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Applied Numerical Mathematics

www.elsevier.com/locate/apnum

An analysis of the finite-difference method for one-dimensional Klein–Gordon equation on unbounded domain ^✩

Houde Han

, Zhiwen Zhang

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR China

a r t i c l e i n f o a b s t r a c t

Article history:

Received 4 September 2007

Received in revised form 21 October 2008 Accepted 24 October 2008

Available online 6 November 2008 Keywords:

Artificial Boundary Condition (ABC) Unbounded domain

Energy method Fast algorithm

Discrete Artificial Boundary Condition (DABC)

The numerical solution of the one-dimensional Klein–Gordon equation on an unbounded domain is analyzed in this paper. Two artificial boundary conditions are obtained to reduce the original problem to an initial boundary value problem on a bounded computational domain, which is discretized by an explicit difference scheme. The stability and convergence of the scheme are analyzed by the energy method. A fast algorithm is obtained to reduce the computational cost and a discrete artificial boundary condition (DABC) is derived by the Z-transform approach. Finally, we illustrate the efficiency of the proposed method by several numerical examples.

©2008 IMACS. Published by Elsevier B.V. All rights reserved.

1. Introduction

The Klein–Gordon equation arises in relativistic quantum mechanics and field theory, which is of great importance for the high energy physicists [15], and is used to model many different phenomena, including the propagation of dislocations in crystals and the behavior of elementary particles. The one-dimensional Klein–Gordon equation is given by the following partial differential equation:

¯ h²∂²u

∂t² − ¯^h2c2∂²u

∂x² +^m2c4u= ^f(x,t), ∀^x∈R¹, t>0, (1.1)

where u=^u(x,t) represents the wave density at position x and time t, ^h¯ is the Planck constant, c and m are particle velocity and particle mass, respectively.

There are a lot of studies on the numerical solution of initial and initial-boundary problems of the linear or nonlinear Klein–Gordon equation. For example, Khalifa and Elgamal [22] developed a numerical scheme based on a finite element method for the nonlinear Klein–Gordon equation with Dirichlet boundary condition on a bounded domain, which shows the overflow solution as expected. Duncan [6] analyzed three finite difference approximations of the initial nonlinear Klein–

Gordon equation, showed they are directly related to symplectic mappings and tested the schemes on the traveling wave and periodic breather problems over long time intervals. However, when we wish to solve the Klein–Gordon numerically on an unbounded domain, these methods will face essential difficulties. Since the unboundedness of the physical domain in our problem, the standard finite element method or finite difference method can’t be used directly.

✩ This work was supported by The NSFC Project. No. 10471073.

*

E-mail addresses:hhan@math.tsinghua.edu.cn(H. Han),zhangzhiwen02@mails.tsinghua.edu.cn(Z. Zhang).

0168-9274/$30.00©2008 IMACS. Published by Elsevier B.V. All rights reserved.

doi:10.1016/j.apnum.2008.10.005

The artificial boundary condition (ABC) method is a powerful approach to reduce the problems on the unbounded domain to a bounded computational domain. In the well-known paper of Engquist and Majda [12], absorbing boundary conditions using Padé approximation of a pseudo-differential operator on a line-type boundary for the wave equation are derived.

Higdon [20,21] developed radiation boundary conditions for the numerical modeling of dispersive waves. By specifying the wavenumber and frequency parameters, the boundary conditions based on compositions of simple first-order differen- tial operators was got. His formulas can be applied without modification to higher-dimensional problem. Low-order local ABCs may have low accuracy, whereas high-order local ABCs are usually hard to implement because they typically in- volve high-order derivatives [13]. The local ABCs may generate some nonphysical reflection at the artificial boundary and the well-posedness of the resulting truncated initial-boundary problem is still open in general. Nonlocal artificial bound- ary conditions have the potential of being more accurate than the local ones. Han and Zheng [19] obtained three kinds of exact nonreflecting boundary conditions for exterior problems of wave equations in two and three-dimensional space by an approach based on Duhamel’s principle. X. Antoine and C. Besse [2] obtained a nonreflecting boundary conditions for the one-dimensional Schrödinger equation. X. Antoine, C. Besse and V. Mouysset [3] also generalized their approach to simulate the two-dimensional Schrödinger equation using nonreflecting boundary conditions. Han and Huang [16], Han, Yin and Huang [18] derived the exact nonreflecting boundary conditions for two- and three-dimensional Schrödinger equations.

Han and Yin [17] also derived the exact nonreflecting boundary conditions for two- and three-dimensional Klein–Gordon equations. Generally speaking, the exact boundary conditions require more computational cost. In order to overcome this disadvantage, a fast algorithm is given in our paper.

The organization of this article is the following: In Section 2 we introduce two artificial boundaries and find the artificial boundary conditions, then reduce the original problem to an equivalent problem on the bounded computational domain. In Section 3, a finite-difference scheme for the reduced problem is given and its stability and convergence are analyzed. A fast algorithm is obtained in Section 4 to reduce the computational cost. In Section 5, a discrete artificial boundary condition (DABC) is derived by the Z-transform approach. Some numerical results will be given in Section 6 to demonstrate the accuracy and efficiency of the proposed methods.

2. The artificial boundary condition

In this section, we study the numerical approximation of a dispersive wave solutionu(x,t), to the equation with a source term on an unbounded domain. More precisely, we consider the following linear Klein–Gordon equation onR¹× [⁰,T]^:

∂²u

∂t² −^a2∂²u

∂x² +^b2u= ^f(x,t), ∀^x∈R¹, t∈ [⁰,T], (2.1)

u|t=0=ϕ₀(x), ∀^x∈R¹, (2.2)

u_t|t=⁰=ϕ1(x), ∀^x∈R¹. (2.3)

Herea,bare two real constants. Assumeϕ0(x),ϕ1(x)and f(x,t)satisfying: Supp{ϕ0(x)} ⊂ [^xl,x_r]^{, Supp}{ϕ1(x)} ⊂ [^xl,x_r]^and Supp{^f(x,t)} ⊂ [^xl,xr] × [⁰,T]. For simplicity of the deduction, we takex_l= −^{1 andx}r=^1.

In order to reduce the problem (2.1)–(2.3) to a bounded computational domain, we introduce two artificial boundaries, Σ_r=

(x,t)|^x=¹, 0^tT , Σl=

(x,t)|^x= −¹, 0^tT , which divide R¹× [⁰,T]into three parts,

D_l=

(x,t)| −∞^x−¹, 0^tT , Di=

(x,t)| −^1x1, 0^tT , D_r=

(x,t)|^1x+∞, 0^tT .

The bounded domain D_i is our computational domain. Consider the restriction of u(x,t) on the unbounded domain D_r. u(x,t) satisfies,

∂²u

∂t² −^a2∂²u

∂x² +^b2u=⁰, ∀(x,t)∈^Dr, (2.4)

u|Σr=^u(1,t)≡^g1(t), (2.5)

u|t=0=⁰, x¹, (2.6)

u_t|t=0=⁰, x¹. (2.7)

Since u(1,t) is unknown, the problem (2.4)–(2.7) is incomplete, which cannot be solved independently. If u|Σr =^u(1,t)≡ g₁(t)is given, the problem above has a unique solution. Let

U(x,s)=L u(x,t)

= +∞

0

e^−stu(x,t)dt, Res>0,

denotes the Laplace transform of the unknown solutionu(x,t). By (2.4)–(2.7) it satisfies,

s²U(x,s)−^a2Uxx(x,s)+^b2U(x,s)=⁰, 1<x<+∞, (2.8) U(1,s)=^G(s)≡L

g1(t)

= +∞

0

e^−stg1(t)dt, (2.9)

U(x,s)<+∞, x→ +∞. (2.10)

Eq. (2.8) is a second-order linear ODE with constant coefficient, its general solution is given by U(x,s)=^c1(s)exp

−

√s²+^b2

a (x−¹) +^c2(s)exp √

s²+^b2

a (x−¹) . The condition (2.10) implies c₂(s)≡0, and we obtain

U(x,s)=^c1(s)exp

−

√s²+^b2

a (x−¹) , (2.11)

here the roots with positive real parts are taken. The partial derivative with respect tox yields,

∂U(x,s)

∂x = −

√s²+^b2

a c1(s)exp

−

√s²+^b2

a (x−¹) . (2.12)

Combining (2.11) and (2.12) on the artificial boundary Σ_r, we arrive at

∂U(1,s)

∂x = −

√s²+^b2

a U(1,s)= − ¹ a√

s²+^b2

s²U(1,s)+^b2U(1,s)

. (2.13)

By the table of Laplace transform (see page 1108 of [14]), we obtain L⁻¹

1

√s²+^b2 = ^J0(bt),

L⁻¹

s²U(1,s)+^b2U(1,s)

= ∂²u(1,t)

∂t² +^b2u(1,t).

Then, by the convolution theorem of Laplace transforms, from (2.13) we obtain:

∂u(1,t)

∂x = −¹_a t

0

J₀(bt−^bτ)

∂²u(1,τ)

∂τ^{2 +b}

2u(1,τ)

dτ

= −¹_a∂u(1,t)

∂t −^b_a t

0

J₀(bt−^bτ)∂u(1,τ)

∂t dτ₋^b

2

a t

0

J₀(bt−^bτ)u(1,τ)dτ

= −¹_a∂u(1,t)

∂t −^b_a² t

0

J₀(bt−^bτ)+ ^J0(bt−^bτ)

u(1,τ)dτ.

We get the artificial boundary condition of the problem (2.1)–(2.3) on Σ_r. Similarly we can get the artificial boundary condition onΣ_l, witht∈ [⁰,T]. Hence, we can reduce the initial boundary value problem on the computational domain D_i.

∂²u

∂t² −^a2∂²u

∂x² +^b2u= ^f(x,t), ∀(x,t)∈ [−¹,1] × [⁰,T], (2.14)

u|t=0=ϕ₀(x), u_t|t=0=ϕ₁(x), ∀^x∈ [−¹,1], (2.15)

∂u(1,t)

∂t +^a∂u(1,t)

∂x = −^b2 t

0

J₀(bt−^bτ)+ ^J0(bt−^bτ)

u(1,τ)dτ, (2.16)

∂u(−¹,t)

∂t −^a∂u(−¹,t)

∂x = −^b2 t

0

J₀(bt−^bτ)+ ^J0(bt−^bτ)

u(−¹,τ)dτ. (2.17)

Let J(x)=^J0(x)+^J0(x), whereJ(x)is a special function. By some basic recursion formulas of Bessel functions (see page 242 of [1]), we have

J(x):=^J0(x)+^J0(x)=^J0(x)−

J₁(x)= ¹ 2

J0(x)+^J2(x) .

The artificial boundary conditions (2.16), (2.17) will have two simple forms:

∂u(1,t)

∂t +^a∂u(1,t)

∂x = −^b2 t

0

J(bt−^bτ)u(1,τ)dτ, (2.18)

∂u(−¹,t)

∂t −^a∂u(−¹,t)

∂x = −^b2 t

0

J(bt−^bτ)u(−¹,τ)dτ. (2.19)

Moreover, letF(x) denotes the primitive ofJ(x):

F(x)= x

0

J(s)ds= ∞ k=0

(−¹)^k k!^k!(2k+²)

x 2

2k+1

+ ∞ k=0

(−¹)^kx k!(k+²)!(2k+³)

x 2

2k+3

.

Integrating by parts in Eqs. (2.18) and (2.19), we obtain anther two equivalent forms of boundary conditions. The new forms will bring some convenience for the proof of stability and convergence.

∂u(1,t)

∂t +^a∂u(1,t)

∂x = −^b t

0

F(bt−^bτ)∂u(1,τ)

∂τ ^dτ. (2.20)

∂u(−¹,t)

∂t −^a∂u(−¹,t)

∂x = −^b

t

0

F(bt−^bτ)∂u(−¹,τ)

∂τ ^dτ. (2.21)

Next, we discuss the uniqueness and stability estimate of the reduced problem (2.14)–(2.17). Multiply Eq. (2.1) by ^∂_∂^u_t and integrate with respect tox∈ [−¹,1]^{for fixedt}∈(0,T], using the boundary condition (2.16), (2.17), we get

d dt

1 2 1

−¹

∂u(x,t)

∂t

2

+^a2

∂u(x,t)

∂x

2

+^b2u2(x,t)

dx

+^a2∂u(1,t)

∂t B u(1,t)

+^a2∂u(−¹,t)

∂t B

u(−¹,t)

= 1

−¹

f(x,t)∂u(x,t)

∂t dx. (2.22)

Here B

ν(t)

=¹ a

dν(t) dt +^b2

a t

0

J₀(bt−^bτ)+ ^J0(bt−^bτ) ν(τ)dτ.

We introduce two auxiliary functions W⁽¹⁾(x,t) and W⁽²⁾(x,t), which satisfy the following problems, respectively:

W_tt⁽¹⁾−^a2Wxx⁽¹⁾+^b2W(1)=⁰, ∀(x,t)∈^Dr, W⁽¹⁾|Σr=^u(1,t), 0^tT,

W⁽¹⁾|t=0=⁰, 1<x<+∞, W_t⁽¹⁾|t=0=⁰, 1<x<+∞

and

W_tt⁽²⁾−^a2Wxx⁽²⁾+^b2W(2)=⁰, ∀(x,t)∈^Dl, W⁽²⁾|Σ_l=^u(−¹,t), 0^tT,

W⁽²⁾|t=0=⁰, −∞<x<−¹, W_t⁽²⁾|t=0=⁰, −∞<x<−¹.

Through a similar analysis, we obtain,

a²∂u(1,t)

∂t B u(1,t)

=_dt^d 1

2 +∞

1

∂W⁽¹⁾(x,t)

∂t

2

+^a2

∂W⁽¹⁾(x,t)

∂x

2

+^b2

W⁽¹⁾(x,t)2 dx

, (2.23)

and

a²∂u(−¹,t)

∂t B

u(−¹,t)

= ^d dt

1 2

−¹

−∞

∂W⁽²⁾(x,t)

∂t

2

+^a2

∂W⁽²⁾(x,t)

∂x

2

+^b2

W⁽²⁾(x,t)2 dx

. (2.24)

We introduceE(t)and F(t) as following, E(t)= ¹

2 1

−1

∂u(x,t)

∂t

2

+^a2

∂u(x,t)

∂x

2

+^b2u2(x,t)

dx

+ ¹₂

+∞

1

∂W⁽¹⁾(x,t)

∂t

2

+^a2

∂W⁽¹⁾(x,t)

∂x

2

+^b2

W⁽¹⁾(x,t)2 dx

+ ¹ 2

−¹

−∞

∂W⁽²⁾(x,t)

∂t

2

+^a2

∂W⁽²⁾(x,t)

∂x

2

+^b2

W⁽²⁾(x,t)2

dx, (2.25)

F(t)= ¹₂ 1

−¹

f(x,t)2

dx. (2.26)

Combining (2.22)–(2.26) and using Cauchy–Schwarz inequality, we obtain, d

dtE(t)E(t)+F(t), 0^tT. (2.27)

Using the Gronwall inequality from inequality (2.27) and noticing that, E(0)=¹

2 1

−¹

ϕ1(x)2

+^a2 ϕ₁(x)2

+^b2

ϕ0(x)2 dx.

we get the stability estimate for the solution of the reduced problem (2.14)–(2.17):

1

−1

∂u(x,t)

∂t

2

+^a2

∂u(x,t)

∂x

2

+^b2u2(x,t)

dx

1

−¹

ϕ₁(x)2

+^a2 ϕ₁(x)2

+^b2

ϕ₀(x)2 dx+

t

0

1

−¹ e^t−τ

f(x,τ)2

dx dτ. (2.28)

Then, we obtain the following stability estimate of the reduced problem (2.14)–(2.17).

Theorem 2.1.The reduced problem(2.14)–(2.17)has at most one solution u(x,t)on the bounded computational domain Di, and u(x,t)continuously depends on the initial value{ϕ0(x),ϕ1(x)}^{, and f}(x,t).

From Theorem 2.1, we know that the reduced problem (2.14)–(2.17) is equivalent to the original problem (2.1)–(2.3).

Namely, the solution of the reduced problem (2.14)–(2.17) is the restriction of the solution of original problem (2.1)–(2.3) on the bounded domain D_i, vice versa.

3. Analysis of the difference scheme

In this section, we consider the finite difference approximation of the reduced problem (2.14)–(2.17) on the bounded domain D_i. We divide the domain D_i by a set of lines parallel to thex- and t-axes to form a grid. We writeh=¹/I and τ₌T/N for the line spacings, whereI and N are two positive integers. The crossing pointsΩ_h^τ are called the grid points,

Ω_h^τ=

(x_i,t_n)|^xi= −¹+^ih, i=⁰,1, . . . ,2I; ^tn=ⁿτ, n=⁰,1, . . . ,T/τ.

SupposeU= {^uni |^0i2I,n⁰} is a grid function on Ω_h^τ. For the simplicity, assume the constantsa=^b=1. Introduce the following notations [24]:

uⁿ_i−1/2= ¹₂

uⁿ_i−1+^uni

, uⁿ_i^−1/2=¹₂

uⁿ_i +^uni⁻¹

,

δxuⁿ_i−1/2= ¹ h

uⁿ_i −^un_i−1

, δtuⁿ_i^−1/2= ¹ τ

uⁿ_i −^un_i⁻¹ , δ⁰_xuⁿ_i = ¹

2h

uⁿ_i+1−^un_i−1

, δ_t⁰uⁿ_i = ¹ 2τ

uⁿ_i⁺¹−^un_i⁻¹ , δ²_xuⁿ_i = ¹

h²

uⁿ_i+1−^2un_i +^un_i−1

, δ_t²uⁿ_i = ¹ τ²

uⁿ_i⁺¹−^2un_i +^un_i⁻¹ .

We denote the value of the solutionu(x,t) at the grid point(xi,tn)byUⁿ_i and f_iⁿ= ^f(xi,tn). Using the Taylor expansion, it follows from (2.14), (2.15), (2.21), (2.20) that:

δ_t²U_iⁿ−δ²_xUⁿ_i +^Uni = ^fiⁿ+^Rni, 1^i2I−¹, n¹, (3.1)

U⁰_i =φ0(ih), 0^i2I, (3.2)

U¹_i =^U0_i +τφ1(ih)+^S1i, 0^i2I, (3.3)

δtUⁿ_2I^+1/2+δxUⁿ_2I−1/2= − n

m=¹

δ⁰_tU_2I^m

mτ

(m−1)τ

F(nτ₋s)ds+^Pn2I, n², (3.4)

δtUⁿ₀^+1/2−δxUⁿ_1/2= − n

m=1

δ_t⁰U^m₀

mτ

(m−1)τ

F(nτ₋s)ds+^Q0ⁿ, n². (3.5)

If the solutionu(x,t)is smooth enough, there exists a constant C, such that Rⁿ_i^C

h²+τ², S¹_i^Cτ², Pⁿ_2I^C(h+τ), Q₀^nC(h+τ).

Omitting the truncation errors in (3.1)–(3.5), we construct a finite difference scheme of the reduced problem (2.14)–(2.17):

δ_t²uⁿ_i −δ²_xuⁿ_i +^uni = ^fiⁿ, 1^i2I−¹, n¹, (3.6)

u⁰_i =φ0(ih), 0^i2I, (3.7)

u¹_i =^u0i +τφ1(ih), 0^i2I, (3.8)

δtuⁿ_2I^+1/2+δxuⁿ_2I−1/2= − n

m=¹

δ⁰_tu^m_2I

mτ

(m−1)τ

F(nτ₋s)ds, n², (3.9)

δtuⁿ₀^+1/2−δxuⁿ_1/2= − n

m=1

δ_t⁰u^m₀

mτ

(m−¹)τ

F(nτ₋s)ds, n². (3.10)

This is an explicit scheme with global boundary conditions. In the following, we consider the stability and convergence of the scheme. Multiplying (3.6) by 2hδ⁰_tuⁿ_i and summing up forifrom 1 to 2I−1, we obtain

2I−1

i=¹

2hδ_t⁰uⁿ_i

δ_t²uⁿ_i −δ_x²uⁿ_i +^un_i

=

2I−1

i=¹

2hδ⁰_tuⁿ_i f_iⁿ. (3.11)

After some calculation, we get

2I−1

i=¹

2hδ_t⁰uⁿ_iδ_t²uⁿ_i = ^h τ

2I−1

i=¹

δtuⁿ_i^+1/22

−

δtuⁿ_i^−1/22

. (3.12)

Using the summation by parts formula and the boundary condition (3.9)–(3.10), we have

−

2I−¹

i=1

2hδ⁰_tuⁿ_iδ²_xuⁿ_i =^h τ

2I−¹

i=0

δ_xuⁿ_i+⁺₁¹_/2δ_xuⁿ_i+1/2−^h τ

2I−¹

i=0

δ_xuⁿ_i+1/2δ_xuⁿ_i+⁻₁¹_/2+²δ_xuⁿ_1/2δ_t⁰uⁿ₀−²δ_xuⁿ_2I−1/2δ_t⁰uⁿ_2I

=^h τ

2I−¹

i=0

δ_xuⁿ_i+⁺₁¹_/2δ_xuⁿ_i+1/2−^h τ

2I−¹

i=0

δ_xuⁿ_i+1/2δ_xuⁿ_i+⁻₁¹_/2

+²δ⁰_tuⁿ₀

δtuⁿ₀^+1/2+ n

m=¹

δ_t⁰u^m₀

mτ

(m−1)τ

F(nτ₋s)ds

+²δ⁰_tuⁿ_2I

δtuⁿ_2I^+1/2+ n

m=¹

δ_t⁰u^m_2I

mτ

(m−1)τ

F(nτ₋s)ds

≡^I1+^I2+^I3. (3.13)

Notice that, I₂=²δ⁰_tuⁿ₀

δtuⁿ₀^+1/2+ n

m=1

δ_t⁰u^m₀

mτ

(m−1)τ

F(nτ₋s)ds

=²δ⁰_tuⁿ₀

δ_t⁰uⁿ₀+ n

m=1

δ_t⁰u^m₀

mτ

(m−¹)τ

F(nτ₋s)ds

+²δ_t⁰uⁿ₀

δtuⁿ₀^+1/2−δ_t⁰uⁿ₀

≡^I2^A+^I2^B. Similarly,

I₃≡^I3^A+^I3^B.

Using the linear interpolation to construct a continuous function ^u˜(1,t), which satisfies ^u˜t(1,t)=δ_t⁰u^m₀, t∈ [^mτ₋τ,mτ). According to the stability estimate of the continuous case in Section 2, we obtain

I₂^A=²δ_t⁰uⁿ₀

δ⁰_tuⁿ₀+ n

m=¹

δ⁰_tu^m₀

mτ

(m−1)τ

F(nτ₋s)ds

⁰. (3.14)

It is easy to check, I₂^B=²δ_t⁰uⁿ₀

δtuⁿ₀^+1/2−δ_t⁰uⁿ₀

=¹ 2

δtuⁿ₀^+1/22

−

δtuⁿ₀^−1/22 . By the same technique, we obtain that I₃^A is also nonnegative, and

I₃^B=²δ_t⁰uⁿ_2I

δ_tuⁿ_2I^+1/2−δ_t⁰uⁿ_2I

=¹₂

δ_tuⁿ_2I^+1/22

−

δ_tuⁿ_2I^−1/22 . The third term in the left of Eq. (3.11) is easy,

2I−1

i=¹

2hδ_t⁰uⁿ_iuⁿ_i = ^h τ

2I−1

i=¹

uⁿ_iuⁿ_i⁺¹− ^h τ

2I−1

i=¹

uⁿ_i⁻¹uⁿ_i. (3.15)

We introduce an auxiliary quantity E˜, E˜≡ ^s

2h

δtuⁿ₀^+1/22

+^h

2I−1

i=¹

δtuⁿ_i^+1/22

+ ^s 2h

δtuⁿ_2I^+1/22

+^h

2I−1

i=⁰

δxuⁿ_i+⁺₁¹_/2δxuⁿ_i+1/2+^h

2I−1

i=¹ uⁿ_iuⁿ_i⁺¹

= ^s 2h

δtuⁿ₀^+1/22

+^h

2I−1

i=¹

δtuⁿ_i^+1/22

+ ^s 2h

δtuⁿ_2I^+1/22

+^h

2I−1

i=⁰

δxuⁿ_i+⁺₁¹_/^/₂²2

−^h 4

2I−1

i=⁰

δxuⁿ_i+⁺₁¹_/2−δxuⁿ_i+1/22

+^h

2I−1

i=¹

uⁿ_i^+1/22

−^hτ² 4

2I−1

i=¹

δtuⁿ_i^+1/22

=₂^sh

δtuⁿ₀^+1/22

+

1−^s2−τ² 4 h

2I−¹

i=1

δtuⁿ_i^+1/22

+₂^sh

δtuⁿ_2I^+1/22

+^s2h

2I−¹

i=1

δtuⁿ_i^+1/22

−^hs₄²

2I−¹

i=0

δ_tuⁿ_i+⁺₁¹_/2−δ_tuⁿ_i+1/22

+^h

2I−¹

i=0

δ_xuⁿ_i+⁺₁¹_/^/₂²2

+^h

2I−¹

i=1

uⁿ_i^+1/22

αh 2I

i=0

δ_tuⁿ_i^+1/22

+^h

2I−¹

i=0

δ_xuⁿ_i+⁺₁¹_/^/₂²2

+^h

2I−¹

i=1

uⁿ_i^+1/22

>0, (3.16)