference scheme for a singular perturbation problem, Num. Math. 56 (1990), pp. 675–693. [10] N. Kopteva, On the uniform in small parameter convergence of a weighted scheme for the onedimensional time dependent convection-diffusion equation, Comp. Math. and Math. Phy. 37 (1997), pp. 1173–1180. [11] J.J.H. Miller, E. O’Riordan, G.I. Shishkin and L.P. Shishkina, Fitted mesh methods for problems with parabolic boundary layers, Math. Proc. of Royal Irish Acad. 98A (1998), pp. 173–190. [12] G.I. Shishkin, On finite difference fitted schemes for singularly perturbed boundary value problems with a parabolic boundary layer , Journal of Mathematical Analysis and Applications 208 (1997), pp. 181–204.