Symmetric Regularized Long Wave Equations with Damping Term

Boundary Value Problems

Volume 2010, Article ID 781750,16pages doi:10.1155/2010/781750

Research Article

A Linear Difference Scheme for Dissipative

Symmetric Regularized Long Wave Equations with Damping Term

Jinsong Hu,

Youcai Xu,

and Bing Hu

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

2School of Mathematics, Sichuan University, Chengdu 610064, China

Correspondence should be addressed to Youcai Xu,[email protected] Received 24 August 2010; Accepted 14 November 2010

Academic Editor: V. Shakhmurov

Copyrightq2010 Jinsong Hu et al. 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.

We study the initial-boundary problem of dissipative symmetric regularized long wave equations with damping term by finite difference method. A linear three-level implicit finite difference scheme is designed. Existence and uniqueness of numerical solutions are derived. It is proved that the finite difference scheme is of second-order convergence and unconditionally stable by the discrete energy method. Numerical simulations verify that the method is accurate and efficient.

1. Introduction

A symmetric version of regularized long wave equationSRLWE, utρxuux−uxxt0,

ρtux0, 1.1

has been proposed to model the propagation of weakly nonlinear ion acoustic and space charge waves1 . The sec²solitary wave solutions are

ux, t 3 v²−1

v sec²1 2

v²−1

v² x−vt, ρx, t 3

v²−1 v² sec²1

2

v²−1

v² x−vt.

1.2

The four invariants and some numerical results have been obtained in 1 , where vis the velocity,v²>1. Obviously, eliminatingρfrom1.1, we get a class of SRLWE:

utt−uxx1 2

u²

xt−uxxtt0. 1.3

Equation1.3is explicitly symmetric in thex andt derivatives and is very similar to the regularized long wave equation that describes shallow water waves and plasma drift waves 2, 3 . The SRLW equation also arises in many other areas of mathematical physics 4–6 . Numerical investigation indicates that interactions of solitary waves are inelastic7 ; thus, the solitary wave of the SRLWE is not a solution. Research on the wellposedness for its solution and numerical methods has aroused more and more interest. In8 , Guo studied the existence, uniqueness, and regularity of the numerical solutions for the periodic initial value problem of generalized SRLW by the spectral method. In9 , Zheng et al. presented a Fourier pseudospectral method with a restraint operator for the SRLWEs and proved its stability and obtained the optimum error estimates. There are other methods such as pseudospectral method, finite difference method for the initial-boundary value problem of SRLWEssee9–

15 . In applications, the viscous damping effect is inevitable, and it plays the same important role as the dispersive effect. Therefore, it is more significant to study the dissipative symmetric regularized long wave equations with the damping term

utρx−υuxxuux−uxxt0, 1.4

ρtuxγρ0, 1.5

whereυ, γare positive constants,υ >0 is the dissipative coefficient, andγ >0 is the damping coefficient. Equations 1.4-1.5are a reasonable model to render essential phenomena of nonlinear ion acoustic wave motion when dissipation is considered. Existence, uniqueness, and wellposedness of global solutions to 1.4-1.5 are presented see 16–20 . But it is difficult to find the analytical solution to 1.4-1.5, which makes numerical solution important.

To authors’ knowledge, the finite difference method to dissipative SRLWEs with damping term1.4-1.5has not been studied till now. In this paper, we propose linear three level implicit finite difference scheme for1.4-1.5with

ux,0 u0x, ρx,0 ρ0x, x∈xL, xR , 1.6

and the boundary conditions

uxL, t uxR, t 0, ρxL, t ρxR, t 0, t∈0, T . 1.7 We show that this difference scheme is uniquely solvable, convergent, and stable in both theoretical and numerical senses.

Lemma 1.1. Suppose thatu0∈H¹,ρ0∈L2, the solution of1.4–1.7satisfiesuL2 ≤C,uxL2 ≤ C,ρL2≤C, anduL∞ ≤C, whereCis a generic positive constant that varies in the context.

Proof. Let

Et u²_L2ux²_L2ρ²

L2 _xR

xL

u²dx _xR

xL

ux²dx _xR

xL

ρ²dx, t∈0, T . 1.8

Multiplying1.4byuand integrating overxL, xR , we have _xR

xL

uutuρx−υuuxxu²ux−uuxxt

dx0. 1.9

According to

_xR

xL

uutdx 1 2

d dt

_xR

xL

u²dx, _xR

xL

uρxdxuρ|^x_x^R_L− _xR

xL

ρdu− _xR

xL

uxρdx,

− _xR

xL

uuxxdx−uux|^x_x^R_{L xR}

xL

uxdu _xR

xL

ux²dx, _xR

xL

u²uxdx 1

3u³|^x_x^R_L 0,

− _xR

xL

uuxxtdx−uuxt|^x_x^R_{L xR}

xL

uxtdu 1 2

d dt

_xR

xL

ux²dx,

1.10

we get

d dt

_xR

xL

u²u²_x dx−2

_xR

xL

uxρdx2υ _xR

xL

ux²dx0, 1.11

Then, multiplying1.5byρand integrating overxL, xR , we have _xR

xL

ρρtρuxγρ²

dx0. 1.12

By

_xR

xL

ρρtdx 1 2

d dt

_xR

xL

ρ²dx, 1.13

we get

d dt

_xR

xL

ρ²dx2 _xR

xL

uxρdx2γ _xR

xL

ρ²dx0. 1.14

Adding1.14to1.11, we obtain

d dt

_xR

xL

u²u²_xρ²

dx−2υ _xR

xL

ux²dx−2γ _xR

xL

ρ²dx≤0. 1.15

SoEtis decreasing with respect tot, which implies thatEt u²_L2ux²_L2ρ²_L2≤E0, t ∈ 0, T . Then, it indicates that uL2 ≤ C,uxL2 ≤ C, andρL2 ≤ C. It is followed from Sobolev inequality thatuL∞ ≤C.

2. Finite Difference Scheme and Its Error Estimation

Lethandτbe the uniform step size in the spatial and temporal direction, respectively. Denote xj xLjhj 0,1,2, . . . , J,tn nτn 0,1,2, . . . , N,N T/τ ,uⁿ_j ≈ uxj, tn,ρⁿ_j ≈ ρxj, tn, andZ⁰_h{u uj|u0uJ 0, j0,1,2, . . . , J}. We define the difference operators as follows:

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⁻¹_j

2τ ,

uⁿ_j uⁿ¹_j uⁿ⁻¹_j

2 , uⁿ, vⁿh

J−1

j0

uⁿ_jv_jⁿ, uⁿ²uⁿ, uⁿ, uⁿ_∞ max

0≤j≤J−1

uⁿ_j. 2.1

Then, the average three-implicit finite difference scheme for the solution of1.4–1.7is as follow:

uⁿ_j

t− uⁿ_j

xx t ρ_jⁿ

x−υ uⁿ_j

xx1 3

uⁿ_j uⁿ_j

x

uⁿ_juⁿ_j

x

0, 2.2

ρ_jⁿ

t

uⁿ_j

xγρⁿ_j 0, 2.3

u⁰_j u0

xj

, ρ⁰_j ρ0

xj

, 0≤j≤J, 2.4

uⁿ₀ uⁿ_J 0, ρⁿ₀ ρⁿ_J 0, 1≤n≤N. 2.5

Lemma 2.1. Summation by parts follows [12,21] that for any two discrete functionsu, v∈Z⁰_h uj

x, vj

−

uj, vj

x

,

vj, uj

xx

−

vj

x, uj

x

. 2.6

Lemma 2.2discrete Sobolev’s inequality 12,21 . There exist two constantsC1 andC2 such that

uⁿ_∞≤C1uⁿC2uⁿ_x. 2.7 Lemma 2.3discrete Gronwall inequality12,21 . Suppose thatwk,ρkare nonnegative functions and ρkis nondecreasing. IfC >0 and

wk≤ρk Cτ k−1

l0

wl. 2.8

Thenwk≤ρke^Cτk.

Theorem 2.4. Ifu0∈H¹,ρ0∈L2, then the solution of 2.2–2.5satisfies

uⁿ ≤C, uⁿ_x ≤C, ρⁿ≤C, uⁿ_∞≤C n1,2, . . . , N. 2.9 Proof. Taking an inner product of 2.2 with 2uⁿ_j i.e., uⁿ¹_j uⁿ⁻¹_j and considering the boundary condition2.5andLemma 2.1, we obtain

1 2τ

uⁿ¹²−uⁿ⁻¹²

1 2τ

uⁿ¹x ²−uⁿ⁻¹_x ²

ρⁿ_j

x,2uⁿ_j

−υ uⁿ_j

xx,2uⁿ_j

P,2uⁿ_j 0,

2.10

whereP 1/3uⁿ_juⁿ_jx uⁿ_juⁿ_jx . Since ρⁿ_j

x,2uⁿ_j −

ρⁿ_j,2 uⁿ_j

x

,

uⁿ_j

xx,2uⁿ_j

−2uⁿ_x²,

P,2uⁿ_j 2

3h J−1 j0

uⁿ_j

uⁿ_j

x

uⁿ_juⁿ_j

x

uⁿ_j

1 12

J−1

j0

uⁿ_j

uⁿ¹_j1 uⁿ⁻¹_j1 −uⁿ¹_j−1 −uⁿ⁻¹_j−1 uⁿ_j1

uⁿ¹_j1 uⁿ⁻¹_j1

−uⁿ_j−1

uⁿ¹_j−1 uⁿ⁻¹_j−1

×

uⁿ¹_j uⁿ⁻¹_j 1

12

J−1

j0

uⁿ_j uⁿ_j1

uⁿ¹_j1 uⁿ⁻¹_j1

uⁿ¹_j uⁿ⁻¹_j

−1 12

J−1 j0

uⁿ_j uⁿ_j−1

uⁿ¹_j uⁿ⁻¹_j

uⁿ¹_j−1 uⁿ⁻¹_j−1 0,

2.11

we obtain 1 2τ

uⁿ¹²−uⁿ⁻¹²

1 2τ

uⁿ¹_x ²−uⁿ⁻¹_x ²

− ρ_jⁿ,2

uⁿ_j

x

2υuⁿ_x²0. 2.12

Taking an inner product of2.3with 2ρⁿ_j i.e., ρⁿ¹_j ρⁿ⁻¹_j , we obtain 1

2τ

ρⁿ¹²−ρⁿ⁻¹²

uⁿ_j

x,2ρⁿ_j

2γρⁿ_j² 0. 2.13

Adding2.12to2.13, we have

uⁿ¹²−uⁿ⁻¹²uⁿ¹x ²−uⁿ⁻¹x ²ρⁿ¹²−ρⁿ⁻¹² 2τ

ρⁿ_j,2 uⁿ_j

x

−

uⁿ_j

x,2ρⁿ_j

−4υτuⁿ_x²−4γτρⁿ_j².

2.14

Since

ρⁿ_j,2 uⁿ_j

x

ρⁿ_j, uⁿ¹_j

x

uⁿ⁻¹_j

x

≤ρⁿ²1 2

uⁿ¹_x ²uⁿ⁻¹_x ²

,

− uⁿ_j

x,2ρⁿ_j −

uⁿ_j

x, ρⁿ¹_j ρ_jⁿ⁻¹

≤ uⁿx²1 2

ρⁿ¹²ρⁿ⁻¹²

.

2.15

Equation2.14can be changed to

uⁿ¹²−uⁿ⁻¹²uⁿ¹_x ²−uⁿ⁻¹_x ²ρⁿ¹²−ρⁿ⁻¹²

≤Cτ

uⁿ¹x ²uⁿ_x²uⁿ⁻¹x ²ρⁿ¹²ρⁿ²ρⁿ⁻¹²

.

2.16

LetAⁿuⁿ¹²uⁿ²uⁿ¹_x ²uⁿ_x²ρⁿ¹²ρⁿ², and2.16is changed to Aⁿ−Aⁿ⁻¹≤Cτ

AⁿAⁿ⁻¹

. 2.17

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

Aⁿ−Aⁿ⁻¹≤CτAⁿ⁻¹. 2.18

Summing up2.18from 1 ton, we have

Aⁿ≤A⁰Cτ n−1

l0

A^l. 2.19

FromLemma 2.3, we obtainAⁿ ≤ C, which implies that,uⁿ ≤ C,uⁿ_x ≤ C, andρⁿ ≤ C.

ByLemma 2.2, we obtainuⁿ_∞≤C.

Theorem 2.5. Assume thatu⁰∈H²,ρ⁰∈H¹, the solution of difference scheme2.2–2.5satisfies:

ρⁿ_x≤C, uⁿ_xx ≤C, uⁿ_x∞≤C, ρⁿ

∞≤C n1,2, . . . , N. 2.20 Proof. Differentiating backward2.2–2.5with respect tox, we obtain

uⁿ_j

x t− uⁿ_j

xxx t ρⁿ_j

x x−υ uⁿ_j

xxx1 3

uⁿ_j uⁿ_j

x

uⁿ_juⁿ_j

x

x 0, 2.21

ρⁿ_j

x t uⁿ_j

xxγ ρⁿ_j

x0, 2.22

u⁰_j

xu0,x

xj

,

ρ⁰_j

xρ0,x

xj

, 0≤j≤J, 2.23

uⁿ₀

x

uⁿ_J

x0,

ρⁿ₀

x

ρⁿ_J

x0, 0≤n≤N. 2.24

Computing the inner product of2.21with 2uⁿ_x i.e., uⁿ¹_x uⁿ⁻¹_x and considering2.24and Lemma 2.1, we obtain

1 2τ

uⁿ¹x ²−uⁿ⁻¹x ²

1 2τ

uⁿ¹xx ²−uⁿ⁻¹xx ²

ρⁿ_{x x},2uⁿ_x

−υ

uⁿ_xxx,2uⁿ_x

R,2uⁿ_x 0,

2.25

whereR 1/3uⁿ_juⁿ_jx uⁿ_juⁿ_{jx x}. It follows fromTheorem 2.4that uⁿ_j≤C

j 0,1,2, . . . , J

. 2.26

By the Schwarz inequality andLemma 2.1, we get R,2uⁿ_x

2 3

uⁿ_j

uⁿ_j

x

uⁿ_juⁿ_j

x

x, uⁿ_x −2

3

uⁿ_j uⁿ_j

x

uⁿ_juⁿ_j

x, uⁿ_xx −2

3h J−1 j0

uⁿ_j

uⁿ_j

x

uⁿ_juⁿ_j

x

uⁿ_j

xx

≤ 2 3Ch

J−1

j0

uⁿ_j

x

· uⁿ_j

xx

≤Cuⁿ_x²uⁿ_xx²

≤C

uⁿ¹x ²uⁿ⁻¹x ²uⁿ¹xx ²uⁿ⁻¹xx ²

.

2.27

Noting that ρⁿ_{x x},2uⁿx

−

2uⁿ_{x x}, ρxⁿ

−

ρⁿx, uⁿ¹_{x x} uⁿ⁻¹_{x x}

≤ρxⁿ²1 2

uⁿ¹xx ²uⁿ⁻¹xx ²

, uⁿ_xxx,2uⁿ_x

−2uⁿ_xx²,

2.28

it follows from2.25that

uⁿ¹_x ²−uⁿ⁻¹_x ²uⁿ¹_xx ²−uⁿ⁻¹_xx ²

≤ −4υτuⁿ_xx²Cτ

uⁿ¹x ²uⁿ⁻¹x ²uⁿ¹xx ²uⁿ⁻¹xx ²ρⁿ_x² .

2.29

Computing the inner product of2.22with 2ρⁿ_xi.e.,ρ_xⁿ¹ρⁿ⁻¹_x and considering2.24and Lemma 2.1, we obtain

1 2τ

ρⁿ¹x ²−ρ_xⁿ⁻¹²

uⁿ_{x x}, ρⁿ_x

2γρⁿ_x²0. 2.30 Since

uⁿ_xx,2ρⁿ_x

uⁿ_xx, ρⁿ¹_x ρⁿ⁻¹_x

≤ uⁿ_xx²1 2

ρ_xⁿ¹²ρⁿ⁻¹_x ²

, 2.31

then2.30is changed to

ρ_xⁿ¹−ρⁿ⁻¹_x ≤ −4γτρⁿ_x²Cτ

uⁿ_xx²ρ_xⁿ¹²ρⁿ⁻¹_x ²

. 2.32

Adding2.29to2.32, we have

uⁿ¹_x ²−uⁿ⁻¹_x ²uⁿ¹_xx ²−uⁿ⁻¹_xx ²ρⁿ¹_x ²−ρ_xⁿ⁻¹²

≤ −4υτuⁿ_xx²−4γτρⁿ_x² Cτ

uⁿ¹x ²uⁿ⁻¹x ²uⁿ_xx²uⁿ¹xx ²uⁿ⁻¹xx ²ρⁿ¹x ²ρⁿ_x²ρxⁿ⁻¹²

≤Cτ

uⁿ¹x ²uⁿ⁻¹_x ²uⁿ_xx²uⁿ¹_xx ²uⁿ⁻¹_xx ²ρⁿ¹_x ²ρ_xⁿ²ρⁿ⁻¹_x ²

. 2.33 LetingBⁿuⁿ¹_x ²uⁿ_x²uⁿ¹_xx ²uⁿ_xx²ρ_xⁿ¹²ρⁿ_x², we obtainBⁿ−Bⁿ⁻¹≤CτBⁿ Bⁿ⁻¹. Choosing suitableτwhich is small enough to satisfy 1−Cτ >0, we get

Bⁿ−Bⁿ⁻¹≤CτBⁿ⁻¹. 2.34

Summing up2.34from 1 ton, we have

Bⁿ≤B⁰Cτ n−1

l0

B^l. 2.35

By Lemma 2.3, we get Bⁿ ≤ C, which implies that ρⁿ_x ≤ C,uⁿ_xx ≤ C. It follows from Theorem 2.4andLemma 2.2thatuⁿ_x∞≤C,ρⁿ_∞≤C.

3. Solvability

Theorem 3.1. The solutionuⁿof2.2–2.5is unique.

Proof. Using the mathematical induction, clearly, u⁰,ρ⁰ are uniquely determined by initial conditions2.4. then select appropriate second-order methodssuch as the C-N Schemes and calculate u¹ and ρ¹ i.e. u⁰, ρ⁰, and u¹, ρ¹ are uniquely determined. Assume that u⁰, u¹, . . . , uⁿandρ⁰, ρ¹, . . . , ρⁿare the only solution, now consideruⁿ¹andρⁿ¹in2.2and 2.3:

1

2τuⁿ¹_j − 1 2τ

uⁿ¹_j

xx−υ 2

uⁿ¹_j

xx1 6

uⁿ_j uⁿ¹_j

x

uⁿ_juⁿ¹_j

x

0, 3.1

1

2τρⁿ¹_j γ

2ρⁿ¹_j 0. 3.2

Taking an inner product of3.1withuⁿ¹, we have 1

2τ

uⁿ¹² 1 2τ

uⁿ¹_x ²υ 2

uⁿ¹_x ²1 6h

J−1 j0

uⁿ_j

uⁿ¹_j

x

uⁿ_juⁿ¹_j

x

uⁿ¹_j 0. 3.3

Since 1 6h

J−1 j0

uⁿ_j

uⁿ¹_j

x

uⁿ_juⁿ¹_j

x

uⁿ¹_j

1 12

J−1 j0

uⁿ_j

uⁿ¹_j1 −uⁿ¹_j−1

uⁿ_j1uⁿ¹_j1 −uⁿ_j−1uⁿ¹_j−1 uⁿ¹_j

1 12

J−1 j0

uⁿ_juⁿ¹_j uⁿ¹_j1 uⁿ_j1uⁿ¹_j uⁿ¹_j1

− 1 12

J−1

j0

uⁿ_j−1uⁿ¹_j−1uⁿ¹_j uⁿ_juⁿ¹_j−1uⁿ¹_j 0,

3.4

then it holds

1 2τ

uⁿ¹²

1

2τ υ 2

uⁿ¹x ² 0. 3.5

Taking an inner product of3.2withρⁿ¹and adding to3.5, we have 1

2τ

uⁿ¹²

1

2τ υ 2

uⁿ¹_x ²

1

2τ γ 2

ρⁿ¹²0, 3.6

which implies that3.1-3.2have only zero solution. So the solutionuⁿ¹_j andρⁿ¹_j of2.2–

2.5is unique.

4. Convergence and Stability

Let vx, t and ∅x, t be the solution of problem 1.4–1.7; that is, v_jⁿ uxj, tn, ∅ⁿ_j ρxj, tn, then the truncation of the difference scheme2.2–2.5is

r_jⁿ vⁿ_j

t− vⁿ_j

xx t

∅ⁿ_j

x−υ vⁿ_j

xx1 3

vⁿ_j vⁿ_j

x

v_jⁿvⁿ_j

x

, 4.1

sⁿ_j

∅ⁿ_j

t

vⁿ_j

xγ∅ⁿ_j. 4.2

Making use of Taylor expansion, it holds|r_jⁿ||sⁿ_j|Oτ²h²ifh, τ → 0.

Theorem 4.1. Assume thatu0 ∈ H¹,ρ0 ∈L2, then the solutionuⁿandρⁿin the senses of norms · _∞and · L², respectively, to the difference scheme2.2–2.5converges to the solution of problem 1.4–1.7and the order of convergence isOτ²h².

Proof. Subtracting2.2from4.1subtracting2.3from4.2, and lettingeⁿ_j v_jⁿ−uⁿ_j,ηⁿ_j

∅ⁿ_j −ρⁿ_j, we have

r_jⁿ eⁿ_j

t− eⁿ_j

xx t η_jⁿ

x−υ eⁿ_j

xxQ, 4.3

sⁿ_j η_jⁿ

t

e_jⁿ

xγηⁿ_j, 4.4

where

Q 1 3

v_jⁿ vⁿ_j

x−uⁿ_j uⁿ_j

x

1

3

v_jⁿvⁿ_j

x− uⁿ_juⁿ_j

x

. 4.5

Computing the inner product of4.3with 2eⁿ, we get

eⁿ¹²−eⁿ⁻¹²e_xⁿ¹²−eⁿ⁻¹_x ²−4υτeⁿ_x² 2τ

r_jⁿ,2eⁿ_j

− ηⁿ_j

x,2eⁿ_j

−

Q,2eⁿ_j .

4.6

According to

− Q,2eⁿ_j

−2 3h

J−1 j0

vⁿ_j

vⁿ_j

x−uⁿ_j uⁿ_j

x

eⁿ_j −2

3h

J−1

j0

vⁿ_jvⁿ_j

x− uⁿ_juⁿ_j

x

eⁿ_j

−2 3h

J−1 j0

vⁿ_j

eⁿ_j

xe_jⁿ uⁿ_j

x

eⁿ_j 2

3h J−1

j0

vⁿ_jvⁿ_j −uⁿ_juⁿ_j eⁿ_j

x

−2 3h

J−1 j0

vⁿ_j

eⁿ_j

xe_jⁿ uⁿ_j

x

eⁿ_j 2

3h J−1

j0

eⁿ_jvⁿ_j uⁿ_jeⁿ_j eⁿ_j

x,

4.7

it follow fromLemma 1.1, Theorems2.4, and2.5that vⁿ_j≤C, vⁿj≤C,

uⁿ_j

x

≤C, uⁿ_j≤C

j0,1,2, . . . , J

. 4.8

By the Schwarz inequality, we obtain

− Q,2eⁿ

≤ 2 3Ch

J−1 j0

eⁿ_j

x

eⁿ_j

·eⁿj 2 3Ch

J−1 j0

eⁿ_jeⁿj

· eⁿ_j

x

≤Ceⁿ_x²eⁿ²eⁿ²

≤C

eⁿ¹²eⁿ²eⁿ⁻¹²eⁿ¹_x ²eⁿ⁻¹_x ²

.

4.9

Since

r_jⁿ,2eⁿ_j

r_jⁿ, eⁿ¹_j eⁿ⁻¹_j

≤ rⁿ²1 2

eⁿ¹²eⁿ⁻¹²

,

− ηⁿ_j

x,2eⁿ_j

ηⁿ_j,2 eⁿ_j

x

≤ηⁿ²1 2

exⁿ¹²eⁿ⁻¹_x ²

,

4.10

it follows from4.9–4.10and4.6that eⁿ¹²−eⁿ⁻¹²eⁿ¹_x ²−eⁿ⁻¹_x ²

≤2τrⁿCτ

eⁿ¹²eⁿ²eⁿ⁻¹²eⁿ¹x ²eⁿ⁻¹x ²η² .

4.11

Computing the inner product of4.4with 2ηⁿ, we obtain ηⁿ¹²−ηⁿ⁻¹²2τ

sⁿ_j,2ηⁿ_j

−2τ eⁿ_j

x,2ηⁿ_j

−2γτηⁿ²

≤Cτ

ηⁿ¹²ηⁿ⁻¹²eⁿ_x²

2τsⁿ².

4.12

Adding4.12to4.11, we have

eⁿ¹²−eⁿ⁻¹²eⁿ¹_x ²−e_xⁿ⁻¹²ηⁿ¹² ηⁿ⁻¹²

≤2τrⁿ²2τsⁿ²Cτ

eⁿ¹²eⁿ²eⁿ⁻¹²eⁿ¹_x ² eⁿ⁻¹x ²eⁿ_x²ηⁿ¹²ηⁿ²ηⁿ⁻¹²

.

4.13

Leting

Dⁿeⁿ²eⁿ¹²eⁿ_x²eⁿ¹x ²ηⁿ²ηⁿ¹², 4.14

we get

Dⁿ−Dⁿ⁻¹≤2τrⁿ²2τsⁿ²Cτ

Dⁿ¹Dⁿ

. 4.15

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

Dⁿ−Dⁿ⁻¹≤CτDⁿ⁻¹Cτrⁿ²Cτsⁿ². 4.16 Summing up4.16from 1 ton, we have

Dⁿ≤D⁰Cτ n

l1

r^l2Cτ n

l1

s^l2Cτ n−1

l0

D^l. 4.17

Select appropriate second-order methodssuch as the C-N Schemes, and calculateu¹ and ρ¹, which satisfies

D⁰O

τ²h²2

. 4.18