Convergence analysis

5.3 Fundamental studies

5.3.1 Convergence analysis

Consider equation (5.1) with the convective coefficients p. q and r and the reactive coefficient eas constants. To perform the convergence analysis of the schemes assume a= 1 subject to the initial condition

φ(x, y, z,0) =φ₀(x, y, z), (x, y, z)∈Ω∪∂Ω (5.18) and with the boundary condition

φ(x, y, z, t) = constant, (x, y, z)∈∂Ω, t >0, (5.19) where φ₀(x, y, z) is a known smooth function and ∂Ω is the boundary of the finite domain Ω.

Thence as discussed in the previous section, our finite difference scheme for this problem is the one given by (5.8) but with constant coefficients.

Lemma 5.3.1. There exists a lower bound C to all the coefficients appearing in equation (5.8) at all grid points, where C is a real number.

Equivalently, there exists a real constant C such that C .V, where V may be any of the coefficients appearing in equation (5.8).

5.3. Fundamental studies 103

Proof. Since all of the coefficients A, B, C, P, Q, R, D, E, F, H, K, L, M, N, O,S, T,U, G and W, appearing in equation (5.8) are real numbers, they have finite values, so the proof of the above lemma follows.

Next, let us introduce some useful notations.

For the sake of our discussions in the current section, let us assume that h =

△x =△y = △z and τ = △t.

Denote the grid points as (xi, yj, zk), where xi = xa + ih, yj = ya + jh, zk

=z_a +kh and [x_a, x_b] × [y_a,y_b] ×[z_a, z_b] is the domain under consideration with the index sets given as

I_x = {i|i = 0,1,2, . . . , I}, I_x^o = {i|i = 1,2, . . . , I −1}, where I = (x_b − x_a)/h+ 1,

Iy ={j|j = 0,1,2, . . . , J},

I_y^o ={j|j = 1,2, . . . , J−1}, where J = (y_b−y_a)/h+ 1, I_z ={k|k= 0,1,2, . . . , K},

I_z^o ={k|k= 1,2, . . . , K−1}, where K = (z_b−z_a)/h+ 1.

For grid functions of the formφ ={φⁿijk|(i, j, k)∈Ix×Iy×Iz, n = 0,1,2, . . . , N} (N is the total number of time steps), we have the following finite difference operators:

δ_xφⁿ_ijk = _2h¹ (φⁿ_i+1jk−φⁿ_i−1jk), i6=I;

δx²φⁿijk = ¹

h²(φⁿ_i+1jk−2φⁿijk+φⁿi−1jk), i∈Ix^o; δyφⁿijk = _2h¹ (φⁿ_ij+1k−φⁿij−1k), j 6=J;

δ_y²φⁿ_ijk = ¹

h²(φⁿ_ij+1k−2φⁿ_ijk+φⁿ_ij−1k), j ∈I_y^o; δ_zφⁿ_ijk = _2h¹(φⁿ_ijk+1−φⁿ_ijk−1), k6=K;

δz²φⁿijk = ¹

h²(φⁿ_ijk+1−2φⁿijk+φⁿijk−1), k∈Iz^o;

∇hφⁿ_ijk = (δ_x, δ_y, δ_z)φⁿ_ijk, ∇h²φⁿ_ijk = (δ_x², δ²_y, δ_z²)φⁿ_ijk, i6=I, j 6=J, k 6=K;

∇²_h²φⁿ_ijk = (δ_x²δ²_y, δ_y²δ²_z, δ_z²δ²_x)φⁿ_ijk, i6=I, j 6=J, k 6=K;

δ_t⁺φⁿ_ijk = ¹_τ(φⁿ⁺¹_ijk −φⁿ_ijk), n6=N; δ_tφⁿ_ijk = _2τ¹ (φⁿ⁺¹_ijk −φⁿ⁻¹_ijk ), n6= 0, N;

δt²φⁿijk = ¹

τ²(φⁿ⁺¹_ijk −2φⁿijk+φⁿ⁻¹_ijk ), n6= 0, N;

Ahφⁿ_ijk ={1 +^h₁₂²(δ_x²−pδ_x+δ_y²−qδ_y+δ²_z−rδ_z)}φⁿ_ijk; i∈ I_x^o, j ∈I_y^o, k∈I_z^o.

104 HOC schemes for the transient 3D convection-diffusion-reaction equations

Further, let Φⁿ_ijk be the numerical approximation ofφ(x_i, y_j, z_k, t_n), then utiliz- ing the average parameterς in equation (5.8), coefficients of the formAδx²φⁿijk

can be replaced by ςAδx²φⁿ⁺¹_ijk + (1−ς)Aδx²φⁿijk. Likewise, we replace for the other terms as well.

With these replacements, the finite difference scheme can now be written as Ahδ_t⁺Φⁿ_ijk+ (−ςA)δ²_xΦⁿ⁺¹_ijk − {(1−ς)A}δ_x²Φⁿ_ijk+ (−ςB)δ_y²Φⁿ⁺¹_ijk

− {(1−ς)B}δy²Φⁿijk+ (−ςC)δz²Φⁿ⁺¹_ijk − {(1−ς)C}δ²zΦⁿijk+ (ςP)δxΦⁿ⁺¹_ijk

− {(ς −1)P}δ_xΦⁿ_ijk+ (ςQ)δ_yΦⁿ⁺¹_ijk − {(ς −1)Q}δ_yΦⁿ_ijk+ (ςR)δ_zΦⁿ⁺¹_ijk

− {(ς −1)R}δ_zΦⁿ_ijk+ (−ςD)δ_xδ_yΦⁿ⁺¹_ijk − {(1−ς)D}δ_xδ_yΦⁿ_ijk + (−ςE)δyδzΦⁿ⁺¹_ijk − {(1−ς)E}δyδzΦⁿijk+ (−ςF)δzδxΦⁿ⁺¹_ijk

− {(1−ς)F}δ_zδ_xΦⁿ_ijk+ (ςH)δ_xδ_y²Φⁿ⁺¹_ijk − {(ς −1)H}δ_xδ_y²Φⁿ_ijk + (ςK)δx²δyΦⁿ⁺¹_ijk − {(ς−1)K}δx²δyΦⁿijk+ (−ςL)δx²δy²Φⁿ⁺¹_ijk

− {(1−ς)L}δ²_xδ_y²Φⁿ_ijk+ (ςM)δ_yδ²_zΦⁿ⁺¹_ijk − {(ς−1)M}δ_yδ²_zΦⁿ_ijk + (ςN)δ_y²δ_zΦⁿ⁺¹_ijk − {(ς −1)N}δ_y²δ_zΦⁿ_ijk+ (−ςO)δ_y²δ_z²Φⁿ⁺¹_ijk

− {(1−ς)O}δy²δz²Φⁿijk+ (ςS)δx²δzΦⁿ⁺¹_ijk − {(ς −1)S}δx²δzΦⁿijk

+ (ςT)δ_xδ²_zΦⁿ⁺¹_ijk − {(ς −1)T}δ_xδ²_zΦⁿ_ijk+ (−ςU)δ_z²δ_x²Φⁿ⁺¹_ijk

− {(1−ς)U}δz²δx²Φⁿijk+ (ςG)Φⁿ⁺¹_ijk − {(ς−1)G}Φⁿijk =W

(5.20)

Define the initial approximations as

Φ⁰ijk=φ₀(xi, yj, zk), (5.21) where i∈I_x, j∈I_y, k∈I_z and φ₀(x_i, y_j, z_k) =φ(x_i, y_j, z_k,0)

Also consider that,

Φ¹_ijk =φ₀(x_i, y_j, z_k) +τ φ₁(x_i, y_j, z_k), (i, j, k)∈I_x×I_y×I_z, (5.22) where φⁿ_ijk =φ(x_i, y_j, z_k, t_n) and φ₁(x_i, y_j, z_k) = φ_t(x_i, y_j, z_k,0).

5.3. Fundamental studies 105

Remark 5.3.2. By virtue of lemma 5.3.1, it follows that there exists a real constantKsuch thatK.|ς|ξⁿ⁺¹ andK.|ς−1|ξⁿ, whereξijk can be either of the coefficients A, B, C, P, Q, R, D, E, F, H, K, L, M, N, O, S, T, U orG.

Now, let us consider the space, S := {φ = φ_ijk|(i, j, k) ∈ I_x ×I_y × I_z} ⊆ R^{(I+1)×(J+1)×(K+1)} and the inner product and discrete norms defined overS as hφ1, φ2i=h³

I−1P

i=1 J−1P

j=1 K−1P

k=1

(φ1)ijk(φ2)ijk, where φ1, φ2∈ S,

kφk=hφ, φi¹², kδ_xφk= 8h³

XI−1

i=0

XJ−1

j=1 K−1X

k=1

|δ_xφ_ijk|²

!¹₂ ,

kδ_zφk= 8h³ XI−1

i=1

XJ−1

j=1 K−1X

k=0

|δ_zφ_ijk|²

!¹₂ ,

kδ_x²φk= h⁶

I−1

X

i=0

XJ−1

j=1 K−1X

k=1

|δ_x²φ_ijk|²

!¹₂ ,

kδz²φk= h⁶

I−1

X

i=1 J−1

X

j=1 K−X1

k=0

|δz²φijk|²

!¹₂ ,

kδx²δy²φk= h¹² XI−1

i=0

XJ−1

j=1 K−1X

k=1

|δx²δy²φijk|²

!¹₂ ,

kδz²δx²φk= h¹² XI−1

i=1

XJ−1

j=1 K−1X

k=0

|δz²δ²xφijk|²

!¹₂ ,

kφk∞= max

(i,j,k)∈I_x×I_y×I_z|φ_ijk|, kδyφk= 8h³

I−1

X

i=1 J−1

X

j=0 K−X1

k=1

|δyφijk|²

!¹₂ , k∇hφk= kδ_xφk²+kδ_yφk² +kδ_zφk²¹₂

, kδ_y²φk= h⁶

XI−1

i=1

XJ−1

j=0 K−1X

k=1

|δ²_yφ_ijk|²

!¹₂ , k∇h²φk=

kδx²φk²+kδy²φk²+kδ²zφk²¹₂ , kδ²yδz²φk= h¹²

I−1

X

i=1 J−1

X

j=0 K−X1

k=1

|δy²δz²φijk|²

!¹₂ , k∇²_h²φk=

kδ_x²δ_y²φk²+kδ²_yδ_z²φk²+kδ²_zδ_x²φk²¹₂ .

Lemma 5.3.3. If φⁿ⁻¹, φⁿ, φⁿ⁺¹ ∈ S, then we have 1

τ(hA^hφⁿ⁺¹, φⁿ⁺¹i − hA^hφⁿ, φⁿi)≤ hA^hδt⁺φⁿ, δt⁺φⁿi (5.23)

106 HOC schemes for the transient 3D convection-diffusion-reaction equations

Proof. Since,

hAhδ_t⁺φⁿ, δ_t⁺φⁿi=hAh

φⁿ⁺¹−φⁿ

τ , φⁿ⁺¹−φⁿ

τ i

= 1

τ²hAh(φⁿ⁺¹−φⁿ), (φⁿ⁺¹−φⁿ)i[∵Ah is linear]

= 1

τ²{hAhφⁿ⁺¹, φⁿ⁺¹i − hAhφⁿ⁺¹, φⁿi

− hA^hφⁿ, φⁿ⁺¹i+hA^hφⁿ, φⁿi}

Similarly, we have

hAhδ_t⁺φⁿ⁻¹, δ_t⁺φⁿ⁻¹i = 1

τ²{hAhφⁿ, φⁿi − hAhφⁿ, φⁿ⁻¹i

− hA^hφⁿ⁻¹, φⁿi+hA^hφⁿ⁻¹, φⁿ⁻¹i}

Therefore, using the definition ofAh, property of symmetry ofhφⁿ, φⁿ⁺¹iand considering the unit vectors as φ

kφk², the proof of the lemma follows.

In order to analyse convergence of the difference scheme, let us define the local truncation error rⁿ ∈ S as follows

rⁿijk = A^hδ⁺t φⁿijk+ (−ςA)δx²φⁿ⁺¹_ijk − {(1−ς)A}δx²φⁿijk+ (−ςB)δy²φⁿ⁺¹_ijk

− {(1−ς)B}δ²_yφⁿ_ijk+ (−ςC)δ_z²φⁿ⁺¹_ijk − {(1−ς)C}δ²_zφⁿ_ijk+ (ςP)δ_xφⁿ⁺¹_ijk (5.24)

− {(ς−1)P}δ_xφⁿ_ijk+ (ςQ)δ_yφⁿ⁺¹_ijk − {(ς −1)Q}δ_yφⁿ_ijk+ (ςR)δ_zφⁿ⁺¹_ijk

− {(ς−1)R}δzφⁿijk+ (−ςD)δxδyφⁿ⁺¹_ijk − {(1−ς)D}δxδyφⁿijk+ (−ςE)δyδzφⁿ⁺¹_ijk

− {(1−ς)E}δ_yδ_zφⁿ_ijk+ (−ςF)δ_zδ_xφⁿ⁺¹_ijk − {(1−ς)F}δ_zδ_xφⁿ_ijk+ (ςH)δ_xδ²_yφⁿ⁺¹_ijk

− {(ς−1)H}δxδy²φⁿijk+ (ςK)δx²δyφⁿ⁺¹_ijk − {(ς −1)K}δx²δyφⁿijk+ (−ςL)δx²δy²φⁿ⁺¹_ijk

− {(1−ς)L}δ_x²δ_y²φⁿ_ijk+ (ςM)δ_yδ_z²φⁿ⁺¹_ijk − {(ς−1)M}δ_yδ²_zφⁿ_ijk+ (ςN)δ_y²δ_zφⁿ⁺¹_ijk

− {(ς−1)N}δ_y²δ_zφⁿ_ijk+ (−ςO)δ_y²δ²_zφⁿ⁺¹_ijk − {(1−ς)O}δ_y²δ²_zφⁿ_ijk+ (ςS)δ_x²δ_zφⁿ⁺¹_ijk

− {(ς−1)S}δx²δzφⁿijk+ (ςT)δxδ²zφⁿ⁺¹_ijk − {(ς −1)T}δxδ²zφⁿijk+ (−ςU)δz²δ²xφⁿ⁺¹_ijk

− {(1−ς)U}δ_z²δ²_xφⁿ_ijk+ (ςG)φⁿ⁺¹_ijk − {(ς−1)G}φⁿ_ijk+W, where (i, j, k)∈I_x^o×I_y^o×I_z^o, and 1≤n≤N −1.

5.3. Fundamental studies 107

Using Taylor Series expansion, we have

r_ijk^o = φ¹_ijk−φ₀(x_i, y_j, z_k)

τ −φ₁(x_i, y_j, z_k), ∀(i, j, k)∈I_x^o×I_y^o×I_z^o (5.25) Also we have the following

|δx⁺rijk⁰ |.τ², |δy⁺rijk⁰ |.τ², |δz⁺r⁰ijk|.τ², |rⁿijk|.τ², (5.26) where (i, j, k)∈I_x^o×I_y^o×I_z^o and 0≤n≤N.

Let the error function eⁿ ∈ S be defined as

eⁿ_ijk =φⁿ_ijk−Φⁿ_ijk, (i, j, k)∈I_x×I_y ×I_z, 0≤n ≤N. (5.27)

Lemma 5.3.4. For any grid function φ∈ S, we have

|hA^hφ, φi| ≤ kφk² (5.28) Proof. From the defintion of Ahφ and use of diiferential operators, it follows that

|hA^hφ, φi| ≤ |hφ, φi|

Now, the proof of the above lemma follows from the definition of inner product and Cauchy-Schwarz inequality.

Theorem 5.3.5. If φ ∈ C^6,6,6,3([xa, xb]×[ya, yb]×[za, zb]×[0, T]), and if h and τ are sufficiently small, then the solution of the finite difference problem (5.20)-(5.21)-(5.22) converges to the solution of the initial continuous problem (5.1)-(5.18)-(5.19) with order O(τ²+h⁴) in the discreteL² norm for Φⁿ, i.e.

keⁿk+k∇heⁿk+k∇h²eⁿk+k∇²_h²eⁿk.τ²+h⁴, 0≤n≤N. (5.29) Proof. Let us adopt an induction argument to proof that (5.29) holds for every

108 HOC schemes for the transient 3D convection-diffusion-reaction equations

non-negative integer less thanN. From equation (5.20) and remark (5.3.2), it is obvious that (5.29) holds for n= 0. For (i, j, k)∈Ix^o×Iy^o×Iz^o, we have

|e¹ijk|=|φ¹ijk−Φ¹ijk|=|τ rijk⁰ |.τ³, |δt⁺e⁰ijk|=|r⁰ijk|.τ², and (5.30a)

|δ_x⁺e¹_ijk|.τ|δ_x⁺r⁰_ijk|. τ³ .τ², (5.30b) Likewise, we have

|δy⁺e¹ijk|.τ² and |δ⁺ze¹ijk|.τ² (5.30c) Equation (5.30) clearly implies that (5.29) holds forn = 1. Using the triangle inequality, when τ and h are sufficiently small, we have

|Φ¹ijk| ≤ |φ¹ijk|+|e¹ijk| ≤Co+C(τ²+h²)≤1 +Co, (i, j, k)∈Ix×Iy×Iz (5.31) where C_o = max

0≤t≤T kφ(·,·,·, t)kL^∞(Ω).

We assume that (5.29) is valid for 0≤n ≤m−1< N−1, then we will show that it is still valid for n = m. Now, adding (5.20) and (5.24), we have the following error equation for the error functioneⁿ∈ S

A^hδt⁺eⁿijk+ (−ςA)δx²eⁿ⁺¹_ijk − {(1−ς)A}δx²eⁿijk+ (−ςB)δ²yeⁿ⁺¹_ijk

− {(1−ς)B}δ_y²eⁿ_ijk+ (−ςC)δ_z²eⁿ⁺¹_ijk − {(1−ς)C}δ_z²eⁿ_ijk+ (ςP)δ_xeⁿ⁺¹_ijk

− {(ς−1)P}δxeⁿijk+ (ςQ)δyeⁿ⁺¹_ijk − {(ς −1)Q}δyeⁿijk+ (ςR)δzeⁿ⁺¹_ijk

− {(ς−1)R}δ_zeⁿ_ijk+ (−ςD)δ_xδ_yeⁿ⁺¹_ijk − {(1−ς)D}δ_xδ_yeⁿ_ijk+ (−ςE)δ_yδ_zeⁿ⁺¹_ijk

− {(1−ς)E}δ_yδ_zeⁿ_ijk+ (−ςF)δ_zδ_xeⁿ⁺¹_ijk − {(1−ς)F}δ_zδ_xeⁿ_ijk+ (ςH)δ_xδ_y²eⁿ⁺¹_ijk

− {(ς−1)H}δxδy²eⁿijk+ (ςK)δx²δyeⁿ⁺¹_ijk − {(ς −1)K}δx²δyeⁿijk+ (−ςL)δ²xδ²yeⁿ⁺¹_ijk

− {(1−ς)Lⁿ_ijk}δ_x²δ_y²eⁿ_ijk+ (ςM)δ_yδ_z²eⁿ⁺¹_ijk − {(ς −1)M}δ_yδ_z²eⁿ_ijk+ (ςN)δ²_yδ_zeⁿ⁺¹_ijk

− {(ς−1)N}δ²yδzeⁿijk+ (−ςO)δy²δz²eⁿ⁺¹_ijk − {(1−ς)O}δy²δz²eⁿijk+ (ςS)δx²δzeⁿ⁺¹_ijk

− {(ς−1)S}δ²_xδ_zeⁿ_ijk+ (ςT)δ_xδ²_zeⁿ⁺¹_ijk − {(ς −1)T}δ_xδ_z²eⁿ_ijk+ (−ςU)δ²_zδ_x²eⁿ⁺¹_ijk

− {(1−ς)U}δ_z²δ_x²eⁿ_ijk+ (ςG)eⁿ⁺¹_ijk − {(ς −1)G}eⁿ_ijk=rⁿ_ijk, (i, j, k)∈I_x^o×I_y^o×I_z^o

Assuming τ and h to be sufficiently small and considering (5.29) for 1≤n ≤

5.3. Fundamental studies 109

m−1, we have the following estimate

kΦⁿk^∞ ≤ kφⁿk^∞+keⁿk^∞≤Co+ 1, 1≤n≤m−1. (5.32) Let us consider the case when ς = 0.5. Throughout this section, we will con- sider the case only where ς = 0.5.

Now, computing the inner product of (5.32) withδ_t⁺eⁿand considering remark (5.3.2) and lemma (5.3.3), we have

1

τ(hA^heⁿ⁺¹, eⁿ⁺¹i − 1

τhA^heⁿ, eⁿi+Kh(δx²+δy²+δ²z+δx+δy+δz+δxδy +δyδz

+δ_zδ_x+δ_xδ_y²+δ_x²δ_y+δ²_xδ_y²+δ_yδ²_z +δ_y²δ_z +δ_y²δ²_z+δ_zδ²_x+δ_z²δ_x+δ_z²δ_x²+ 1)eⁿ⁺¹, δ_t⁺eⁿi ≤ hrⁿ, δ⁺_t eⁿi+Kh(δ²_x+δ_y²+δ_z²+δ_x+δ_y+δ_z +δ_xδ_y+δ_yδ_z+δ_zδ_x (5.33) +δxδ²y+δx²δy+δx²δy²+δyδz²+δ²yδz+δ²yδz²+δzδx²+δ²zδx+δ²zδx²+ 1)eⁿ, δ⁺t eⁿi Upon simplifying and transposing, we have

(hAheⁿ⁺¹, eⁿ⁺¹i+Kτ²(kδ²_xeⁿ⁺¹k² +kδ_y²eⁿ⁺¹k²+kδ²_zeⁿ⁺¹k²+kδ_xeⁿ⁺¹k² +kδ_yeⁿ⁺¹k²+kδ_zeⁿ⁺¹k²+kδ_xδ_yeⁿ⁺¹k²+kδ_yδ_zeⁿ⁺¹k²+kδ_zδ_xeⁿ⁺¹k² +kδxδy²eⁿ⁺¹k²+kδx²δyeⁿ⁺¹k²+kδ²xδy²eⁿ⁺¹k²+kδyδ²zeⁿ⁺¹k²+kδ²yδzeⁿ⁺¹k² +kδ_y²δ_z²eⁿ⁺¹k²+kδ_zδ_x²eⁿ⁺¹k²+kδ_z²δ_xeⁿ⁺¹k²+kδ_z²δ_x²eⁿ⁺¹k²+keⁿ⁺¹k²)

≤ τ²(krⁿk²− keⁿ⁺¹k²+keⁿk²) +Kτ²(kδ_x²eⁿk²+kδ_y²eⁿk²

+kδz²eⁿk²+kδxeⁿk²+kδyeⁿk²+kδzeⁿk² +kδxδyeⁿk²+kδyδzeⁿk²+kδzδxeⁿk² +kδ_xδ_y²eⁿk²+kδ_x²δ_yeⁿk²+kδ_x²δ_y²eⁿk²+kδ_yδ²_zeⁿk²+kδ²_yδ_zeⁿk²+kδ_y²δ_z²eⁿk² +kδzδx²eⁿk²+kδz²δxeⁿk² +kδz²δx²eⁿk²+keⁿk²) + (hA^heⁿ, eⁿi (5.34) Define Gⁿ as follows:

Gⁿ = (hA^heⁿ, eⁿi+τ²keⁿk²+Kτ²(kδx²eⁿk² +kδy²eⁿk²+kδ²zeⁿk²+kδxeⁿk² + kδ_yeⁿk²+kδ_zeⁿk²+kδ_xδ_yeⁿk²+kδ_yδ_zeⁿk²+kδ_zδ_xeⁿk²+kδ_xδ_y²eⁿk² + kδ²xδyeⁿk²+kδ²xδy²eⁿk²+kδyδz²eⁿk²+kδy²δzeⁿk²+kδy²δz²eⁿk²+kδzδx²eⁿk² + kδ²_zδ_xeⁿk²+kδ_z²δ²_xeⁿk²+keⁿk²), 0≤n≤m−1 (5.35)

110 HOC schemes for the transient 3D convection-diffusion-reaction equations

Then equation (5.34) can be written as

Gⁿ⁺¹−Gⁿ.τ²(krⁿk²) +Kτ²(Gⁿ⁺¹+Gⁿ), 0≤n < m−1. (5.36) This together with Gronwall’s inequality [22, 38] gives

Gⁿ⁺¹ .[G⁰+τ² Xn+1

l=1

(kr^lk²)]e^4CT. (5.37)

Equation (5.37), concurrently with lemma 5.3.4 and equation (5.30) gives the following estimates

ke^mk+k∇he^mk+k∇_h²e^mk+k∇²_h²e^mk+ some other terms .τ²+h⁴. (5.38) which ultimately shows that, by principle of mathematical induction, we have

keⁿk+k∇^heⁿk+k∇h²eⁿk+k∇²_h²eⁿk.τ²+h⁴, 0≤n≤N. (5.39) This completes the proof.