Error Analysis

Chapter 6 6.4. Error Analysis Theorem 6.4.3. Let uandU be respectively the continuous and the numerical solutions of the IBVPs (6.1.1) and (6.3.3) and satisfying sufficient compatibility condition at the corners. Then, we have the following bound

maxi,n |(u−U)(xi, tn)| ≤C[∆t+N⁻²] for all (xi, tn)∈G^N,∆t, (6.4.2) where U(x_i, t_n) =U_iⁿ.

Proof. We prove the theorem through the following steps. We first prove the result on the interval [0, τ], i.e., the time discretization parameter n varies from 0 to p. Let η_iⁿ = uⁿ_i − U_iⁿ be the truncation error in the computed solution at each mesh point (xi, tn). We write the scheme (6.3.1) as

δ_tU_iⁿ+L^Nε U_iⁿ=−bⁿ_iφ_b(x_i, t_n−p) +f_iⁿ i= 1, . . . , N −1, n = 1, . . . , p.

Therefore, the truncation error of the above scheme can be written in the following way as given in [49] and [17],

δtη_iⁿ+L^Nε η_iⁿ=χⁿ_1,i+χⁿ_2,i, for (xi, tn)∈G^N,∆t₁ , whereχⁿ_1,i and χⁿ_2,i as follows,

χⁿ_1,i:=L^Nε uⁿ_i −(L^εu)ⁿ_i and χⁿ_2,i :=δtuⁿ_i −(ut)ⁿ_i.

With this splitting of the truncation error we can decompose the errorηasη=φ+ψ. Here the functionφⁿ_i is, for each fixedn= 0, . . . , p, the solution of the discrete two-point boundary-value problem





L^Nε φⁿ_i =χⁿ_1,i for i= 1, . . . , N −1, φⁿ₀ =φⁿ_N = 0,

(6.4.3) and ψ_iⁿ, the solution of the discrete parabolic problem











δtψ_iⁿ+L^Nε ψ_iⁿ=χⁿ_2,i−δtφⁿ_i for i= 1, . . . , N −1, ψ₀ⁿ=ψ_Nⁿ = 0 forn = 1, . . . , p,

ψ_i⁰ =−φ⁰_i fori= 0, . . . , N.

(6.4.4)

Equation (6.4.3) is a sequence of two-point boundary-value problems that has been discretized usingL^Nε , withχⁿ_1,i playing the role of truncation error and can be bounded using the technique from [7]. The problem (6.1.1) exhibits regular boundary layers and the same is true for the equation (6.4.3), therefore, the error bound derived in [7] can be invoked for all temporal levels:

|φⁿ_i| ≤CN⁻², for all i, n ≤p, (6.4.5)

Chapter 6 6.4. Error Analysis with the assumption that N⁻¹ ≫ √

ε and the fact that our problem exhibits regular boundary layers.

All it remains is to bound the other error component ψ. By Lemma 6.4.1 and the discrete maximum principle (Lemma 6.4.2), we get the following bound for the error componentψ,

|ψ_iⁿ| = C

maxi |φ⁰_i|+ max

i,n |χⁿ_2,i−δtφⁿ_i|

for i, n≤p.

Using the bounds ofχⁿ_2,i and (6.4.5), we obtain

|ψ_iⁿ| = C

N⁻² + ∆t+ max

i,n |δtφⁿ_i|

fori, n ≤p. (6.4.6) It remains to bound the termδtφ in (6.4.6). Using the assumption thata(x) is indepen- dent oft, the definition (6.4.3) implies that δtφ satisfies

( L^Nε (δtφ)ⁿ_i =δtχⁿ_1,i, for i= 1, . . . , N −1,

(δtφ)ⁿ₀ = (δtφ)ⁿ_N = 0. (6.4.7)

To analyze the above sequence of two-point boundary-value problems (6.4.7), observe that the right-hand side of the above equation can be written as,

δtχⁿ_1,i = 1

∆t χⁿ_1,i−χⁿ_1,i⁻¹

= 1

∆t (L^Nε uⁿ_i −(L^εu)ⁿ_i)−(L^Nε uⁿ_i⁻¹−(L^εu)ⁿ_i⁻¹)

= 1

∆t (L^Nε uⁿ_i − L^Nε uⁿ_i⁻¹)−((L^εu)ⁿ_i −(L^εu)ⁿ_i⁻¹) .

Let ˆLεu = −εu_xx and ˆL^Nε uⁿ_i = −εδ_x²uⁿ_i. That is, ˆL^Nε u is the discretization of the continuous operator ˆLεu. Then one can write the above formula as

δ_tχⁿ_1,i= 1

∆t Z tn

tn−1

Lˆ^Nε

∂

∂tu(x_i, t)−Lˆε

∂

∂tu(x_i, t)

.

By using the Peano kernel theorem as in [38], and following the argument given in [17] we obtain the same estimate onδ_tχⁿ_1,i as the corresponding truncation error bounds arising in [7] for a standard reaction-diffusion two-point boundary-value problem. Now analyzing the problem the same way as (6.4.3) we obtain the following bound forδtφⁿ_i,

|δtφⁿ_i| ≤CN⁻² for all i, n ≤p. (6.4.8) Combining (6.4.5), (6.4.6) and (6.4.8), we get

maxi,n |(u−U)(xi, tn)| ≤ C[∆t+N⁻²], for all (xi, tn)∈G^N,∆t₁ , (6.4.9)

Chapter 6 6.4. Error Analysis whereU(xi, tn) =U_iⁿ.

For t ≥ τ, it is not possible to follow the above argument because the delay term, u(x, t−τ), is explicitly unknown fort ≥τ. For this reason, we examine the detailed proof of the estimate for the difference between the numerical solution U and the solution u itself over the interval [τ,2τ]. The proof follows the same approach of [2] in which fitted piecewise-uniform mesh is used for the analysis.

Consider the following singularly perturbed delay parabolic equation on the domain G2 = (0,1)×(τ,2τ],











 ∂

∂t +L^ε

u(x, t) =−b(x, t)u(x, t−τ) +f(x, t), (x, t) ∈G2

u(x, τ) =u(x, t(p)), x∈Ω,

u(0, t) = φ0(t), u(1, t) =φ1(t), t∈[τ,2τ].

(6.4.10)

We discretize (6.4.10) by means of the backward-Euler scheme for the time derivative, and the central difference for the space derivative. Hence the discretization takes the form,













δt+L^Nε

U(xi, tn)≡ δtU_iⁿ−εδ_x²U_iⁿ+aU =−bU_i^n−p+f(xi, tn), (xi, tn) ∈G^N,∆t₂ , U(xi, tn) =U1(xi, tn), (xi, tn)∈G^N,∆t₁ ,

U(0, tn) =φ0(tn), U(1, tn) =φ1(tn), tn∈Λ^p_2,t,

(6.4.11) where U1 is the numerical solution calculated on G^N,∆t₁ . The solution u of (6.4.10) is decomposed into the regular and the singular components u = y+z. The regular componentyis further decomposed intoy=y0+εy1, wherey0 andy1 solve the following problems:







∂y₀

∂t (x, t) +ay0(x, t) =−by0(x, t−τ) +f(x, t), (x, t)∈G2

y0(x, t) =u(x, t), (x, t)∈Ω×[0, τ],

and 











 ∂

∂t +L^ε

y1(x, t) =−by1(x, t−τ) +∂²y0

∂x² (x, t), (x, t)∈G2, y1(x, t) = 0, (x, t)∈Ω×[0, τ],

y1(0, t) =y1(1, t) = 0, t∈[τ,2τ].

Chapter 6 6.4. Error Analysis For the above definition ofy0 and y1, the regular component y satisfies,













 ∂

∂t +Lε

y(x, t) =−by(x, t−τ) +f(x, t), (x, t)∈G₂, y(x, t) =u(x, t), (x, t)∈Ω×[0, τ]

y(0, t) =y0(0, t), y(1, t) =y0(1, t), t∈[τ,2τ].

The singular component z satisfies,











 ∂

∂t +L^ε

z(x, t) =−bz(x, t−τ), (x, t)∈G2, z(x, t) = 0, (x, t)∈Ω×[0, τ]

z(0, t) =φl(t)−y0(0, t), z(1, t) =φr(t)−y0(1, t), t∈[τ,2τ].

The singular component z can be further decomposed into zl and zr, where zl and zr

are the corresponding to the left-hand and the right-hand layers, respectively:

z =z_l+z_r, wherezl and zr satisfies the following PDEs,











 ∂

∂t +L^ε

zl(x, t) =−bzl(x, t−τ), (x, t)∈G2, zl(x, t) = 0, (x, t)∈Ω×[0, τ],

zl(0, t) =φl(t)−y0(0, t), zl(1, t) = 0, t∈[τ,2τ].

and 











 ∂

∂t +L^ε

zr(x, t) =−bzr(x, t−τ), (x, t)∈G2, z_r(x, t) = 0, (x, t)∈Ω×[0, τ],

z_r(0, t) = 0, z_r(1, t) =φ_r(t)−y₀(1, t), t ∈[τ,2τ].

Similarly, the numerical solution U of (6.4.11) is decomposed into the regular and the singular components in an analogous manner to the decomposition of the solution uof (6.4.10). Thus

U =Y +Z, whereY is the solution of the following problem













δt+L^Nε

Y(xi, tn) =−bY(xi, tn−p) +f, (xi, tn)∈G^N,∆t₂ , Y(x_i, t_n) =U_τ(x_i, t_n), (x_i, t_n)∈G^N,∆t₁ ,

Y(0, tn) =y(0, tn), Y(1, tn) =y(1, tn) tn ∈Λ^p_2,t.

Chapter 6 6.4. Error Analysis From the above equation the singular component approximation Z satisfies,













δt+L^Nε

Z(xi, tn) = −bZ(xi, tn−p), (xi, tn)∈G^N,∆t₂ , Z(xi, tn) = 0, (xi, tn)∈G^N,∆t₁ ,

Z(0, tn) =φl(tn)−y(0, tn), Z(1, tn) =φr(tn)−y(1, tn) tn∈Λ^p_2,t. Therefore, the error at the node (xi, tn) can be written in the following way,

(U−u) (x_i, t_n) = (Y −y) (x_i, t_n) + (Z−z) (x_i, t_n), thus

|(U −u) (xi, tn)| ≤ |(Y −y) (xi, tn)|+|(Z −z) (xi, tn)|,

from the above inequality it is enough to bound the regular and the singular component error with an optimal bound. The truncation error of the regular component can be written as

δ_t+L^Nε

(Y −y) =−bY(x_i, t_n−p) +f − δ_t+L^Nε

y

=b(y(xi, tn−p)−Y(xi, tn−p)) + ∂

∂t +L^ε

− δt+L^Nε

y

=b(u(xi, tn−p)−U1(xi, tn−p)) + ∂

∂t +L^ε

− δt+L^Nε

y, therefore, we have

δt+L^Nε

(Y −y) = −b(u(xi, tn−p)−U1(xi, tn−p))−ε ∂²

∂x² −δ_x²

y+ ∂

∂t −δt

y.

Now taking the modulus and using (6.4.9) for the first part it reduces to,

| δt+L^Nε

(Y −y)(xi, tn)| ≤C(N⁻²+ ∆t) +ε

∂²

∂x² −δ_x²

y +

∂

∂t −δt

y

. Using Taylor series expansions it is easy to show that,

| δt+L^Nε

(Y −y)(xi, tn)| ≤C

N⁻²+ ∆t+ (hi+1+hi)²ε

∂⁴y

∂x⁴

∞

+ ∆t

∂²y

∂t²

∞

. Applying Lemma 6.3.1, and the estimates of the derivatives given in (6.2.4), we obtain

| δ_t+L^Nε

(Y −y)(x_i, t_n)| ≤C(N⁻²+ ∆t), for (x_i, t_n)∈G^N,∆t₂ . Now using the fact that the discrete operator δt+L^Nε

satisfies the discrete maxi- mum principle (Lemma 6.4.2) and the inverse operator is uniformly bounded, the above inequality can be reduced to,

|(Y −y)(xi, tn)| ≤C(N⁻²+ ∆t), for (xi, tn)∈G^N,∆t₂ . (6.4.12)

Chapter 6 6.4. Error Analysis To estimate error of the the singular component, we decompose Z as the way its continuous counterpart z is decomposed,

Z =Zl+Zr,

where Zl and Zr are the left and the right part layers of the approximate solutions respectively, that are defined as













δt+L^Nε

Zl =−bZl(xi, tn−p), (xi, tn)∈G^N,∆t₂ , Z_l(x_i, t_n) = 0, (x_i, t_n)∈G^N,∆t₁ ,

Zl(0, tn) =φl(tn)−y(0, tn), Zl(1, tn) = 0 tn∈Ω^N₂^τ,

and 













δt+L^Nε

Zr =−bZr(xi, tn−p), (xi, tn)∈G^N,∆t₂ , Zr(xi, tn) = 0, (xi, tn)∈G^N,∆t₁ ,

Zr(0, tn) = 0, Zr(1, tn) =φr(tn)−y(1, tn), tn∈Ω^N₂ . The error can then be written in the form

(Z−z) (x_i, t_n) = (Z_l−z_l) (x_i, t_n) + (Z_r−z_r) (x_i, t_n), (x_i, t_n)∈G^N,∆t₂ ,

and the errorsZl−zl andZr−zr, associated with boundary layers of Γl and Γr respec- tively, can be estimated separately. Consider the error Zl−zl,

δt+L^Nε

(Zl−zl) = ∂

∂t +L^ε

− δt+L^Nε

zl

=−ε ∂²

∂x² −δ_x²

z_l+ ∂

∂t −δ_t

z_l.

Taking the modulus and using the Taylor series expansions on time, we obtain

| δt+L^Nε

(Zl−zl)| ≤ C

N⁻²+ ∆t+ε

∂²

∂x² −δ_x²

zl

∞

+ ∆t

∂²zl

∂t²

∞

,

≤ C

N⁻²+ ∆t+ε

∂²

∂x² −δ_x²

zl

∞

.

By fixingt, the lateral part of the above inequality can be seen as the truncation error of the reaction-diffusion two-point boundary-value problem as in [7] corresponding to the left-hand layer part. By this observation the truncation error in space can be analyzed the same way as [7, Lemmas 8, 9], the only difference is that there it is given for both sides layers but here, we need only for the left layer part, hence we obtain

| δt+L^Nε

(Zl−zl)(xi, tn)| ≤C(N⁻²+ ∆t), (xi, tn)∈G^N,∆t₂ . (6.4.13)

Chapter 6 6.5. Numerical Results