# High Efficient Numerical Methods for Viscous and Nonviscous Wave Problems

Authors: Xiujie Lv; Jinggang Qin; Tongke Wang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper is concerned with accurate and efficient numerical methods for solving viscous
and nonviscous wave problems. The paper first introduces a new second-order PR-ADI like
scheme. For an efficient simulation, the scheme is also extended to a high-order compact PRADI
like method. Both of them have the advantages of unconditional stability, less impact of
the perturbing terms on the accuracy, and being convenient to compute the boundary values
of the intermediates. Besides this, the compact scheme has high-order accuracy and costs less
in computational time. Numerical results are presented to show the accuracy and efficiency of
the new algorithms.


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## Body

## 1. Introduction

In this paper, we consider the following differential problem with initial and boundary conditions on domainΩ=[0,1]2:
(1)γ1∂u∂t+γ2∂2u∂t2-a▵u-q∂∂t▵u=S(x,y,t),(x,y)∈Ω,t∈(0,T],(2)u(x,y,0)=u0(x,y),∂u∂t(x,y,0)=ψ(x,y),(x,y)∈Ω,(3)u(x,y,t)=φ(x,y,t),(x,y)∈∂Ω,t∈(0,T],
where γ1,γ2,a, and q are nonnegative constants, and aγ2>γ1q, S denotes the source, and u0,ψ, and φ are given functions, and ∂Ω is the boundary of the domain Ω.Equations (1)–(3) govern various physical phenomena by choosing the coefficients. When γ2=0 and q=0, the equation represents standard heat transfer, and when γ1=0 and q=0, the equation represents standard wave equations. Since there are many efficient algorithms known for them, we exclude their consideration.Some conventional numerical approaches for solving wave equations introduce auxiliary variables to rewrite the equations as first-order hyperbolic systems [1–3]; however, these approaches introduce new unknowns which results in an increase in the number of variables in the discrete problems. Thus, there are advantages in keeping the formulation (1)–(3) involving the second time-derivative and a scalar unknown. It has been the case that it is hard to construct methods combining good stability with high accuracy with this formulation [4].This paper is concerned with numerical solutions to viscous and nonviscous nonhomogeneous wave problems. The wave equation is often solved by explicit time-stepping schemes, which require to choose a time step size sufficiently small to satisfy the stability condition and to reduce numerical dispersion as well. As is well known, the alternating direction implicit (ADI) schemes [3–14] are unconditionally stable and only need to solve a sequence of tridiagonal linear systems. In 1955, Peaceman and Rachford Jr. [15] presented a method to solve two-dimensional parabolic equations. When constructing the difference scheme, they first evaluated the derivative with respect to x implicitly, the derivative with respect to y explicitly and used a time step of Δt/2 to get a scheme, then took another half step with explicit in x and implicit in y to get another scheme. The final two-step scheme, called Peaceman and Rachford (PR) scheme, is unconditionally stable and second-order accurate. Alternative conventional approaches for solving wave equations introduce an auxiliary variable to rewrite the equation as a first-order hyperbolic system. With these approaches one introduces new unknowns, which result in an increase in the number of variables in the discrete problems. Thus, there are good reasons to try to keep the formulation involving the second time-derivative and a scalar unknown. Lim et al. [11] introduce a stable three-level locally one-dimensional (LOD) method for solving (1)–(3), which is second-order in space.In the context of high order finite difference methods, compact schemes feature high-order accuracy and small stencils. Recently, there has been a renewed interest in the development and the application of compact finite difference methods for the numerical solution of the differential equations [8–10, 12, 14]. To obtain satisfactory higher order numerical results with reasonable computational cost, there have been attempts to develop higher order compact ADI methods.In this paper, we propose a new set of ADI methods for viscous and nonviscous wave problems. In Section3, we obtain a new PR ADI-like scheme with truncation error O(Δt2+h2) and analyze the error estimate by discrete energy method. In Section 4, we further extend the method to compact PR ADI-like scheme with truncation error O(Δt2+h4). In Section 5, we provide two numerical examples to illustrate the effectiveness of the scheme. We also compare the two methods with some other schemes and the results show that they have much higher efficiency.


## 2. Notation

The domainΩ is divided into a mesh by points xi=ih,yj=jh(i,j=0,…,N), denoted by Ωh, where h=1/N is the spatial mesh size in both x and y directions. Let tn=nΔt,tn+1/2=(n+1/2)Δt, with Δt being the time increment. Denote the values of u(xi,yj,tn) and S(xi,yj,tn+1/2) by ui,jn and Si,jn+1/2, respectively. Denote also the difference solutions to ui,jn and vi,jn by Ui,jn and Vi,jn, respectively.Assume thatVh={w∣w={wi,j}∈Ωh, and w|∂Ω=0}. For w={wi,j}∈Vh and v={vi,j}∈Vh, denote by
(4)δtvi,jn=vi,jn+1-vi,jnΔt,δxvi,jn=vi+1,jn-vi,jnh,δyvi,jn=vi,j+1n-vi,jnh,δx2vi,jn=vi+1,jn-2vi,jn+vi-1,jnh2,δy2vi,jn=vi,j+1n-2vi,jn+vi,j-1nh2.
For ∀v,w∈Vh, define their discrete inner product and norms as follows:
(5)(v,w)=∑i,j=1N-1‍vi,jwi,jh2,∥v∥=(v,v),∥δxv∥=∑i=0N-1‍∑j=1N-1‍(δxvi,j)2h2,∥δyv∥=∑i=1N-1‍∑j=0N-1‍(δyvi,j)2h2,∥δxδyv∥=∑i=0N-1‍∑j=0N-1‍(δxδyvi,j)2h2,|v|H1=∥δxv∥2+∥δyv∥2.


## 3. The New Second-Order ADI Scheme for Two-Dimensional Problems

In this section, we derive a new second-order ADI method for the numerical solution of the differential problems (1)–(3).


### 3.1. Construction of the New ADI Scheme

Introducingv=γ1u+γ2(∂u/∂t), we can rewrite (1) as follows:
(6)∂v∂t-α▵v-β▵u=S,(x,y)∈Ω,t∈(0,T],(7)v=γ1u+γ2∂u∂t,(8)u(x,y,0)=u0(x,y),v(x,y,0)=γ1u0(x,y)+γ2ψ(x,y),(x,y)∈Ω,(9)u(x,y,t)=φ(x,y,t),v(x,y,t)=γ1φ(x,y,t)+γ2φt(x,y,t),(x,y)∈∂Ω,t∈(0,T],
where α=q/γ2,β=a-(γ1q/γ2).Applying Crank-Nicolson implicit discretization to (6), we have
(10)Vi,jn+1-Vi,jnΔt-α(δx2+δy2)Vi,jn+1+Vi,jn2-β(δx2+δy2)Ui,jn+1+Ui,jn2=Si,jn+1/2,Vi,jn+1+Vi,jn2=γ1Ui,jn+1+Ui,jn2+γ2Ui,jn+1-Ui,jnΔt.From (10), we obtain
(11)Vi,jn+1-Δt2α(δx2+δy2)Vi,jn+1-Δt2β(δx2+δy2)Ui,jn+1=Vi,jn+Δt2α(δx2+δy2)Vi,jn+Δt2β(δx2+δy2)Ui,jn+ΔtSi,jn+1/2,(12)Ui,jn+1=Δtγ1Δt+2γ2(Vi,jn+1+Vi,jn)-γ1Δt-2γ2γ1Δt+2γ2Ui,jn.
Let
(13)λ1=Δtγ1Δt+2γ2,λ2=γ1Δt-2γ2γ1Δt+2γ2,c=α+βλ1,γ=12β(1-λ2),
then (11) can be reorganized as
(14)Vi,jn+1-cΔt2(δx2+δy2)Vi,jn+1=Vi,jn+cΔt2(δx2+δy2)Vi,jn+γΔt(δx2+δy2)Ui,jn+ΔtSi,jn+1/2.
Adding perturbing term (c2Δt2/4)δx2δy2(Vi,jn+1-Vi,jn) to the left hand side of (14), we factor (14) as
(15)(1-cΔt2δx2)(1-cΔt2δy2)Vi,jn+1=(1+cΔt2δx2)(1+cΔt2δy2)Vi,jn+γΔt(δx2+δy2)Ui,jn+ΔtSi,jn+1/2.
Introducing the intermediate variable Vi,jn+1/2, we obtain the PR ADI-like scheme as follows:
(16)(1-cΔt2δx2)Vi,jn+1/2=(1+cΔt2δy2)Vi,jn+γ(Δt2δy2+1c)Ui,jn+Δt2Si,jn+1/2,(17)(1-cΔt2δy2)Vi,jn+1=(1+cΔt2δx2)Vi,jn+1/2+γ(Δt2δy2-1c)Ui,jn+Δt2Si,jn+1/2,(18)Ui,jn+1=λ1(Vi,jn+1+Vi,jn)-λ2Ui,jn.
From (16), (17), we get the boundary equation as follows:
(19)Vi,jn+1/2=Vi,jn+1+Vi,jn2+γcUi,jn-cΔt4δy2(Vi,jn+1-Vi,jn).
But usually we compute the boundary values of the intermediate variable using the following simple equality:
(20)Vi,jn+1/2=Vi,jn+1+Vi,jn2+γcUi,jn,
where i=0,N,j=0,…,N-1.We see that schemes (16)–(20) can solve differential problems (1)–(3). In addition, following the idea of Douglas ([2, 3]), a Douglas-like scheme is as follows:
(21)(1-cΔt2δx2)(Vi,jn+1/2-Vi,jn)=cΔt(δx2+δy2)Vi,jn+γΔt(δx2+δy2)Ui,jn+ΔtSi,jn+1/2,(22)(1-cΔt2δy2)(Vi,jn+1-Vi,jn)=Vi,jn+1/2-Vi,jn.
The intermediate value of Vi,jn+1/2 on the boundary is easy to be obtained from (22). PR ADI-like scheme (16)–(18) and Douglas-like scheme (21)-(22) are all derived from equation (15), but PR ADI-like scheme has more efficiency, which will be found in numerical experiment.


### 3.2. Error Estimate

We have derived a new kind of ADI scheme in Section3.1. We further analyze the error of the scheme starting from (15). Expanding (15), we get
(23)Vi,jn+1-Vi,jnΔt-α(δx2+δy2)Vi,jn+1+Vi,jn2-β(δx2+δy2)Ui,jn+1+Ui,jn2+c2Δt24δx2δy2Vi,jn+1-Vi,jnΔt=Si,jn+1/2.
Discretizing (6) as above and letting ξ=u-U,η=v-V, we obtain the error equations as follows:
(24)ηi,jn+1-ηi,jnΔt-α(δx2+δy2)ηi,jn+1+ηi,jn2-β(δx2+δy2)ξi,jn+1+ξi,jn2+c2Δt24δx2δy2ηi,jn+1-ηi,jnΔt=Ri,jn+1/2,(25)ηi,jn+1+ηi,jn2=γ1ξi,jn+1+ξi,jn2+γ2ξi,jn+1-ξi,jnΔt+R~i,jn+1/2,
where the truncation errors
(26)Ri,jn+1/2=Δt2(124∂3v∂t3-α8(∂4v∂t2∂x2+∂4v∂t2∂y2)c-β8(∂4u∂t2∂x2+∂4u∂t2∂y2)c+c2Δt24∂5v∂t∂x2∂y2)(xi,yj,tn+1/2)-αh212(∂4v∂x4+∂4v∂y4)(xi,yj,tn+1/2)-βh212(∂4u∂x4+∂4u∂y4)(xi,yj,tn+1/2)+O(Δt4+Δt2h2+h4),R~i,jn+1/2=Δt2(18∂2v∂t2-γ18∂2u∂t2-γ224∂3u∂t3)(xi,yj,tn+1/2)+O(Δt4).
Hence there exist positive constants C1,C2 such that
(27)|Ri,jn+1/2|≤C1(Δt2+h2),|R~i,jn+1/2|≤C2Δt2.Lemma 1 (see [1]).
Forw∈Vh, the following equalities hold:
(28)-∑i=1N-1‍∑j=1N-1‍wi,j(δx2wi,j)h2=∑i=0N-1‍∑j=1N-1‍(δxwi,j)2h2=∥δxw∥2,-∑i=1N-1‍∑j=1N-1‍wi,j(δy2wi,j)h2=∑i=1N-1‍∑j=0N-1‍(δywi,j)2h2=∥δyw∥2.From Lemma1 and Cauchy-Schwarz inequality, multiplying ((ηi,jn+1+ηi,jn)/2)h2 to both sides of (24), computing the inner product, it is easy to follow that
(29)I1=(ηn+1-ηnΔt,ηn+1+ηn2)=12Δt(∥ηn+1∥2-∥ηn∥2),I21=-(αδx2ηn+1+ηn2,ηn+1+ηn2)=α∥δxηn+1+ηn2∥2,I22=-(αδy2ηn+1+ηn2,ηn+1+ηn2)=α∥δyηn+1+ηn2∥2,I31=-(βδx2ξn+1+ξn2,ηn+1+ηn2)=-β(δx2ξn+1+ξn2,ηn+1+ηn2)=-β(δx2ξn+1+ξn2,cγ1ξn+1+ξn2+γ2ξn+1-ξnΔt+R~n+1/2)=βγ1∥δxξn+1+ξn2∥2+βγ22Δt(∥δxξn+1∥2-∥δxξn∥2)-[-β(δxξn+1+ξn2,δxR~n+1/2)]≥βγ22Δt(∥δxξn+1∥2-∥δxξn∥2)-∥βδxξn+1+ξn2∥∥δxR~n+1/2∥≥βγ22Δt(∥δxξn+1∥2-∥δxξn∥2)-β4(2∥δxξn+1∥2+2∥δxξn∥2+∥δxR~n+1/2∥2),I32≥βγ22Δt(∥δyξn+1∥2-∥δyξn∥2)-β4(2∥δyξn+1∥2+2∥δyξn∥2+∥δyR~n+1/2∥2),I4=(c2Δt24δx2δy2ηn+1-ηnΔt,ηn+1+ηn2)=12ΔtcΔt24(∥δxδyηn+1∥2-∥δxδyηn∥2),I5=(Rn+1/2,ηn+1+ηn2)≤14∥Rn+1/2∥2+12(∥ηn+1∥2+∥ηn∥2).
Multiplying 2Δt to both sides of the result derived from (29), we have
(30)(∥ηn+1∥2-∥ηn∥2)+2αΔt(∥δx(ηn+1+ηn)∥2+∥δy(ηn+1+ηn)∥2)+βγ2(∥δxξn+1∥2+∥δyξn+1∥2-∥δxξn∥2-∥δyξn∥2)+cΔt24(∥δxδyηn+1∥2-∥δxδyηn∥2)≤Δt2(β∥δxR~n+1/2∥2+β∥δyR~n+1/2∥2+∥Rn+1/2∥2)+Δt(∥ηn+1∥2+∥ηn∥2)+β(∥δxξn+1∥2+∥δxξn∥2+∥δyξn+1∥2+∥δyξn∥2).
Summing for n to both sides of (30), we obtain
(31)∥ηn∥2+βγ2(∥δxξn∥2+∥δyξn∥2)≤Δt2∑l=1n-1‍(∥Rl+1/2∥2+β∥δxR~l+1/2∥2+β∥δyR~l+1/2∥2)+Δt∑l=1n-1‍(2∥ηl∥2+β∥δxξl∥2+β∥δyξl∥2),
noticing that ξ0=η0=0, by Gronwall Lemma, we obtain the following theorem.Theorem 2.
Assume thatu,v are the accurate solutions of (6)–(9) with sufficient smoothness and U,V are the difference solutions of (16)–(18). Let ξ=u-U,η=v-V, then there exists a positive constant K independent of Δt and h, such that
(32)maxn≤[T/Δt](∥ηn∥+∥δxξn∥+∥δyξn∥)≤K(Δt2+h2).


## 3.1. Construction of the New ADI Scheme

Introducingv=γ1u+γ2(∂u/∂t), we can rewrite (1) as follows:
(6)∂v∂t-α▵v-β▵u=S,(x,y)∈Ω,t∈(0,T],(7)v=γ1u+γ2∂u∂t,(8)u(x,y,0)=u0(x,y),v(x,y,0)=γ1u0(x,y)+γ2ψ(x,y),(x,y)∈Ω,(9)u(x,y,t)=φ(x,y,t),v(x,y,t)=γ1φ(x,y,t)+γ2φt(x,y,t),(x,y)∈∂Ω,t∈(0,T],
where α=q/γ2,β=a-(γ1q/γ2).Applying Crank-Nicolson implicit discretization to (6), we have
(10)Vi,jn+1-Vi,jnΔt-α(δx2+δy2)Vi,jn+1+Vi,jn2-β(δx2+δy2)Ui,jn+1+Ui,jn2=Si,jn+1/2,Vi,jn+1+Vi,jn2=γ1Ui,jn+1+Ui,jn2+γ2Ui,jn+1-Ui,jnΔt.From (10), we obtain
(11)Vi,jn+1-Δt2α(δx2+δy2)Vi,jn+1-Δt2β(δx2+δy2)Ui,jn+1=Vi,jn+Δt2α(δx2+δy2)Vi,jn+Δt2β(δx2+δy2)Ui,jn+ΔtSi,jn+1/2,(12)Ui,jn+1=Δtγ1Δt+2γ2(Vi,jn+1+Vi,jn)-γ1Δt-2γ2γ1Δt+2γ2Ui,jn.
Let
(13)λ1=Δtγ1Δt+2γ2,λ2=γ1Δt-2γ2γ1Δt+2γ2,c=α+βλ1,γ=12β(1-λ2),
then (11) can be reorganized as
(14)Vi,jn+1-cΔt2(δx2+δy2)Vi,jn+1=Vi,jn+cΔt2(δx2+δy2)Vi,jn+γΔt(δx2+δy2)Ui,jn+ΔtSi,jn+1/2.
Adding perturbing term (c2Δt2/4)δx2δy2(Vi,jn+1-Vi,jn) to the left hand side of (14), we factor (14) as
(15)(1-cΔt2δx2)(1-cΔt2δy2)Vi,jn+1=(1+cΔt2δx2)(1+cΔt2δy2)Vi,jn+γΔt(δx2+δy2)Ui,jn+ΔtSi,jn+1/2.
Introducing the intermediate variable Vi,jn+1/2, we obtain the PR ADI-like scheme as follows:
(16)(1-cΔt2δx2)Vi,jn+1/2=(1+cΔt2δy2)Vi,jn+γ(Δt2δy2+1c)Ui,jn+Δt2Si,jn+1/2,(17)(1-cΔt2δy2)Vi,jn+1=(1+cΔt2δx2)Vi,jn+1/2+γ(Δt2δy2-1c)Ui,jn+Δt2Si,jn+1/2,(18)Ui,jn+1=λ1(Vi,jn+1+Vi,jn)-λ2Ui,jn.
From (16), (17), we get the boundary equation as follows:
(19)Vi,jn+1/2=Vi,jn+1+Vi,jn2+γcUi,jn-cΔt4δy2(Vi,jn+1-Vi,jn).
But usually we compute the boundary values of the intermediate variable using the following simple equality:
(20)Vi,jn+1/2=Vi,jn+1+Vi,jn2+γcUi,jn,
where i=0,N,j=0,…,N-1.We see that schemes (16)–(20) can solve differential problems (1)–(3). In addition, following the idea of Douglas ([2, 3]), a Douglas-like scheme is as follows:
(21)(1-cΔt2δx2)(Vi,jn+1/2-Vi,jn)=cΔt(δx2+δy2)Vi,jn+γΔt(δx2+δy2)Ui,jn+ΔtSi,jn+1/2,(22)(1-cΔt2δy2)(Vi,jn+1-Vi,jn)=Vi,jn+1/2-Vi,jn.
The intermediate value of Vi,jn+1/2 on the boundary is easy to be obtained from (22). PR ADI-like scheme (16)–(18) and Douglas-like scheme (21)-(22) are all derived from equation (15), but PR ADI-like scheme has more efficiency, which will be found in numerical experiment.


## 3.2. Error Estimate

We have derived a new kind of ADI scheme in Section3.1. We further analyze the error of the scheme starting from (15). Expanding (15), we get
(23)Vi,jn+1-Vi,jnΔt-α(δx2+δy2)Vi,jn+1+Vi,jn2-β(δx2+δy2)Ui,jn+1+Ui,jn2+c2Δt24δx2δy2Vi,jn+1-Vi,jnΔt=Si,jn+1/2.
Discretizing (6) as above and letting ξ=u-U,η=v-V, we obtain the error equations as follows:
(24)ηi,jn+1-ηi,jnΔt-α(δx2+δy2)ηi,jn+1+ηi,jn2-β(δx2+δy2)ξi,jn+1+ξi,jn2+c2Δt24δx2δy2ηi,jn+1-ηi,jnΔt=Ri,jn+1/2,(25)ηi,jn+1+ηi,jn2=γ1ξi,jn+1+ξi,jn2+γ2ξi,jn+1-ξi,jnΔt+R~i,jn+1/2,
where the truncation errors
(26)Ri,jn+1/2=Δt2(124∂3v∂t3-α8(∂4v∂t2∂x2+∂4v∂t2∂y2)c-β8(∂4u∂t2∂x2+∂4u∂t2∂y2)c+c2Δt24∂5v∂t∂x2∂y2)(xi,yj,tn+1/2)-αh212(∂4v∂x4+∂4v∂y4)(xi,yj,tn+1/2)-βh212(∂4u∂x4+∂4u∂y4)(xi,yj,tn+1/2)+O(Δt4+Δt2h2+h4),R~i,jn+1/2=Δt2(18∂2v∂t2-γ18∂2u∂t2-γ224∂3u∂t3)(xi,yj,tn+1/2)+O(Δt4).
Hence there exist positive constants C1,C2 such that
(27)|Ri,jn+1/2|≤C1(Δt2+h2),|R~i,jn+1/2|≤C2Δt2.Lemma 1 (see [1]).
Forw∈Vh, the following equalities hold:
(28)-∑i=1N-1‍∑j=1N-1‍wi,j(δx2wi,j)h2=∑i=0N-1‍∑j=1N-1‍(δxwi,j)2h2=∥δxw∥2,-∑i=1N-1‍∑j=1N-1‍wi,j(δy2wi,j)h2=∑i=1N-1‍∑j=0N-1‍(δywi,j)2h2=∥δyw∥2.From Lemma1 and Cauchy-Schwarz inequality, multiplying ((ηi,jn+1+ηi,jn)/2)h2 to both sides of (24), computing the inner product, it is easy to follow that
(29)I1=(ηn+1-ηnΔt,ηn+1+ηn2)=12Δt(∥ηn+1∥2-∥ηn∥2),I21=-(αδx2ηn+1+ηn2,ηn+1+ηn2)=α∥δxηn+1+ηn2∥2,I22=-(αδy2ηn+1+ηn2,ηn+1+ηn2)=α∥δyηn+1+ηn2∥2,I31=-(βδx2ξn+1+ξn2,ηn+1+ηn2)=-β(δx2ξn+1+ξn2,ηn+1+ηn2)=-β(δx2ξn+1+ξn2,cγ1ξn+1+ξn2+γ2ξn+1-ξnΔt+R~n+1/2)=βγ1∥δxξn+1+ξn2∥2+βγ22Δt(∥δxξn+1∥2-∥δxξn∥2)-[-β(δxξn+1+ξn2,δxR~n+1/2)]≥βγ22Δt(∥δxξn+1∥2-∥δxξn∥2)-∥βδxξn+1+ξn2∥∥δxR~n+1/2∥≥βγ22Δt(∥δxξn+1∥2-∥δxξn∥2)-β4(2∥δxξn+1∥2+2∥δxξn∥2+∥δxR~n+1/2∥2),I32≥βγ22Δt(∥δyξn+1∥2-∥δyξn∥2)-β4(2∥δyξn+1∥2+2∥δyξn∥2+∥δyR~n+1/2∥2),I4=(c2Δt24δx2δy2ηn+1-ηnΔt,ηn+1+ηn2)=12ΔtcΔt24(∥δxδyηn+1∥2-∥δxδyηn∥2),I5=(Rn+1/2,ηn+1+ηn2)≤14∥Rn+1/2∥2+12(∥ηn+1∥2+∥ηn∥2).
Multiplying 2Δt to both sides of the result derived from (29), we have
(30)(∥ηn+1∥2-∥ηn∥2)+2αΔt(∥δx(ηn+1+ηn)∥2+∥δy(ηn+1+ηn)∥2)+βγ2(∥δxξn+1∥2+∥δyξn+1∥2-∥δxξn∥2-∥δyξn∥2)+cΔt24(∥δxδyηn+1∥2-∥δxδyηn∥2)≤Δt2(β∥δxR~n+1/2∥2+β∥δyR~n+1/2∥2+∥Rn+1/2∥2)+Δt(∥ηn+1∥2+∥ηn∥2)+β(∥δxξn+1∥2+∥δxξn∥2+∥δyξn+1∥2+∥δyξn∥2).
Summing for n to both sides of (30), we obtain
(31)∥ηn∥2+βγ2(∥δxξn∥2+∥δyξn∥2)≤Δt2∑l=1n-1‍(∥Rl+1/2∥2+β∥δxR~l+1/2∥2+β∥δyR~l+1/2∥2)+Δt∑l=1n-1‍(2∥ηl∥2+β∥δxξl∥2+β∥δyξl∥2),
noticing that ξ0=η0=0, by Gronwall Lemma, we obtain the following theorem.Theorem 2.
Assume thatu,v are the accurate solutions of (6)–(9) with sufficient smoothness and U,V are the difference solutions of (16)–(18). Let ξ=u-U,η=v-V, then there exists a positive constant K independent of Δt and h, such that
(32)maxn≤[T/Δt](∥ηn∥+∥δxξn∥+∥δyξn∥)≤K(Δt2+h2).


## 4. Generalized Compact ADI Scheme

In this section, we extend the method to compact ADI scheme for the numerical solution of the wave problems (1)–(3).


### 4.1. Construction of the Compact ADI Scheme

Introducing(33)fxv=∂2v∂x2,fyv=∂2v∂y2,fxu=∂2u∂x2,fyu=∂2u∂y2.
We can rewrite (6) as follows:
(34)∂v∂t-α(fxv+fyv)-β(fxu+fyu)=S(x,y,t).Using the fourth-order compact finite difference discretization [6, 8], we discretize (33) as follows:
(35)Ax(fx)i,jn≔112(fx)i-1,jn+56(fx)i,jn+112(fx)i+1,jn=δx2Ui,jn,Ay(fy)i,jn≔112(fy)i,j-1n+56(fy)i,jn+112(fy)i,j+1n=δy2Ui,jn.The Crank-Nicolson implicit discretization of (34) reads
(36)Vi,jn+1-Vi,jn▵t-α((fxv)i,jn+1+(fxv)i,jn+(fyv)i,jn+1+(fyv)i,jn)2-β((fxu)i,jn+1+(fxu)i,jn+(fyu)i,jn+1+(fyu)i,jn)2=gi,jn+1/2,
where i,j=1,2,…,N-1. Multiplying (36) by Δt, applying to its both sides with AxAy, and using the fact that the two operators Ax and Ay commute with each other, we obtain the implicit approximation of (6)–(9)
(37)AxAy(Vi,jn+1-Vi,jn)-αΔt2(AyAx(fxv)i,jn+1+AyAx(fxv)i,jnαΔt2cc+AxAy(fyv)i,jn+1+AxAy(fyv)i,jn)-βΔt2(AyAx(fxu)i,jn+1+AyAx(fxu)i,jn-βΔt2cc+AxAy(fyu)i,jn+1+AxAy(fyu)i,jn)=ΔtAxAySi,jn+1/2,(38)Vi,jn+1+Vi,jn2=γ1Ui,jn+1+Ui,jn2+γ2Ui,jn+1-Ui,jnΔt.
Let
(39)λ1=Δtγ1Δt+2γ2,λ2=γ1Δt-2γ2γ1Δt+2γ2,c=α+βλ1,γ=12β(1+λ2),
then (37) can be conveniently written as
(40)(AxAy-cΔt2Ayδx2-cΔt2Axδy2)Vi,jn+1=(AxAy+cΔt2Ayδx2+cΔt2Axδy2)Vi,jn+γΔt(Ayδx2+Axδy2)Ui,jn+ΔtAxAySi,jn+1/2.
Adding perturbing term (c2Δt2/4)δx2δy2(Vi,jn+1-Vi,jn) to the left hand side of (40) and introducing the intermediate variable Vi,jn+1/2, we obtain the compact PR ADI-like scheme as follows:
(41)(Ax-cΔt2δx2)Vi,jn+1/2=(Ay+cΔt2δy2)Vi,jn+γ(Δt2δy2+1cAy)Ui,jn+Δt2AySi,jn+1/2,(42)(Ay-cΔt2δy2)Vi,jn+1=(Ax+cΔt2δx2)Vi,jn+1/2+γ(Δt2δy2-1cAy)Ui,jn+Δt2AySi,jn+1/2,(43)Ui,jn+1=λ1(Vi,jn+1+Vi,jn)-λ2Ui,jn.
From (41) and (42), we obtain the boundary equation as follows:
(44)AxVi,jn+1/2=AyVi,jn+1+Vi,jn2+γcAyUi,jn-cΔt4δy2(Vi,jn+1-Vi,jn).
But usually we compute the boundary values of the intermediate using the following simple equality:
(45)Vi,jn+1/2=Vi,jn+1+Vi,jn2+γcUi,jn,
where i=0,N,j=1,N-1. In addition, following the idea of Douglas [2, 3], we can get a compact Douglas scheme as follows:
(46)(Ax-cΔt2δx2)(Vi,jn+1/2-Vi,jn)=cΔt(Ayδx2+Axδy2)Vi,jn+γΔt(Ayδx2+Axδy2)Ui,jn+ΔtAxAygi,jn+1/2,(47)(Ay-cΔt2δy2)(Vi,jn+1-Vi,jn)=Vi,jn+1/2-Vi,jn.
The intermediate value of Vi,jn+1/2 on the boundary is easy to be obtained from (47).


### 4.2. Error Estimate

We have derived a kind of compact PR ADI-like scheme in Section4.1. In the following, we further analyze the error of the scheme. Expanding (41)-(42), we get
(48)AxAyVi,jn+1-Vi,jnΔt-α(Ayδx2+Axδy2)Vi,jn+1+Vi,jn2-β(Ayδx2+Axδy2)Ui,jn+1+Ui,jn2+c2Δt24δx2δy2Vi,jn+1-Vi,jnΔt=AxAySi,jn+1/2.
Noting that AxUi,jn=(1+(h2/12)δx2)Ui,jn,AyUi,jn=(1+(h2/12)δy2)Ui,jn, (48) can be written as
(49)(1+h212δx2)(1+h212δy2)Vi,jn+1-Vi,jnΔt-α((1+h212δy2)δx2+(1+h212δx2)δy2)Vi,jn+1+Vi,jn2-β((1+h212δy2)δx2+(1+h212δx2)δy2)Ui,jn+1+Ui,jn2+c2Δt24δx2δy2Vi,jn+1-Vi,jnΔt=(1+h212δx2)(1+h212δy2)Si,jn+1/2.
Discretizing (6) as above and letting ξ=u-U,η=v-V, we obtain the error equations as follows:
(50)(1+h212δx2)(1+h212δy2)ηi,jn+1-ηi,jnΔt-α((1+h212δy2)δx2+(1+h212δx2)δy2)ηi,jn+1+ηi,jn2-β((1+h212δy2)δx2+(1+h212δx2)δy2)ξi,jn+1+ξi,jn2+c2Δt24δx2δy2ηi,jn+1-ηi,jnΔt=Ri,jn+1/2,(51)ηi,jn+1+ηi,jn2=γ1ξi,jn+1+ξi,jn2+γ2ξi,jn+1-ξi,jnΔt+R~i,jn+1/2,
where the truncation errors
(52)Ri,jn+1/2=Δt2(124AxAy∂2v∂t3-α8(Ay∂4v∂t2∂x2+Ax∂4v∂t2∂y2)c-β8(Ay∂4u∂t2∂x2+Ax∂4u∂t2∂y2)c+c2Δt44∂5v∂t∂x2∂y2c-αh4240(Ay∂6v∂x6+Ax∂6v∂y6)c-βh4240(Ay∂6u∂x6+Ax∂6u∂y6))(xi,yj,tn+1/2)+O(Δt4+Δt2h4+h6),R~i,jn+1/2=Δt2(18∂2v∂t2-γ18∂2u∂t2-γ224∂3u∂t3)(xi,yj,tn+1/2)+O(Δt4).
Hence there exist positive constants C′,C′′ such that
(53)|Ri,jn+1/2|≤C′(Δt2+h4),|R~i,jn+1/2|≤C′′Δt2.Lemma 3 (see [1]).
Forw∈Vh, the following inequalities hold:
(54)14h2∥δxw∥2≤∥w∥2,14h2∥δyw∥2≤∥w∥2,14h2∥δxδyw∥2≤∥δxw∥2,14h2∥δxδyw∥2≤∥δyw∥2.From Lemmas1 and 3 and Cauchy-Schwarz inequality, multiplying ((ηi,jn+1+ηi,jn)/2)h2 to both sides of (50) and computing the inner product, we have
(55)I1=((1+h212δx2)(1+h212δy2)δtηn,ηn+1+ηn2)=(δtηn,ηn+1+ηn2)+h212(δx2δtηn,ηn+1+ηn2)+h212(δy2δtηn,ηn+1+ηn2)+h4144(δx2δy2δtηn,ηn+1+ηn2)=12Δt(∥ηn+1∥2-∥ηn∥2-h224Δt(∥δxηn+1∥2-∥δxηn∥2)12Δtcc-h224Δt(∥δyηn+1∥2-∥δyηn∥2)12Δtcc+h4288Δt(∥δxδyηn+1∥2-∥δxδyηn∥2)),I21=-(α(1+h212δy2)δx2ηn+1+ηn2,ηn+1+ηn2)=α∥δxηn+1+ηn2∥2-αh212∥δxδyηn+1+ηn2∥2≥αh26∥δxδyηn+1+ηn2∥2≥0,I22=-(α(1+h212δx2)δy2ηn+1+ηn2,ηn+1+ηn2)=α∥δyηn+1+ηn2∥2-αh212∥δxδyηn+1+ηn2∥2≥αh26∥δxδyηn+1+ηn2∥2≥0,I31=-(β(1+h212δy2)δx2ξn+1+ξn2,ηn+1+ηn2)=-(βδx2ξn+1+ξn2,ηn+1+ηn2)-βh212(δx2δy2ξn+1+ξn2,ηn+1+ηn2)=βγ1(δxξn+1+ξn2,δxξn+1+ξn2)+βγ2(δxξn+1+ξn2,δxξn+1-ξnΔt)+β(δxξn+1+ξn2,δxR~n+1/2)-βγ1h212(δxδyξn+1+ξn2,δxδyξn+1+ξn2)-βγ2h212(δxδyξn+1+ξn2,δxδyξn+1-ξnΔt)-βh212(δxδyξn+1+ξn2,δxδyR~n+1/2)≥βγ22Δt(∥δxξn+1∥2-∥δxξn∥2)-h212βγ22Δt(∥δxδyξn+1∥2-∥δxδyξn∥2)-β4(∥δxξn+1∥2+∥δxξn∥2+∥δxR~n+1/2∥2)-βh248(∥δxδyξn+1∥2+∥δxδyξn∥2+∥δxδyR~n+1/2∥2)≥βγ22Δt(∥δxξn+1∥2-∥δxξn∥2)-h212βγ22Δt(∥δxδyξn+1∥2-∥δxδyξn∥2)-β3(2∥δxξn+1∥2+2∥δxξn∥2+∥δxR~n+1/2∥2),I32=-(β(1+h212δx2)δy2ξn+1+ξn2,ηn+1+ηn2)≥βγ22Δt(∥δyen+1∥2-∥δyen∥2)-h212βγ22Δt(∥δxδyen+1∥2-∥δxδyen∥2)-β3(2∥δyξn+1∥2+2∥δyξn∥2+∥δyR~n+1/2∥2),I4=c2Δt4(δx2δy2(ηn+1-ηn),ηn+1+ηn2)=c2Δt8(∥δxδyηn+1∥2-∥δxδyηn∥2),I5=(Rn+1/2,ηn+1+ηn2)≤14(∥Rn+1/2∥2+2∥ηn+1∥2+2∥ηn∥2).Multiplying2Δt to both sides of the result derived from (55), we have
(56)(∥ηn+1∥2-h212∥δxηn+1∥2-h212∥δyηn+1∥2+h4144∥δxδyηn+1∥2)+βγ2(∥δxξn+1∥2-h212∥δxδyξn+1∥2+∥δyξn+1∥2c-h212∥δxδyξn+1∥2)+c2Δt24∥δxδyηn+1∥2≤(∥ηn∥2-h212∥δxηn∥2-h212∥δyηn∥2+h4144∥δxδyηn∥2)+βγ2(∥δxξn∥2-h212∥δxδyξn∥2c+∥δyξn∥2-h212∥δxδyξn∥2)+2βΔt3(2∥δxξn+1∥2+2∥δxξn∥2+∥δxR~n+1/2∥2+2∥δyξn+1∥2+2∥δyξn∥2+∥δyR~n+1/2∥2)+c2Δt24∥δxδyηn∥2+Δt2(∥Rn+1/2∥2+2∥ηn+1∥2+2∥ηn∥2).
Summing for n to both sides of (56), and by Lemma 3, we obtain
(57)13∥ηn∥2+2βγ23(∥δxξn∥2+∥δyξn∥2)+h4144∥δxδyξn∥2≤2Δt∑l=1n-1‍∥ηn∥2+8βΔt3∑l=1n-1‍(∥δxξn∥2+∥δyξn∥2)+Δt2∑l=1n-1‍∥Rn+1/2∥2+2βΔt3∑l=1n-1‍(∥δxR~n+1/2∥2+∥δyR~n+1/2∥2).
Noticing that ξ0=η0=0, by Gronwall Lemma, we obtain the following theorem.Theorem 4.
Assume thatu,v are the accurate solutions of (6)–(9) with sufficient smoothness and U,V are the difference solutions of (41)–(43). Let ξ=u-U,η=v-V, then there exists a positive constant K′ independent of Δt and h, such that
(58)maxn≤[T/Δt](∥ηn∥+∥δxξn∥+∥δyξn∥+h4144∥δxδyξn∥)≤K′(Δt2+h4).


## 4.1. Construction of the Compact ADI Scheme

Introducing(33)fxv=∂2v∂x2,fyv=∂2v∂y2,fxu=∂2u∂x2,fyu=∂2u∂y2.
We can rewrite (6) as follows:
(34)∂v∂t-α(fxv+fyv)-β(fxu+fyu)=S(x,y,t).Using the fourth-order compact finite difference discretization [6, 8], we discretize (33) as follows:
(35)Ax(fx)i,jn≔112(fx)i-1,jn+56(fx)i,jn+112(fx)i+1,jn=δx2Ui,jn,Ay(fy)i,jn≔112(fy)i,j-1n+56(fy)i,jn+112(fy)i,j+1n=δy2Ui,jn.The Crank-Nicolson implicit discretization of (34) reads
(36)Vi,jn+1-Vi,jn▵t-α((fxv)i,jn+1+(fxv)i,jn+(fyv)i,jn+1+(fyv)i,jn)2-β((fxu)i,jn+1+(fxu)i,jn+(fyu)i,jn+1+(fyu)i,jn)2=gi,jn+1/2,
where i,j=1,2,…,N-1. Multiplying (36) by Δt, applying to its both sides with AxAy, and using the fact that the two operators Ax and Ay commute with each other, we obtain the implicit approximation of (6)–(9)
(37)AxAy(Vi,jn+1-Vi,jn)-αΔt2(AyAx(fxv)i,jn+1+AyAx(fxv)i,jnαΔt2cc+AxAy(fyv)i,jn+1+AxAy(fyv)i,jn)-βΔt2(AyAx(fxu)i,jn+1+AyAx(fxu)i,jn-βΔt2cc+AxAy(fyu)i,jn+1+AxAy(fyu)i,jn)=ΔtAxAySi,jn+1/2,(38)Vi,jn+1+Vi,jn2=γ1Ui,jn+1+Ui,jn2+γ2Ui,jn+1-Ui,jnΔt.
Let
(39)λ1=Δtγ1Δt+2γ2,λ2=γ1Δt-2γ2γ1Δt+2γ2,c=α+βλ1,γ=12β(1+λ2),
then (37) can be conveniently written as
(40)(AxAy-cΔt2Ayδx2-cΔt2Axδy2)Vi,jn+1=(AxAy+cΔt2Ayδx2+cΔt2Axδy2)Vi,jn+γΔt(Ayδx2+Axδy2)Ui,jn+ΔtAxAySi,jn+1/2.
Adding perturbing term (c2Δt2/4)δx2δy2(Vi,jn+1-Vi,jn) to the left hand side of (40) and introducing the intermediate variable Vi,jn+1/2, we obtain the compact PR ADI-like scheme as follows:
(41)(Ax-cΔt2δx2)Vi,jn+1/2=(Ay+cΔt2δy2)Vi,jn+γ(Δt2δy2+1cAy)Ui,jn+Δt2AySi,jn+1/2,(42)(Ay-cΔt2δy2)Vi,jn+1=(Ax+cΔt2δx2)Vi,jn+1/2+γ(Δt2δy2-1cAy)Ui,jn+Δt2AySi,jn+1/2,(43)Ui,jn+1=λ1(Vi,jn+1+Vi,jn)-λ2Ui,jn.
From (41) and (42), we obtain the boundary equation as follows:
(44)AxVi,jn+1/2=AyVi,jn+1+Vi,jn2+γcAyUi,jn-cΔt4δy2(Vi,jn+1-Vi,jn).
But usually we compute the boundary values of the intermediate using the following simple equality:
(45)Vi,jn+1/2=Vi,jn+1+Vi,jn2+γcUi,jn,
where i=0,N,j=1,N-1. In addition, following the idea of Douglas [2, 3], we can get a compact Douglas scheme as follows:
(46)(Ax-cΔt2δx2)(Vi,jn+1/2-Vi,jn)=cΔt(Ayδx2+Axδy2)Vi,jn+γΔt(Ayδx2+Axδy2)Ui,jn+ΔtAxAygi,jn+1/2,(47)(Ay-cΔt2δy2)(Vi,jn+1-Vi,jn)=Vi,jn+1/2-Vi,jn.
The intermediate value of Vi,jn+1/2 on the boundary is easy to be obtained from (47).


## 4.2. Error Estimate

We have derived a kind of compact PR ADI-like scheme in Section4.1. In the following, we further analyze the error of the scheme. Expanding (41)-(42), we get
(48)AxAyVi,jn+1-Vi,jnΔt-α(Ayδx2+Axδy2)Vi,jn+1+Vi,jn2-β(Ayδx2+Axδy2)Ui,jn+1+Ui,jn2+c2Δt24δx2δy2Vi,jn+1-Vi,jnΔt=AxAySi,jn+1/2.
Noting that AxUi,jn=(1+(h2/12)δx2)Ui,jn,AyUi,jn=(1+(h2/12)δy2)Ui,jn, (48) can be written as
(49)(1+h212δx2)(1+h212δy2)Vi,jn+1-Vi,jnΔt-α((1+h212δy2)δx2+(1+h212δx2)δy2)Vi,jn+1+Vi,jn2-β((1+h212δy2)δx2+(1+h212δx2)δy2)Ui,jn+1+Ui,jn2+c2Δt24δx2δy2Vi,jn+1-Vi,jnΔt=(1+h212δx2)(1+h212δy2)Si,jn+1/2.
Discretizing (6) as above and letting ξ=u-U,η=v-V, we obtain the error equations as follows:
(50)(1+h212δx2)(1+h212δy2)ηi,jn+1-ηi,jnΔt-α((1+h212δy2)δx2+(1+h212δx2)δy2)ηi,jn+1+ηi,jn2-β((1+h212δy2)δx2+(1+h212δx2)δy2)ξi,jn+1+ξi,jn2+c2Δt24δx2δy2ηi,jn+1-ηi,jnΔt=Ri,jn+1/2,(51)ηi,jn+1+ηi,jn2=γ1ξi,jn+1+ξi,jn2+γ2ξi,jn+1-ξi,jnΔt+R~i,jn+1/2,
where the truncation errors
(52)Ri,jn+1/2=Δt2(124AxAy∂2v∂t3-α8(Ay∂4v∂t2∂x2+Ax∂4v∂t2∂y2)c-β8(Ay∂4u∂t2∂x2+Ax∂4u∂t2∂y2)c+c2Δt44∂5v∂t∂x2∂y2c-αh4240(Ay∂6v∂x6+Ax∂6v∂y6)c-βh4240(Ay∂6u∂x6+Ax∂6u∂y6))(xi,yj,tn+1/2)+O(Δt4+Δt2h4+h6),R~i,jn+1/2=Δt2(18∂2v∂t2-γ18∂2u∂t2-γ224∂3u∂t3)(xi,yj,tn+1/2)+O(Δt4).
Hence there exist positive constants C′,C′′ such that
(53)|Ri,jn+1/2|≤C′(Δt2+h4),|R~i,jn+1/2|≤C′′Δt2.Lemma 3 (see [1]).
Forw∈Vh, the following inequalities hold:
(54)14h2∥δxw∥2≤∥w∥2,14h2∥δyw∥2≤∥w∥2,14h2∥δxδyw∥2≤∥δxw∥2,14h2∥δxδyw∥2≤∥δyw∥2.From Lemmas1 and 3 and Cauchy-Schwarz inequality, multiplying ((ηi,jn+1+ηi,jn)/2)h2 to both sides of (50) and computing the inner product, we have
(55)I1=((1+h212δx2)(1+h212δy2)δtηn,ηn+1+ηn2)=(δtηn,ηn+1+ηn2)+h212(δx2δtηn,ηn+1+ηn2)+h212(δy2δtηn,ηn+1+ηn2)+h4144(δx2δy2δtηn,ηn+1+ηn2)=12Δt(∥ηn+1∥2-∥ηn∥2-h224Δt(∥δxηn+1∥2-∥δxηn∥2)12Δtcc-h224Δt(∥δyηn+1∥2-∥δyηn∥2)12Δtcc+h4288Δt(∥δxδyηn+1∥2-∥δxδyηn∥2)),I21=-(α(1+h212δy2)δx2ηn+1+ηn2,ηn+1+ηn2)=α∥δxηn+1+ηn2∥2-αh212∥δxδyηn+1+ηn2∥2≥αh26∥δxδyηn+1+ηn2∥2≥0,I22=-(α(1+h212δx2)δy2ηn+1+ηn2,ηn+1+ηn2)=α∥δyηn+1+ηn2∥2-αh212∥δxδyηn+1+ηn2∥2≥αh26∥δxδyηn+1+ηn2∥2≥0,I31=-(β(1+h212δy2)δx2ξn+1+ξn2,ηn+1+ηn2)=-(βδx2ξn+1+ξn2,ηn+1+ηn2)-βh212(δx2δy2ξn+1+ξn2,ηn+1+ηn2)=βγ1(δxξn+1+ξn2,δxξn+1+ξn2)+βγ2(δxξn+1+ξn2,δxξn+1-ξnΔt)+β(δxξn+1+ξn2,δxR~n+1/2)-βγ1h212(δxδyξn+1+ξn2,δxδyξn+1+ξn2)-βγ2h212(δxδyξn+1+ξn2,δxδyξn+1-ξnΔt)-βh212(δxδyξn+1+ξn2,δxδyR~n+1/2)≥βγ22Δt(∥δxξn+1∥2-∥δxξn∥2)-h212βγ22Δt(∥δxδyξn+1∥2-∥δxδyξn∥2)-β4(∥δxξn+1∥2+∥δxξn∥2+∥δxR~n+1/2∥2)-βh248(∥δxδyξn+1∥2+∥δxδyξn∥2+∥δxδyR~n+1/2∥2)≥βγ22Δt(∥δxξn+1∥2-∥δxξn∥2)-h212βγ22Δt(∥δxδyξn+1∥2-∥δxδyξn∥2)-β3(2∥δxξn+1∥2+2∥δxξn∥2+∥δxR~n+1/2∥2),I32=-(β(1+h212δx2)δy2ξn+1+ξn2,ηn+1+ηn2)≥βγ22Δt(∥δyen+1∥2-∥δyen∥2)-h212βγ22Δt(∥δxδyen+1∥2-∥δxδyen∥2)-β3(2∥δyξn+1∥2+2∥δyξn∥2+∥δyR~n+1/2∥2),I4=c2Δt4(δx2δy2(ηn+1-ηn),ηn+1+ηn2)=c2Δt8(∥δxδyηn+1∥2-∥δxδyηn∥2),I5=(Rn+1/2,ηn+1+ηn2)≤14(∥Rn+1/2∥2+2∥ηn+1∥2+2∥ηn∥2).Multiplying2Δt to both sides of the result derived from (55), we have
(56)(∥ηn+1∥2-h212∥δxηn+1∥2-h212∥δyηn+1∥2+h4144∥δxδyηn+1∥2)+βγ2(∥δxξn+1∥2-h212∥δxδyξn+1∥2+∥δyξn+1∥2c-h212∥δxδyξn+1∥2)+c2Δt24∥δxδyηn+1∥2≤(∥ηn∥2-h212∥δxηn∥2-h212∥δyηn∥2+h4144∥δxδyηn∥2)+βγ2(∥δxξn∥2-h212∥δxδyξn∥2c+∥δyξn∥2-h212∥δxδyξn∥2)+2βΔt3(2∥δxξn+1∥2+2∥δxξn∥2+∥δxR~n+1/2∥2+2∥δyξn+1∥2+2∥δyξn∥2+∥δyR~n+1/2∥2)+c2Δt24∥δxδyηn∥2+Δt2(∥Rn+1/2∥2+2∥ηn+1∥2+2∥ηn∥2).
Summing for n to both sides of (56), and by Lemma 3, we obtain
(57)13∥ηn∥2+2βγ23(∥δxξn∥2+∥δyξn∥2)+h4144∥δxδyξn∥2≤2Δt∑l=1n-1‍∥ηn∥2+8βΔt3∑l=1n-1‍(∥δxξn∥2+∥δyξn∥2)+Δt2∑l=1n-1‍∥Rn+1/2∥2+2βΔt3∑l=1n-1‍(∥δxR~n+1/2∥2+∥δyR~n+1/2∥2).
Noticing that ξ0=η0=0, by Gronwall Lemma, we obtain the following theorem.Theorem 4.
Assume thatu,v are the accurate solutions of (6)–(9) with sufficient smoothness and U,V are the difference solutions of (41)–(43). Let ξ=u-U,η=v-V, then there exists a positive constant K′ independent of Δt and h, such that
(58)maxn≤[T/Δt](∥ηn∥+∥δxξn∥+∥δyξn∥+h4144∥δxδyξn∥)≤K′(Δt2+h4).


## 5. Numerical Experiment

In this section, we exemplify two numerical examples to illustrate the effectiveness of the present ADI scheme. Lele [9] discussed some finite differencing errors for first and second derivatives by establishing the relationship of wave number (for differential equation) and numerical wave number (for difference scheme) and indicated that standard Pade scheme (which is known as compact finite difference scheme in this paper) is a good choice to discrete differential equations. In this paper, we focus on the efficient alternating direction technique for solving second-order wave equations. Because the compact method in spatial direction is standard, the scheme is effective.The computer language used for the programming is Fortran, and the programs are performed on professional computer (CPU is 3.10 GHz, and Memory is 3.47 G). For convenience, we denote the present second-order ADI scheme (16)–(18) by PR, the present compact ADI scheme (41)–(43) by CPR, the second-order ADI scheme (21)–(22) by Dou, the compact ADI scheme (46)–(47) by CDou, and the three-level LOD scheme by LOD.Example 5.
Letγ1=1.0,γ2=1.0,a=1.0,q=1.0, and S=(π2-3π+2)e-πtsin(x+y) in (1), then the true solution is u=e-πtsin(x+y). We chose different grids and compared the accuracy of the computed solutions with other schemes. The solutions were obtained for tmax=2.0. The quantities that we compared are the maximum L2-norm and the total elapsed time (CPU) in seconds.Tables1 and 2 display the L2-norm errors and the CPU time. From Table 1, one can easily see that the error from the present second-order scheme is smaller than that from other schemes, and it converges as fast as other schemes. From Table 2, we can find the superiority of the present compact ADI scheme, which has the highest accuracy and fastest computational efficiency. In Figures 1 and 2, we plot the errors and cost time at various grids. The figures show the superiority of the compact ADI over other schemes (second-order scheme). The CPR has much higher and more efficiency than CDou, which may be the best scheme to solve above questions.Table 1
TheL2-norm and CPU time for the second-order schemes.
Scheme
N
=
40, Δt=h
N
=
60, Δt=h
L
2-norm
CPU time(s)
L
2-norm
CPU time(s)
PR
5.8154
×
1
0
-
4
0.125000
2.5959
×
1
0
-
4
0.406250
Dou
5.7463
×
1
0
-
4
0.125000
2.5751
×
1
0
-
4
0.390625
LOD(θ=0.25)
6.2814
×
1
0
-
3
0.140625
3.9884
×
1
0
-
3
0.437500
LOD(θ=0.5)
6.3956
×
1
0
-
3
0.140625
4.0402
×
1
0
-
3
0.437500Table 2
TheL2-norm and CPU time for the fourth-order schemes.
Scheme
N
=
20, Δt=h2
N
=
40, Δt=h2
L
2-norm
CPU time(s)
L
2-norm
CPU time(s)
CPR
5.6026
×
1
0
-
7
0.375000
3.5894
×
1
0
-
7
5.156250
CDou
5.1636
×
1
0
-
7
0.562500
3.3038
×
1
0
-
7
6.578125
LOD(θ=0.25)
5.1383
×
1
0
-
4
0.390625
1.2695
×
1
0
-
4
5.265625
LOD(θ=0.5)
5.1397
×
1
0
-
4
0.390625
1.2694
×
1
0
-
4
5.265625Figure 1
The error of all schemes for various grid sizes whenΔt=h2.Figure 2
The CPU time of all schemes for various grid sizes whenΔt=h2.Example 6.
Letγ1=1.0,γ2=1.0,a=1.0,q=1.0, and S=π2sin5πxsin2πy[34+(0.5+17π2)cosπ2t+(17-0.5π2)sinπ2t] in (1). The exact solution is u=(1+0.5sinπ2t)sin5πxcos3πy, and the initial and boundary conditions can be obtained based on the exact solution. We chose the grids 25×25 and 50×50, and the time step with Δt=h2. The solutions were obtained for tmax=5.0. We compared the accuracy of the computed solutions from the present ADI scheme and other four ADI schemes. The quantities that we compared are the L2-norm and the total elapsed time (CPU) in seconds.Table3 lists t the total elapsed time (CPU) in seconds and the L2-norm error are listed in Table 3 and shown in Figures 3 and 4. One can see that the error from the same order scheme is very close. From the table and figures, we can see that the present fourth-order scheme not only has high accuracy but also converges as fast as other schemes, especially for CPR scheme.Table 3
TheL2-norm and CPU time for all schemes.
Scheme
N
=
25, Δt=h2
N
=
50, Δt=h2
L
2-norm
CPU time (s)
L
2-norm
CPU time (s)
PR
3.9262
×
1
0
-
2
1.8125
1.0218
×
1
0
-
2
23.7812
CPR
8.0830
×
1
0
-
4
1.9375
5.2720
×
1
0
-
5
24.8906
Dou
3.9262
×
1
0
-
2
1.8281
1.0218
×
1
0
-
2
23.0625
CDou
8.0830
×
1
0
-
4
2.0156
5.2720
×
1
0
-
5
29.0312
LOD
4.2503
×
1
0
-
1
2.0508
6.5634
×
1
0
-
2
25.6705Figure 3
The error of all schemes for various grid sizes whenΔt=h2.Figure 4
The CPU time of all schemes for various grid sizes whenΔt=h2.From Figures1 and 3, it is easily to find that the tested methods degrade to first-order method as the grid gets coarse. In all schemes, we add perturbing term (c2Δt/4)δx2δy2(Vi,jn+1-Vi,jn) that is, an approximation to (c2Δt2/4)(∂5v/∂t∂x2∂y2). When step h becomes large, the time step Δt varies correspondingly because of Δt=h for the second-order schemes or Δt=h2 for the compact schemes. Hence, the effect of perturbing term is enhanced, which results in the schemes seem to degrade to first-order methods as the grid gets coarse. We can choose suitable h and get required accuracy.


## 6. Conclusions

In this paper, we proposed two new ADI schemes for solving nonhomogeneous wave problems. It has been proved that they are unconditionally stable. The compact scheme proposed in this paper is fourth-order accurate in space and second-order accurate in time and allows a considerable saving in computing time. Numerical examples are given to test their high accuracy and to show their superiority over some other schemes in terms of accuracy and computational costs.The present scheme can be applied to the following variable coefficient differential models on domainΩ=[0,1]2:
(59)γ1∂u∂t2+γ2∂2u∂t2-∇·(A∇u)-∂∂t∇·(Q∇u)=S,(x,y)∈Ω,t∈(0,T],u(x,y,0)=u0(x,y),ut(x,y,0)=ψ(x,y),(x,y)∈Ω,u(x,y,t)=φ(x,y,t),(x,y)∈∂Ω,t∈(0,T],
where A=A(x,y) and Q=Q(x,y) are diagonal, nonnegative diffusion tensors.



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*Source: 267106-2013-05-14.xml*