Convergence Analysis

The following theorem ensures that the discrete operatorL^N_ε satisfies the discrete mini- mum principle and therefore, the proposed scheme is ε-uniformly stable.

Theorem 2.6.1. Let us suppose that N ≥N₀ ≥8, where (∥b∥∞+ ∆t⁻¹)

≤ 2^p−1α

N₀^p−1 and N₀

(lnN0)² ≥4σ₀²∥a₀∥∞. (2.6.1) Then, we have

r_i⁻>0, r⁺_i >0, for 1≤i≤N −1,

|r_i⁻|+|r⁺_i |<|r_i⁰|, for 1< i < N −1,

|r₁⁺|<|r⁰₁| and |r⁻_N−1|<|r⁰_N−1|. Proof: From (2.5.4) and (2.5.5), it is clear that

r_i⁻>0, for N/2≤i < N−1, r_i⁺>0, for 1≤i < N/2,

|r_i⁻|+|r⁺_i |<|r_i⁰| for 1 < i < N −1,

|r₁⁺|<|r⁰₁| and |r⁻_N−1|<|r_N⁰₋₁|. Now, we have to prove

r_i⁻>0, for 1 ≤i < N/2, and r⁺_i >0, for N/2≤i < N −1.

From (2.5.4), one can write r⁻_i = 2ε∆t

ˆhh − a_i∆t

ˆh ≥ ∆tN²

16σ²ln²N − ∥a₀∥_∞N∆t

4 = N∆t

16σ² [ N

ln²N − ∥a₀∥∞4σ₀² ]

. (2.6.2)

For 8≤N0 ≤N, it is easy to show that N₀

ln²N₀ ≤ N

ln²N. (2.6.3)

Therefore, by using (2.6.2) and (2.6.3), we get

r⁻_i >0, for 1 ≤i < N/2, when N₀

ln²N₀ ≥ ∥a0∥∞4σ²₀. Now, forN/2≤i < N −1, from (2.5.5), we have

r_i⁺ = 2ε∆t

ˆh_ih_i+1 + a_i+1/2∆t

h_i+1 −b_i+1/2∆t

2 −1

2

≥ 2ε∆t ˆhihi+1

+ α∆t 2hi+1

(x^p_i +x^p_i+1)

− bi+1/2∆t

2 − 1

2

≥ 2ε∆t

ˆh_ih_i+1 + α∆t 2h_i+1

(

x^p_N/2+x^p_N/2+1

)− b_i+1/2∆t

2 − 1

2

≥ 2ε∆t

ˆh_ih_i+1 + α∆t

2h_i+1x^p_N/2+1−b_i+1/2∆t

2 −1

2. (2.6.4)

ForN ≥N₀, one can easily show that x^p_N/2+1≥x^p_N

0/2+1 and also x_N0/2+1=σ+ 21−σ

N₀ = 2 N₀ +σ

( 1− 2

N₀ )

≥ 2

N₀. (2.6.5)

Therefore, by using (2.6.5), we can conclude thatx^p_N

0/2+1≥ ( 2

N₀ )p

and α∆t

2h_i+1x^p_N

0/2+1− b_i+1/2∆t

2 − 1

2 ≥ α∆t

4N₀⁻¹ ( 2

N₀ )p

− ∥b∥∞∆t

2 − 1

2

≥ ∆t 2

[2^p−1α N₀^p−1 −(

∥b∥∞+ ∆t⁻¹)]

. (2.6.6) Hence, from (2.6.4) and (2.6.6), we have

r⁺_i >0, for N/2≤i < N−1, when 2^p−1α N₀^p−1 ≥(

∥b∥∞+ ∆t⁻¹) , which completes the proof.

Now, we define the truncation error as τ_i,ubⁿ⁺¹ =L^N_ε

(ubⁿ⁺¹_i −Ub_iⁿ⁺¹ )

= ∆tτ_i,^x_ubn+1, where

τ_i,^x_ubn+1 =





L^N_cenbuⁿ⁺¹_i −(L_x,εbuⁿ⁺¹) (x_i), for 1≤i < N/2, L^N_midbuⁿ⁺¹_i −(L_x,εbuⁿ⁺¹)_i+1/2, for N/2≤i≤N −1.

Therefore, using the Taylor’s formula one can easily obtain the following lemma.

Lemma 2.6.2. The local truncation error τ_i,ubⁿ⁺¹ at an internal mesh point satisfies

|τ_i,buⁿ⁺¹| ≤C∆t(

h²+h²ε^−3/2exp(−mx_i/√ ε))

, 0< x_i < σ, (2.6.7)

|τ_i,buⁿ⁺¹| ≤C∆t(

εh_i+1+h²_i+1+εh²_i+1+ε^−1/2exp(−mx_i−1/√ ε))

, x_i =σ, (2.6.8) and for σ < x_i <1

|τ_i,buⁿ⁺¹| ≤





C∆t(

εh_i+1+h²_i+1+εh²_i+1+ε^−1/2exp(−mx_i−1/√ ε))

, h_i+1 ≤√ ε, C∆t(

εh_i+1+h²_i+1+εh²_i+1+h⁻_i+1¹ exp(−mx_i−1/√ ε))

, h_i+1 >√ ε.

(2.6.9) Proof. We divided the proof into two cases. Case 1 and Case 2 are corresponding to the proof of the truncation error in the inner and outer regions, respectively.

Case 1 (Inner region): Following the idea given in [42], one can easily show that τ_i,^x_bun+1≤Ch

∫ _xi+1

x_i−1

[ ε

d⁴buⁿ⁺¹ dx⁴ (s)

+

d³buⁿ⁺¹ dx³ (s)

]

ds, for 1 < i < N/2. (2.6.10) Now, we decompose the local truncation errorτ_i,ubⁿ⁺¹ byτ_i,wbⁿ⁺¹ andτ_i,cw

ln+1, whereτ_i,wbⁿ⁺¹ is the local truncation error associated tow(x) and τ_i,wc

ln+1 is the local truncation error associated to w_l(x). From (2.6.10) and the estimates given in Lemma 2.4.2, we obtain

|τ_i,wbⁿ⁺¹| ≤ C∆t[

h²+hε^−1/2(exp(−mx_i−1/√

ε)−exp(−mx_i+1/√ ε))]

, (2.6.11) and

|τ_i,wc

ln+1| ≤C∆thε⁻¹[

exp(−mx_i−1/√

ε)−exp(−mx_i+1/√ ε)]

. (2.6.12)

Therefore, by adding (2.6.11) and (2.6.12), we get

|τ_i,ubⁿ⁺¹| ≤C∆t[h²+hε⁻¹(exp(−mx_i−1/√

ε)−exp(−mx_i+1/√ ε))]

≤C∆t[h²+hε⁻¹exp(−mx_i/√

ε) sinh(mh/√ ε)]. From (2.6.1), one can conclude thatmh/√

ε≤1, thus sinh(mh/√

ε)≤Cmh/√

ε. There- fore,

|τ_i,buⁿ⁺¹| ≤C∆t[

h²+h²ε^−3/2exp(−mx_i/√ ε)]

, for 0 < x_i < σ.

Case 2 (Outer region): For N/2 ≤ i ≤ N −1, one can easily obtain the following truncation error expression

τ_i,^x_bun+1 = 2ε h_i+h_i+1

[h²_i+1 3!

d³ubⁿ⁺¹

dx³ (ξ₁)− h²_i 3!

d³buⁿ⁺¹ dx³ (ξ₂)

]

+a_i+1/2h²_i+1 3!

d³buⁿ⁺¹ dx³ (ξ₁)

−ε 2

[

h_i+1d³buⁿ⁺¹

dx³ (x_i)− h²_i+1 2!

d⁴ubⁿ⁺¹ dx⁴ (ξ₁)

]

− ai+1/2h²_i+1 4

d³buⁿ⁺¹

dx³ (ξ₁),(2.6.13) by using the Taylor’s expansion, wherex_i−1 < ξ₂ < x_i and x_i < ξ₁ < x_i+1.

Now, we are going to divide this case into two subcases, one for h_i+1 ≤ √ ε and another one forh_i+1 >√

ε.

Subcase 1 (h_i+1 ≤ √

ε): Here, also we decompose the local truncation error τ_i,buⁿ⁺¹ by τ_i,wbⁿ⁺¹ and τ_i,cw

ln+1. For N/2 ≤ i ≤ N −1, by using the estimates (2.4.2) in local truncation error expression (2.6.13), we deduce that

|τ_i,wbⁿ⁺¹| ≤ C∆t[

εh_i+1+h²_i+1+εh²_i+1+h_i+1exp(−mx_i−1/√

ε) + exp(−mx_i/√ ε)]

, and

|τ_i,cw

ln+1| ≤C∆tε^−1/2exp(−mx_i−1/√ ε).

Therefore, we can conclude that

|τ_i,ubⁿ⁺¹| ≤C∆t[

εh_i+1+h²_i+1+εh²_i+1+ε^−1/2exp(−mx_i−1/√ ε)]

. Subcase 2 (hi+1 >√

ε): For N/2< i≤ N −1, by using the estimates (2.4.2) in local truncation error expression (2.6.13) and applying the inequalitys^kexp(−s)≤C,s ≥0, we obtain

|τ_i,wbⁿ⁺¹| ≤ C∆t[

εh_i+1+h²_i+1+εh²_i+1+h_i+1exp(−mx_i−1/√

ε) + exp(−mx_i−1/√ ε)]

. Now,

τ_i,cw

ln+1 = ∆t[

L^N_midbuⁿ⁺¹(x_i)−(

L_x,εubⁿ⁺¹)

(x_i−1/2)]

= −ε∆t

2 [w_l^′′(x_i+1) +w^′′_l(x_i)]− ∆t

2 [a_i+1w^′_l(x_i+1) +a_iw^′_l(x_i)]

+∆tr⁻_mid,i[w_l(x_i−1)−w_l(x_i)] + ∆tr⁺_mid,i[w_l(x_i+1)−w_l(x_i)]

+∆tb_i+1

2 [w_l(x_i+1)−w_l(x_i)]. (2.6.14) By using the expression of w_l given in Lemma 2.4.2, we get

ε

2[w^′′_l(x_i+1) +w^′′_l(x_i)]≤Cexp(−µx_i/√

ε), (2.6.15)

1

2[ai+1w_l^′(xi+1) +aiw_l^′(xi)]

≤Cε^−1/2exp(−µxi/√

ε), (2.6.16)

1

2b_i+1[w_l(x_i+1)−w_l(x_i)]

≤Cexp(−µx_i/√

ε). (2.6.17)

From (2.5.7), one can establish the following inequalities r⁻_mid,i= 2ε

h_i(h_i+h_i+1) ≤1, for N/2< i≤N −1, (2.6.18) and

r_mid,i⁺ ≤ 2ε

h_i+1(h_i+h_i+1) +a_i+1/2

h_i+1 + b_i+1/2

2 ≤Ch⁻_i+1¹, for N/2≤i≤N −1. (2.6.19) Therefore, by applying (2.6.15)-(2.6.19) in (2.6.14), we get

|τ_i,cw

ln+1| ≤C∆th⁻_i+1¹ exp(−µx_i−1/√

ε), forN/2< i≤N −1, and hence, forN/2< i≤N −1,

|τ_i,buⁿ⁺¹| ≤C∆t(

εh_i+1+h²_i+1+εh²_i+1+h⁻_i+1¹ exp(−mx_i−1/√ ε))

. From (2.5.7) and by using the inequality exp(s)−1≤Cs , we have

r⁻_mid,N/2[

w_l(x_N/2−1)−w_l(x_N/2)]

≤C 2ε

h(h+H)

[exp(−µx_N/2−1/√

ε)−exp(−µx_N/2/√ ε)]

≤C 2ε

h(h+H)exp(−µx_N/2/√

ε)[exp(µh/√

ε)−1]

≤Cexp(−µx_N/2/√

ε), (2.6.20)

and from (2.6.19), we get r_mid,N/2⁺ [

w_l(x_N/2+1)−w_l(x_N/2)] ≤ C

H

[exp(−µx_N/2+1/√

ε)−exp(−µx_N/2/√ ε)]

≤ C

H exp(−µx_N/2/√

ε)[exp(−µH/√

ε)−1]

≤ CH⁻¹exp(−µx_N/2/√

ε). (2.6.21)

Therefore, by applying (2.6.15)-(2.6.17), (2.6.20) and (2.6.21) in (2.6.14), we conclude

|τ_N/2,cw

ln+1| ≤C∆tε^−1/2exp(−µx_N/2/√

ε). (2.6.22)

From the estimates given in Lemma 2.4.2 and the local truncation error expression (2.6.13), we can obtain

|τ_N/2,wbⁿ⁺¹| ≤ C∆t[

εh_N/2+1+h²_N/2+1+εh²_N/2+1+ε⁻¹h²_N/2+1exp(−mx_N/2/√ ε) +ε^−1/2h²_N/2+1exp(−mx_N/2−1/√

ε)]

, (2.6.23)

Hence, from (2.6.22) and (2.6.23), we get

|τ_N/2,ubⁿ⁺¹| ≤C∆t[

εh_N/2+1+h²_N/2+1+εh²_N/2+1+ε^−1/2exp(−mx_N/2−1/√ ε)]

, which is the required result.

Now, we state some technical lemmas that will be used later.

Lemma 2.6.3. Let S_i be the mesh function defined by S_i =

∏i j=1

(

1 + mh_j

√ε )₋1

, for 1≤i≤N, (with the usual convention that S₀ = 1), where m = √

β. Then there exists a positive constant C such that

L^N_ε S_i ≤









−C∆t

√ε S_i, for 1≤i < N/2,

− C∆t max{√

ε, h_i+1}Si, for N/2≤i≤N −1.

Proof. It is easy to obtain that S_i−S_i−1 = −S_imh_i

√ε . Now, by applying the discrete operatorL^N_ε on S_i, for 1≤i < N/2, we get

L^N_ε S_i = 2ε∆t hi+hi+1

(S_i+1−S_i

hi+1 − S_i−S_i−1 hi

)

+ ∆ta_i

(S_i+1−S_i−1 hi+hi+1

)

−∆tb_iS_i−S_i

≤ − m∆t

√ε+mhS_i [

−m√ ε+ a_i

2 +b_i

√ε+mh m

]

≤ −C∆t

√ε S_i,

and for N/2≤i≤N −1, we obtain L^N_ε S_i = 2ε∆t

h_i+h_i+1

(S_i+1−S_i

h_i+1 − S_i−S_i−1 h_i

)

+ ∆ta_i+1/2

(S_i+1−S_i h_i+1

)

−∆tb_i+1/2S_i+1/2−S_i+1/2

= 2ε∆t

hi+hi+1

(

−m

√εS_i+1+ m

√εS_i )

−∆ta_i+1/2 m

√εS_i+1− ∆tbi+1/2

2 S_i

−

(∆tb_i+1/2

2 S_i+1+S_i+1/2 )

≤ − m∆t

√ε+mhSi

[

ai+1/2+ b_i+1/2 2

√ε+mh_i+1

m −2m√

ε h_i+1 h_i+h_i+1

]

≤ − C∆t max{√

ε, h_i+1}S_i.

Lemma 2.6.4. The mesh function S_i satisfies the following inequality exp(−mx_i/√

ε)≤S_i, for all i, (2.6.24) and there exists a positive constant C, such that

S_N/2 ≤CN⁻². (2.6.25)

Proof. Since exp(−x)≤(1 +x)⁻¹, x≥0, for each j, we have exp

(

−mh_j

√ε )

≤ (

1 + mh_j

√ε )₋1

. (2.6.26)

By multiplying the inequality (2.6.26), for j = 1, 2, · · · , i, we get exp

(

−m

√ε(h1 +h2+· · ·+hi) )

≤ ∏ⁱ

j=1

(

1 + mh_j

√ε )₋1

. (2.6.27)

Therefore, the inequality (2.6.24) follows by using the fact that h₁+h₂ +· · ·+h_i =x_i, in (2.6.27). The second inequality (2.6.25) can be proved by following the idea as given in [89].

Theorem 2.6.5. There exists a positive constantC such that buⁿ⁺¹_i −Ub_iⁿ⁺¹≤





CN⁻²ln²N, for 1≤i < N/2, CN⁻², for N/2≤i≤N −1,

where buⁿ⁺¹(x) is the exact solution of (2.3.2) and {Ub_iⁿ⁺¹} is the numerical solution of the problem (2.5.2).

Proof. We define two discrete functions χ^±₁ by

χ^±₁(x_i) =ϕ₁(x_i)±ψ₁(x_i), for 0 ≤i≤N, (2.6.28) where

ϕ₁(x_i) =C[(

εh_i+1+εh²_i+1+h²_i+1)

(1−x_i) +S_i−1]

, for 0≤i≤N, (2.6.29) and

ψ₁(x_i) =ubⁿ⁺¹_i −Ub_iⁿ⁺¹, for 0≤i≤N. (2.6.30) Thus, from Lemma 2.6.3, we have

L^N_ε χ^±₁(x_i)≤0, for 1≤i≤N −1, χ^±₁(x₀)>0 and χ^±₁(x_N)>0,

for sufficiently large values of C. Hence, by applying the discrete minimum principle, we get

buⁿ⁺¹_i −Ub_iⁿ⁺¹≤ϕ₁(x_i), for 0 ≤i≤N. (2.6.31) Now, from (2.6.31), we can write

buⁿ⁺¹_i −Ub_iⁿ⁺¹ ≤C[

(εH+εH²+H²) +S_N/2−1]

=C [

(εH+εH²+H²) + (

1 + mh

√ε )

S_N/2 ]

, for N/2 ≤ i ≤ N −1. Therefore, by using H ≤ 2N⁻¹, √

ε ≤ N⁻¹, mh/√

ε ≤ C and (2.6.25), we deduce that

buⁿ⁺¹_i −Ub_iⁿ⁺¹≤CN⁻², for N/2≤i≤N−1. (2.6.32)

Next, for 0≤i≤N/2, we choose two discrete functions χ^±₂ as

χ^±₂(xi) = ϕ2(xi)±ψ2(xi), for 0≤i≤N/2, (2.6.33) where

ϕ₂(x_i) =C[

N⁻²(1−x_i) +h²ε⁻¹S_i]

, for 0 ≤i≤N/2, (2.6.34) and

ψ2(xi) =



 b

uⁿ⁺¹_i −Ub_iⁿ⁺¹, for 1≤i≤N/2−1,

0, for i= 0, N/2.

(2.6.35) Thus, from Lemma 2.6.3, we have

L^N_ε χ^±₂(x_i)≤0, for 1≤i≤N/2−1, χ^±₂(x₀)>0 and χ^±₂(x_N/2)>0,

for sufficiently large value ofC and hence, by applying the discrete minimum principle, we get

buⁿ⁺¹_i −Ub_iⁿ⁺¹≤ϕ₂(x_i)≤C [

N⁻² + ( h

√ε )2]

≤C[

N⁻²+N⁻²ln²N] , for 1≤i < N/2.

Theorem 2.6.6. Let u be the exact solution of (2.1.1) and {Uⁿ} be the numerical solution of the fully discrete scheme (2.5.8) at time level t_n = n∆t. Taking ν, ∆t such that0< ν <1 andN^−ν ≤C∆t, there exists a positive constantC which is independent of ε, N, such that

∥u(x_i, t_n)−U_iⁿ∥_∞ ≤





C(∆t+N^−2+νln²N), for 1≤i < N/2, C(∆t+N^−2+ν), for N/2≤i≤N −1.

Proof. The global error at time level t_n =n∆t can be written as

∥u(x_i, t_n)−U_iⁿ∥_∞ ≤ ∥u(x_i, t_n)−buⁿ_i∥_∞+buⁿ_i −Ub_iⁿ

∞+ bU_iⁿ−U_iⁿ

∞. (2.6.36) Now, by using Lemma 2.3.2 and Theorem 2.6.5 in (2.6.36), we get

∥u(x_i, t_n)−U_iⁿ∥_∞ ≤





C∆t(∆t+N^−2+νlnN) + bU_iⁿ−U_iⁿ

∞, for 1≤i < N/2, C∆t(∆t+N^−2+ν) + bU_iⁿ−U_iⁿ

∞, for N/2≤i≤N −1.

(2.6.37) Subtracting (2.5.8) from (2.5.3) and taking the inverse operator(

L^N_ε )₋1

, we get Ub_iⁿ⁺¹−U_iⁿ⁺¹ =(

L^N_ε )₋1

(u(x_i, t_n)−U_iⁿ), for 1 ≤i < N/2. (2.6.38) Now, from (2.5.4) and [91, Theorem 1], we can have (

L^N_ε )₋1

∞ ≤1. Thus, by taking supremum norm in (2.6.38) and using the fact that (

L^N_ε )₋1

∞ ≤1, we can deduce bU_iⁿ⁺¹−U_iⁿ⁺¹

∞ ≤ ∥u(x_i, t_n)−U_iⁿ∥_∞, for 1≤i < N/2. (2.6.39) For then-th time level, we get

bU_iⁿ−U_iⁿ

∞≤u(x_i, t_n−1)−U_iⁿ⁻¹

∞, for 1≤i < N/2. (2.6.40) Now, we replace bU_iⁿ−U_iⁿ

∞ in (2.6.37) by the bound given in (2.6.40) and we obtain

∥u(x_i, t_n)−U_iⁿ∥_∞≤C(∆t+N^−2+νln²N), for 1≤i < N/2. (2.6.41) By following the same approach, we can get the required result forN/2≤i≤N−1.