Two Conservative Difference Schemes for the Generalized Rosenau Equation

Boundary Value Problems

Volume 2010, Article ID 543503,18pages doi:10.1155/2010/543503

Research Article

Two Conservative Difference Schemes for the Generalized Rosenau Equation

Jinsong Hu

and Kelong Zheng

1School of Mathematics and Computer Engineering, Xihua University, Chengdu, Sichuan 610039, China

2School of Science, Southwest University of Science and Technology, Mianyang, Sichuan 621010, China

Correspondence should be addressed to Kelong Zheng,kl [email protected] Received 31 October 2009; Accepted 26 January 2010

Academic Editor: Sandro Salsa

Copyrightq2010 J. Hu and K. Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Numerical solutions for generalized Rosenau equation are considered and two energy conserva- tive finite difference schemes are proposed. Existence of the solutions for the difference scheme has been shown. Stability, convergence, and priori error estimate of the scheme are proved using energy method. Numerical results demonstrate that two schemes are efficient and reliable.

1. Introduction

Consider the following initial-boundary value problem for generalized Rosenau equation:

utuxxxxtux u^p_x 0, x∈0, L , t∈0, T , 1.1

with an initial condition

ux,0 u0x, x∈0, L , 1.2

and boundary conditions

u0, t uL, t 0, u_xx0, t u_xxL, t 0, t∈0, T , 1.3

wherep≥2 is a integer.

When p 2, 1.1 is called as usual Rosenau equation proposed by Rosenau 1 for treating the dynamics of dense discrete systems. Since then, the Cauchy problem of the Rosenau equation was investigated by Park 2 . Many numerical schemes have been proposed, such as C¹-conforming finite element method by Chung and Pani 3 ,

discontinuous Galerkin method by Choo et al. 4 , orthogonal cubic spline collocation method by Manickam 5 , and finite difference method by Chung 6 and Omrani et al.

7 . As for the generalized case, however, there are few studies on theoretical analysis and numerical methods.

It can be proved easily that the problem1.1–1.3has the following conservative law:

Et u²_L2uxx²_L2E0. 1.4

Hence, we propose two conservative difference schemes which simulate conservative law 1.4. The outline of the paper is as follows. In Section 2, a nonlinear difference scheme is proposed and corresponding convergence and stability of the scheme are proved. In Section 3, a linearized difference scheme is proposed and theoretical results are obtained. In Section 4, some numerical experiments are shown.

2. Nonlinear Finite Difference Scheme

Lethandτbe the uniform step size in the spatial and temporal direction, respectively. Denote xj jh0 ≤j ≤J, tnnτ 0 ≤n≤N,uⁿ_j ≈uxj, tn, andZ⁰_h{u uj|u0 uj 0, j 0,1,2, . . . , J}. Define

uⁿ_j

x uⁿ_j1−uⁿ_j

h ,

uⁿ_j

x uⁿ_j −uⁿ_j−1

h ,

uⁿ_j

x uⁿ_j1−uⁿ_j−1 2h ,

uⁿ_j

t uⁿ¹_j −uⁿ_j

τ ,

uⁿ_j

t uⁿ¹_j −uⁿ⁻¹_j

2τ ,

uⁿ_j

xx uⁿ_j1−2uⁿ_j uⁿ_j−1

h² ,

uⁿ_j uⁿ¹_j uⁿ⁻¹_j

2 , u^n1/2_j uⁿ¹_j uⁿ_j

2 ,

uⁿ, vⁿ h

J−1

j0

uⁿ_jvⁿ_j, uⁿ² uⁿ, uⁿ, uⁿ_∞ max

0≤j≤J−1

uⁿ_j,

2.1

and in the paper,Cdenotes a general positive constant which may have different values in different occurrences.

Sinceu^p_x p/p1u^p−1ux u^p_x , then the following finite difference scheme is considered:

uⁿ_j

t

uⁿ_j

xxx xt u^n1/2_j

x p

p1

u^n1/2_{j p−1} u^n1/2_j

x

u^n1/2_{j p}

x

0, 2.2

u⁰_j u0

xj

, 0≤j ≤J−1, 2.3

uⁿ₀ uⁿ_J 0, uⁿ₀

xx

uⁿ_J

xx 0. 2.4

Lemma 2.1see8 . For any two mesh functions,u, v∈Z_h⁰, one has

uj

x, vj

− uj,

vj

x

,

vj, uj

xx

− vj

x, uj

x

,

uj, uj

xx

− uj

x, uj

x

−ux².

2.5

Furthermore, ifuⁿ_0xx uⁿ_Jxx0, then

u_j,

u_j

xxx x

uxx². 2.6

Theorem 2.2. Suppose u₀ ∈ H₀²0, L , then the scheme 2.2–2.4 is conservative for discrete energy, that is,

Eⁿuⁿ²uⁿ_xx²Eⁿ⁻¹· · ·E⁰. 2.7

Proof. Computing the inner product of2.2with 2u^n1/2, according to boundary condition 2.4andLemma 2.1, we have

1 τ

uⁿ¹²− uⁿ²

1 τ

uⁿ¹_xx ²− uⁿ_xx²

u^n1/2_j

x,2u^n1/2_j

P₁,2u^n1/2_j 0,

2.8

where

P₁ p p1

u^n1/2_{j p−1} u^n1/2_j

x

u^n1/2_{j p}

x

. 2.9

According to

u^n1/2_j

x,2u^n1/2_j

0, 2.10

P₁,2u^n1/2_j

2p p1h

J−1

j0

u^n1/2_{j p−1} u^n1/2_j

x

u^n1/2_{j p}

x

u^n1/2_j

p p1

J−1 j0

u^n1/2_{j p}

u^n1/2_j1 −u^n1/2_j−1

u^n1/2_{j1 p}

−

u^n1/2_{j−1 p}

u^n1/2_j

p p1

J−1 j0

u^n1/2_{j1 p−1}

u^n1/2_j

u^n1/2_{j p} u^n1/2_j1

− p p1

J−1 j0

u^n1/2_{j p−1}

u^n1/2_j−1

u^n1/2_{j−1 p} u^n1/2_j

0,

2.11

we obtain

uⁿ¹²− uⁿ²

uⁿ¹_xx ²− uⁿ_xx²

0. 2.12

By the definition ofEⁿ,2.7holds.

To prove the existence of solution for scheme2.2–2.4, the following Browder fixed point Theorem should be introduced. For the proof, see9 .

Lemma 2.3Browder fixed point Theorem. LetHbe a finite dimensional inner product space.

Suppose thatg : H → His continuous and there exists anα > 0 such thatgx, x > 0 for all x∈Hwithxα. Then there existsx^∗∈Hsuch thatgx^∗ 0 andx^∗ ≤α.

Theorem 2.4. There existsuⁿ∈Z_h⁰satisfying the difference scheme2.2–2.4.

Proof. By the mathematical induction, forn≤ N−1, assume thatu⁰, u¹, . . . , uⁿ satisfy2.2–

2.4. Next we prove that there existsuⁿ¹satisfying2.2–2.4.

Define a operatorgonZ_h⁰as follows:

gv 2v−2uⁿ2v_{xxx x}−2uⁿ_{xxx x}τv_x τp p1

v_jp−1 v_j

x

v_jp

x

. 2.13

Taking the inner product of2.13withv, we get

vx, v 0, v_jp−1

v_j

x

v_j^p

x, v

0,

gv, v

2v²−2uⁿ, v 2vxx²−2uⁿ_xx, vxx

≥2v²−2uⁿ · v2vxx²−2uⁿ_xx · vxx

≥2v²−

u²v²

2vxx²−

uxx²vxx²

≥ v²−

uⁿ²uxx²

vxx²

≥ v²−

uⁿ²uⁿ_xx² .

2.14

Obviously, for all v ∈ Z_h⁰, gv, v ≥ 0 with v² uⁿ² uⁿ_xx² 1. It follows from Lemma 2.3that there existsv^∗ ∈ Z⁰_h which satisfiesgv^∗ 0. Letuⁿ¹ 2v^∗−uⁿ, it can be proved thatuⁿ¹is the solution of the scheme2.2–2.4.

Next, we discuss the convergence and stability of the scheme2.2–2.4. Letvx, tbe the solution of problem1.1–1.3,v_jⁿ vxj, tn, then the truncation of the scheme 2.2–

2.4is

r_jⁿ vⁿ_j

t

vⁿ_j

xxx xt v^n1/2_j

x p

p1

v^n1/2_{j p−1} v_j^n1/2

x

v^n1/2_{j p}

x

. 2.15

Using Taylor expansion, we know thatr_jⁿOτ²h²holds ifτ, h → 0.

Lemma 2.5. Suppose thatu ∈ H₀²0, L , then the solution of the initial-boundary value problem 1.1–1.3satisfies

u_L2≤C, ux_L2≤C, u_∞≤C. 2.16

Proof. It follows from1.4that

u_L2≤C, uxx_L2 ≤C. 2.17

Using H ¨older inequality and Schwartz inequality, we get

ux²_{L2 L}

0

u_xu_xdx uu_x|^L₀− _L

0

uu_xxdx− _L

0

uu_xxdx

≤ u_L2· u_xxL2 ≤ 1 2

u²_L2uxx²_L2 .

2.18

Hence,uxL2 ≤C. According to Sobolev inequality, we haveu_∞≤C.

Lemma 2.6Discrete Sobolev’s inequality10 . There exist two constantC1andC2such that

uⁿ_∞≤C₁uⁿC₂uⁿ_x^. 2.19

Lemma 2.7 Discrete Gronwall inequality 10 . Suppose wk, ρk are nonnegative mesh functions andρkis nondecreasing. IfC >0 and

wk≤ρk Cτ k−1

l0

wl, ∀k, 2.20

then

wk≤ρke^Cτk, ∀k. 2.21

Theorem 2.8. Supposeu₀ ∈H₀²0, L , then the solutionuⁿof 2.2satisfiesuⁿ ≤C, uⁿ_x ≤C, which yielduⁿ∞≤Cn1,2, . . . , N.

Proof. It follows from2.7that

uⁿ ≤C, uⁿ_xx ≤C. 2.22

UsingLemma 2.1and Schwartz inequality, we get

uⁿ_x²≤ uⁿuⁿ_xx ≤ 1 2

uⁿ²uⁿ_xx²

≤C. 2.23

According toLemma 2.6, we haveuⁿ∞≤C.

Theorem 2.9. Supposeu0 ∈H₀²0, L , then the solutionuⁿ of the scheme2.2–2.4converges to the solution of problem1.1–1.3and the rate of convergence isOτ²h².

Proof. Subtracting2.15from2.2and lettinge_jⁿv_jⁿ−uⁿ_j, we have

r_jⁿ eⁿ_j

t

e_jⁿ

xxx xt e^n1/2_j

x p

p1

v_j^n1/2_p−1 v^n1/2_j

x

v^n1/2_{j p}

x

− p p1

u^n1/2_{j p−1} u^n1/2_j

x

u^n1/2_{j p}

x

.

2.24

Computing the inner product of2.24with 2e^n1/2, and usinge^n1/2_{j x}, 2e_j^n1/2 0, we get

r_jⁿ,2e^n1/2 1

τ

eⁿ¹²− eⁿ²

1 τ

eⁿ¹_xx ²− eⁿ_xx²

Q₁Q₂,2e^n1/2

, 2.25

where

Q₁ p p1

v^n1/2_{j p−1} v_j^n1/2

x−

u^n1/2_{j p−1} u^n1/2_j

x

,

Q2 p p1

v^n1/2_j p x−

u^n1/2_j p x

.

2.26

According toLemma 2.5,Theorem 2.8, and Schwartz inequality, we have

Q1,2e^n1/2 2p

p1h J−1 j0

v^n1/2_{j p−1} v_j^n1/2

x−

u^n1/2_{j p−1} u^n1/2_j

x

e_j^n1/2

2p p1h

J−1 j0

v^n1/2_{j p−1} e^n1/2_j

xe^n1/2_j

2p p1h

J−1 j0

v^n1/2_{j p−1}

−

u^n1/2_{j p−1} u^n1/2_j

xe^n1/2_j

2p p1h

J−1 j0

v^n1/2_{j p−1} e^n1/2_j

xe^n1/2_j

2p p1h

J−1 j0

e^n1/2_j

p−2 k0

v_j^n1/2_p−2−k

u^n1/2_j k u^n1/2_j

xe^n1/2_j

≤Ch J−1 j0

e_j^n1/2

x

·e^n1/2_j Ch J−1 j0

e^n1/2_{j 2}

≤C

e^n1/2_x ²e^n1/22

≤C

eⁿ¹_x ²eⁿ_x²eⁿ¹²eⁿ²

,

2.27

Q2,2e^n1/2 2p

p1h J−1 j0

v_j^n1/2p x−

u^n1/2_j p x

e^n1/2_j

− 2p p1h

J−1 j0

v_j^n1/2p

−

u^n1/2_j p

e^n1/2_j

x

− 2p p1h

J−1 j0

e^n1/2_{j p−1}

k0

v^n1/2_{j p−1−k}

u^n1/2_j k e^n1/2_j

x

≤C

e^n1/2_x ²e^n1/22

≤C

eⁿ¹_x ²eⁿ_x²eⁿ¹²eⁿ²

.

2.28

Furthermore,

r_jⁿ,2e^n1/2

r_jⁿ, eⁿ¹eⁿ

≤ rⁿ²1 2

eⁿ¹²eⁿ²

. 2.29

Substituting2.27–2.29into2.25, we get eⁿ¹²− eⁿ²

eⁿ¹_xx ²− eⁿ_xx²

≤Cτ

eⁿ¹²eⁿ²eⁿ¹_x ²eⁿ_x²

τrⁿ². 2.30

Similarly to the proof of2.23, we have eⁿ¹_x ² ≤ 1

2

eⁿ¹²eⁿ¹_xx ²

, eⁿ_x²≤ 1 2

eⁿ²eⁿ_xx²

, 2.31

and2.30can be rewritten as eⁿ¹²− eⁿ²

eⁿ¹_xx ²− eⁿ_xx²

≤Cτ

eⁿ¹²eⁿ²eⁿ¹_xx ²eⁿ_xx²

τrⁿ². 2.32

LetBⁿeⁿ²e_xx^{n 2}, then2.32is written as follows:

1−Cτ

Bⁿ¹−Bⁿ ≤2CτBⁿτrⁿ². 2.33

Ifτis sufficiently small which satisfies 1−Cτ >0, then

Bⁿ¹−Bⁿ ≤CτBⁿCτrⁿ². 2.34

Summing up2.34from 0 ton−1, we have

Bⁿ≤B⁰Cτ n−1

l0

r^l2Cτ

n−1

l0

B^l. 2.35

Noticing

τ n−1

l0

r^l2≤nτ max

0≤l≤n−1

r^l2≤T·O

τ²h²2

, 2.36

ande⁰0, we haveB⁰Oτ²h²². Hence

Bⁿ≤O

τ²h²₂ Cτ

n−1

l0

B^l. 2.37

According toLemma 2.7, we getBⁿ≤Oτ²h²², that is,

eⁿ ≤O

τ²h²

, eⁿ_xx ≤O

τ²h²

. 2.38

It follows from2.31that

eⁿ_x ≤O

τ²h²

. 2.39

By usingLemma 2.6, we have

eⁿ_∞≤O

τ²h²

. 2.40

This completes the proof ofTheorem 2.9.

Similarly, the following theorem can be proved.

Theorem 2.10. Under the conditions ofTheorem 2.9, the solution of the scheme2.2–2.4is stable by · ∞.

3. Linearized Finite Difference Scheme

In this section, we propose a linear-implicit finite difference scheme as follows:

uⁿ_j

t uⁿ_j

xxx xt uⁿ_j

x p

p1

uⁿ_jp−1 uⁿ_j

x

uⁿ_jp−1 uⁿ_j

x

0. 3.1

Theorem 3.1. Supposeu0 ∈ H₀²0, L , then the scheme3.1,2.3, and2.4are conservative for discrete energy, that is,

Eⁿ 1 2

uⁿ¹²uⁿ²

1 2

uⁿ¹_xx ²uⁿ_xx²

τh

J−1

j0

uⁿ_j

xuⁿ¹_j Eⁿ⁻¹· · ·E⁰. 3.2

Proof. Computing the inner product of3.1with 2uⁿ, we have 1

2τ

uⁿ¹²−uⁿ⁻¹²

1 2τ

uⁿ¹_xx ²−uⁿ⁻¹_xx ²

uⁿ_j

x,2uⁿ_j

P₂,2uⁿ_j

0, 3.3

where

P₂ p p1

uⁿ_jp−1 uⁿ_j

x

uⁿ_jp−1 uⁿ_j

x

. 3.4

According toLemma 2.1, we get

uⁿ_j

x,2uⁿ_j

uⁿ_j

x, uⁿ¹_j

uⁿ_j

x, uⁿ⁻¹_j

uⁿ_j

x, uⁿ¹_j

− uⁿ_j,

uⁿ⁻¹_j

x

h J−1 j0

uⁿ_j

xuⁿ¹_j −h

J−1

j0

uⁿ⁻¹_j

xuⁿ_j,

3.5

P₂,2uⁿ_j 2p

p1h J−1

j0

uⁿ_jp−1 uⁿ_j

x

uⁿ_jp−1 uⁿ_j

x

uⁿ_j

p p1

J−1 j0

uⁿ_jp−1

uⁿ_j1−uⁿ_j−1

uⁿ_j1p−1

uⁿ_j1−

uⁿ_j−1p−1 uⁿ_j−1

uⁿ_j

p p1

J−1 j0

uⁿ_jp−1

uⁿ_j1uⁿ_j −

uⁿ_j1p−1 uⁿ_j1uⁿ_j

− p p1

J−1

j0

uⁿ_j−1p−1

uⁿ_juⁿ_j−1− uⁿ_jp−1

uⁿ_juⁿ_j−1

0.

3.6

Adding3.3and3.5to3.6, we obtain 1

2τ

uⁿ¹²− uⁿ²

1 2τ

uⁿ¹_xx ²− uⁿ_xx²

h J−1 j0

uⁿ_j

xuⁿ¹_j −h

J−1

j0

uⁿ⁻¹_j

xuⁿ_j 0. 3.7

By the definition ofEⁿ,3.2holds.

Theorem 3.2. The difference scheme3.1is uniquely solvable.

Proof. we use the mathematical induction. Obviously,u⁰ is determined by2.3and we can choose a two-order method to computeu¹e.g., by scheme2.2. Assuming thatu⁰, u¹, . . . , uⁿ are uniquely solvable, consideruⁿ¹in3.1which satisfies

1

2τuⁿ_j 1 2τ

uⁿ_j

xxx x p

2

p1

uⁿ_jp−1 uⁿ¹_j

x

uⁿ_jp−1 uⁿ¹_j

x

0. 3.8

Taking the inner product of3.8withuⁿ¹, we get

1 2τ

uⁿ¹² 1 2τ

uⁿ¹_xx ² ph 2

p1^J−1

j0

uⁿ_jp−1 uⁿ¹_j

x

uⁿ_jp−1 uⁿ¹_j

x

uⁿ¹_j . 3.9

Notice that

ph 2

p1^J−1

j0

uⁿ_jp−1 uⁿ¹_j

x

uⁿ_jp−1 uⁿ¹_j

x

uⁿ¹_j .

p 4

p1^J−1

j0

uⁿ_jp−1

uⁿ¹_j1 −uⁿ¹_j−1

uⁿ_j1p−1

uⁿ¹_j1 −

uⁿ_j−1p−1 uⁿ¹_j−1

uⁿ¹_j 0.

3.10

It follows from3.8that

1 2τ

uⁿ¹² 1 2τ

uⁿ¹_xx ²0. 3.11

That is, there uniquely exists trivial solution satisfying3.8. Hence,uⁿ¹_j in3.1is uniquely solvable.

To discuss the convergence and stability of the scheme3.1, we denote the truncation of the scheme3.1:

r_jⁿ

vⁿ_j

t v_jⁿ

xxx xt v_jⁿ

x p

p1

vⁿ_jp−1 vⁿ_j

x

vⁿ_jp−1 vⁿ_j

x

. 3.12

Using Taylor expansion, we know thatr_jⁿOτ²h²holds ifτ, h → 0.

Theorem 3.3. Supposeu0 ∈ H₀²0, L , then the solution of 3.1satisfiesuⁿ ≤ C, uⁿ_x ≤ C, which yielduⁿ_∞≤Cn1,2, . . . , N.

Proof. It follows from3.2that uⁿ¹²uⁿ²

uⁿ¹_xx ²uⁿ_xx²

C−2τh

J−1

j0

uⁿ_j

xuⁿ¹_j ≤Cτ

uⁿ¹²uⁿ_x²

. 3.13

According to2.23, we have uⁿ¹²uⁿ²

uⁿ¹_xx ²uⁿ_xx²

≤Cτ

uⁿ¹²1

2uⁿ² 1 2uⁿ_xx²

, 3.14

that is,

1−τuⁿ¹² 1−τ

2

uⁿ²uⁿ¹_xx ² 1−τ

2

uⁿ_xx²≤C. 3.15

Ifτis sufficiently small which satisfies 1−τ >0, we get uⁿ¹²uⁿ²

uⁿ¹_xx ²uⁿ_xx²

≤C, 3.16

which yieldsuⁿ ≤C, uⁿ_xx ≤C. According to2.23, we get

uⁿ_x ≤C. 3.17

UsingLemma 2.6, we obtain

uⁿ_∞≤C. 3.18

Theorem 3.4. Supposeu0 ∈ H₀²0, L , then the solution uⁿ of the schemes3.1,2.3, and2.4 converges to the solution of problem1.1–1.3and the rate of convergence isOτ²h².

Proof. Subtracting3.12from3.1and lettinge_jⁿv_jⁿ−uⁿ_j, we have

r_jⁿ

eⁿ_j

t eⁿ_j

xxx xt e_jⁿ

x p

p1

vⁿ_jp−1 vⁿ_j

x

vⁿ_jp−1 vⁿ_j

x

− p p1

uⁿ_jp−1 uⁿ_j

x

uⁿ_jp−1 uⁿ_j

x

.

3.19

Computing the inner product of3.19with 2eⁿ, we get r_jⁿ,2eⁿ

1 2τ

eⁿ¹²−eⁿ⁻¹²

1 2τ

eⁿ¹_xx ²−eⁿ⁻¹_xx ²

e_jⁿ

x,2eⁿ

Q3Q4,2eⁿ , 3.20

where

Q₃ p p1

vⁿ_jp−1 vⁿ_j

x−

uⁿ_jp−1 uⁿ_j

x

,

Q₄ p p1

v_jⁿ_p−1 vⁿ_j

x

− uⁿ_jp−1

uⁿ_j

x

.

3.21

Notice that Q3,2eⁿ

2ph p1

J−1 j0

vⁿ_jp−1 vⁿ_j

x−

uⁿ_jp−1 uⁿ_j

x

eⁿ

2ph p1

J−1 j0

vⁿ_jp−1 eⁿ_j

xeⁿ 2ph p1

J−1

j0

v_jⁿ_p−1

−

uⁿ_jp−1 uⁿ_j

xeⁿ

2ph p1

J−1 j0

vⁿ_jp−1 eⁿ_j

xeⁿ 2ph p1

J−1

j0

e_jⁿ

p−2 k0

v_jⁿ_p−2−k uⁿ_jk

uⁿ_j

xeⁿ

≤Ceⁿ_x²eⁿ²eⁿ²

≤C

eⁿ¹_x ²e_xⁿ⁻¹²eⁿ¹²eⁿ²eⁿ⁻¹²

,

3.22

and similarly

Q4,2eⁿ

≤C

eⁿ¹_x ²eⁿ⁻¹_x ²eⁿ¹²eⁿ²eⁿ⁻¹²

. 3.23

Furthermore, we get r_jⁿ,2eⁿ

r_jⁿ, eⁿ¹eⁿ⁻¹

≤ rⁿ²1 2

eⁿ¹²eⁿ⁻¹²

,

eⁿ_j

x,2eⁿ

eⁿ_j

x, eⁿ¹eⁿ⁻¹

≤ eⁿ_x²1 2

eⁿ¹²eⁿ⁻¹²

.

3.24

Substituting3.22–3.24into3.20, we get eⁿ¹²−eⁿ⁻¹²

eⁿ¹_xx ²−eⁿ⁻¹_xx ²

≤Cτ

eⁿ¹²eⁿ⁻¹²eⁿ²e_xⁿ¹²eⁿ_x²eⁿ⁻¹_x ²

2τrⁿ².

3.25

Similarly to the proof of2.31,3.25can be written as eⁿ¹²eⁿ²

eⁿ¹_xx ²eⁿ_xx²

−

eⁿ²eⁿ⁻¹²

eⁿ_xx²eⁿ⁻¹_xx ²

≤Cτ

eⁿ¹²eⁿ²eⁿ⁻¹²eⁿ¹_xx ²eⁿ_xx²eⁿ⁻¹_xx ²

2τrⁿ².

3.26

LetDⁿ eⁿ¹²eⁿ² eⁿ¹_xx ²eⁿ_xx², then3.26is written as follows:

Dⁿ−Dⁿ⁻¹≤Cτ

eⁿ¹²eⁿ²eⁿ⁻¹²e_xxⁿ¹²e_xx^{n 2}eⁿ⁻¹_xx ²

2τrⁿ²

≤Cτ

DⁿDⁿ⁻¹

2τrⁿ²,

3.27

that is,

1−Cτ

Dⁿ−Dⁿ⁻¹ ≤2CτDⁿ⁻¹2τrⁿ². 3.28

Ifτis sufficiently small which satisfies 1−Cτ >0, then

Dⁿ−Dⁿ⁻¹ ≤CτDⁿ⁻¹Cτrⁿ². 3.29

Summing up3.29from 1 ton, we have

Dⁿ ≤D⁰Cτ n

l1

r^l2Cτ n

l1

D^l. 3.30

Choosing a two-order method to computeu¹e.g., by scheme2.2and noticing

τ n

l1

r^l2≤nτmax

1≤l≤n

r^l2≤T·O

τ²h²2

, 3.31