Research Journal of Applied Sciences, Engineering and Technology 4(19): 3858-3864, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: May 24,
Accepted: June 21, 2012
Published: October 01, 2012
A Numerical Approach for Solving a General Nonlinear Wave Equation
Zainal Abdul Aziz, Nazeeruddin Yaacob, Mohammadreza Askaripour Lahiji,
Mahdi Ghanbari and C.C. Dennis Ling
Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia,
81310 UTM Skudai, Johor, Malaysia
Abstract: An analysis of various numerical schemes and boundary conditions on a general nonlinear wave
equation is considered in this study. In particular, the Lax-Wendroff, Leapfrog and Iterated Crank
Nicholson methods with Dirichlet boundary conditions are used to solve this nonlinear wave equation. The
computation of the solution is made via the reduction of the nonlinear wave equation to the two variable
and three variable systems.
Keywords: Crank-Nicholson method, lax-wendroff method, leapfrog method, solution of nonlinear wave
equation, three variable system, two variable system
INTRODUCTION
There are numerous methods to solve partial
differential equations; especially when the concern is to
generate numerical solutions (Morton and Mayers,
2005). Applications in numerical partial differential
equations are particularly important in gravitationalwave research and numerical relativity (Barry et al.,
1999; Kip, 1997; Will, 1999) and also various
techniques have been devised over the years to solve
such related equations. In this study, we proposed to
discuss three numerical methods of relevance. Turkel
(1974) studied second order schemes for hyperbolic
systems and compared these with respect to stability
properties and errors. Mansour et al. (1996) studied
solutions using standard (Crank-Nicolson) vectorial
beam propagation method for step-index waveguides
containing small oscillations. Pelloni and Dougalis
(2002) stated a fully discrete spectral method for the
numerical solution of the initial and periodic boundary
value problems. An alternating Crank-Nicolson method
is proposed for the numerical solution of phase-field
equations on a dynamically adaptive grid, which
automatically leads to two decoupled algebraic
subsystems, one is linear and the other is semi linear
(Tan and Huang, 2008). Gilles et al. (2000) discussed
the stability and accuracy of second order and fourth
order accurate spatial central discretizations on
staggered grids with a third order accurate spatial
discretization on an unstaggered grid, combined with
second order Leapfrog time integration scheme. A
comparison is made with the standard Lax-Wendroff
algorithm and application to obtaining steady-state
solutions to the equations of inviscid flow by Kolibal
(1992). Wang and Ruuth (2008) studied easily
implementable Variable Step-Size Implicit-Explicit
(VSIMEX) linear multistep methods for timedependent PDEs. A relation between the truncation
error and exact and approximate amplification factors is
derived by Wesseling (1973). In addition, we view the
wave equation as a system of first-order equation.
There are many different options for this system.
In this study, review of the finite difference method
has been proposed. Moreover, the three-variable system
and the two-variable system are considered. Then it is
shown that the nonlinear partial differential equation is
solved via the three methods proposed, with Dirichlet
boundary conditions.
Summary of our approach to designing a finite
difference method: We can systematically create finite
difference methods by separating the treatment of space
and time derivatives, then designing a solver by
choosing/testing:




A time integrator
A discretization for spatial derivatives
Fourier's analysis for classification of the
differential operator
Writing code and testing
SOME CLASSICAL FINITE DIFFERENCE
METHODS
Leapfrog (space and time): We use centered
differencing for both space x and time t (Bourchtein and
Bourchtein, 2010):
u
u
c
t
x
(1)
Corresponding Author: Zainal Abdul Aziz, Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia,
81310 UTM Skudai, Johor, Malaysia
3858
Res. J. Appl. Sci. Eng. Technol., 4(19): 3858-3864, 2012
u mn 1  u mn 1
u n  u mn 1
 c ( m 1
)
2t
2x
where, u  x, t  is the dependent variable, t and
replaced by the discretized centered differences.
Consider the Taylor expansion:
(2)
x are
u ( x, t   )  u ( x, t )  
the independent variables representing time and onedimensional space respectively, c is a real positive
constant. So we have:
u
n 1
m
u
n 1
m
ct n
(um 1  umn 1 )

x
u  2  2 u

 O ( 3 )
2 t 2
t
(9)
We consider the equation:
u
u
 c
t
x
(3)
(10)
and
It is interesting to note that the leapfrog method is a
three-level method. To find the value of the function at
one time step, it is important to know the value of the
function at the previous two-time steps.
Similarly, we know that the leapfrog time stepping
method is only stable for operators with eigen values in
the range (Hesthaven and Warburton, 2008):
 t   i   1, 1
x
c
(5)
(6)
u ( xm , tn 1 )  u ( xm , tn 1 )
 u ( xm1 , tn )  u ( xm1 , tn ) 
 ct 

2
2x


t
 O(t 3 )  c O(x3 )
x
Tmn  O(t 3 )  cO(x3 )
 u c 2 2  2 u

x
2 x 2
(12)
c n
c 2 2 n
(u i 1  u in1 ) 
(u i 1  2u in  u in1 )
2h
2h 2
(13)
This is the approximation of the Eq. (12). This is a
second-order difference equation (in space) which
differentiates with the second-order difference equation
(in time), in series approximation.
Crank-Nicholson method: In numerical analysis, the
Crank-Nicholson method is a finite difference method
used for numerically solving the heat equation and
similar partial differential equations. It is a secondorder method, implicit in time and is numerically stable
(Teukolsky, 2000; Hansen et al., 2004; Bourchtein and
Bourchtein, 2010). The Crank-Nicholson method is
based on central difference in space and the trapezoidal
rule in time, giving second order convergence in time.
Equivalently, it is the average of (7)
forward Euler and
backward Euler in time. For example, in one
dimension, if the partial differential equation is:
Tmn 
x  t so:
u ( x , t   )  u ( x , t )  c
u in 1  u in 
We can perform a full truncation analysis (in space and
time):
We know that if
So, to second order:
and thus:
where, m is the number of data , m  1, 2,3 and with
a condition :
t 
(11)
(4)
where,  is the eigenvalue and i   1 .
The centered difference derivative matrix is a skew
symmetric matrix with eigenvalues:
ic
2 m
s in (
)
x
M
 2u

u
 2u

(c
)  c2
2
t
t
x
x 2
(7)
(8)
Lax-wendroff method: The Lax-Wendroff is a secondorder difference method in both time and space. For this
method, we will approximate our function by the
Taylor expansions where the time derivatives are
u
u  2 u
,
)
 F (u , x , t ,
t
x x 2
(14)
Then, letting u ( i  x , n  t )  u n , the Crank-Nicholson
i
method is the average of the forward Euler method at n
and the backward Euler method at n  1 (Fig. 1)
The Crank-Nicholson stencil for a 1D problem is
given by: Forward Euler:
3859 Res. J. Appl. Sci. Eng. Technol., 4(19): 3858-3864, 2012
u
 2u  2u
 a( 2  2 )
t
x
y
(22)
can be solved with Crank-Nicholson discretization of:
uin, j 1  uin, j 
Fig. 1: Crank-Nicholson method
1 a t
2 ( x ) 2
(uin11, j  uin11, j  uin, j 11  uin, j 11  4uin, j 1 ) 


n
n
n
n
n
  (ui 1, j  ui 1, j  ui , j 1  ui , j 1  4ui , j ) 
(23)
u
n 1
i
u
t
n
i
 Fi n (u , x , t ,
u  u
,
)
x x 2
2
(15)
Assuming that a square grid is used, so that
x  y , then this equation can be simplified by
rearranging them and using:
Backward Euler:
u in  1  u in
u  2u
 F i n 1 ( u , x , t ,
,
)
t
x x 2
 
(16)
u in 1  u in 1  n 1
u  2 u
u  2 u 
  Fi (u , x , t , , 2 )  Fi n (u , x , t , , 2 ) 
t
t x
x x 
2
(17)

(1 2)uin,j1  (uin1,1j  uin1,1j  uin,j11  uin,j11) 
2

The function F should be discretized spatially with
a central difference.
Note that this is an implicit method to get the next
value of u in time and a system of algebraic equations
must be solved. If the partial differential equation is
nonlinear, the discretization will be nonlinear so that
advancing time will involve the solution of a system of
nonlinear algebraic equations.
(1 2)uin, j  (uin1, j  uin1, j  uin, j1  uin, j1)
2
(25)
THREE VARIABLES SYSTEM
Proposition: For some functions u  x, t  , v  x, t  , w  x, t  ,
the partial differential equation u  c 2 u is equivalent
tt
u
 2u
 a
t
x 2
(18)
As an example, for linear diffusion, whose CrankNicholson discretization is given by:
uin 1  uin
a
(uin11  2uin 1  uin11 )  (uin1  2uin  uin1 ) 

t
2( x ) 2
at
2( x) 2
(26)
vt  c 2 w x
(27)
wt  v x
(28)
w (x, 0) = u x (x, 0)
(29)
Proof: First, assume u  c 2u for some function
tt
xx
Set v  ut and w  u x . Then:
(20)
vt  utt  c 2u xx  c 2 wx
 ruin11  (1  2r )uin1  ruin11  ruin1  (1  2r )uin  ruin1
(21)
n
i
ut  v
where,
(19)
Letting:
xx
to the system of equation:
Example: 1D diffusion:
This is a tridiagonal problem, so that u may be
solved by using tridiagonal matrix algorithm.
Example: 2D dimensional
The two-dimensional heat equation:
(24)
For Crank-Nicholson numerical scheme, thus we can
write the scheme as:
Crank-Nicholson:
r 
at
( x)2
u.
(30)
and
wt  u xt  u tx  v x
(31)
Thus, a solution of the wave equation leads to a
solution of the three variables system. Now assume
3860 Res. J. Appl. Sci. Eng. Technol., 4(19): 3858-3864, 2012
wt  v x , vt  c 2 wx and u t  v , with initial condition
ut  vin
w( x,0)  u x ( x,0) . It is easy to see:
u tt  v t  c w x  c u x x
2
2
at time t  0 .
Now, we need to show w  u x for all t , That is:
t
v in  1  v in   v t 
 vx  utx . Hence:
0  u tx  u xt 
(38.b)
Considering equations (36.c) and (36.d), the
following difference equation is obtained:
 ( w  ux ) t  0
We know w
w in 1  w in 1
 Vin
2h
u tt 
(32) (38.a)

2
v tt
(39)
where,
u x
(w  u x )
w


t
t
t
(33)
Nonlinear system: Let us consider a nonlinear wave
equation:
utt  uxx  V (u)  0
w in 1  w in 1
 V i n
2h
vt 
v tt 
(34)
where V (u) is some smooth (potential) function. Define
three-variable system and write up:
ut  v
(35.a)
vt  wx  V (u )
(35.b)
ux  w
w in  1  w in   w t 
utt  wx  V (u)
(36.b)
v t  w x  V  (u )
(36.c)

w tt
2
(41)
where,
w
Lax-wendroff method: Lax-Wendroff approximates u
by the second-order Taylor series, where the derivatives
are replaced by 3 equivalent difference equation:
(36.a)
(40.b)
Considering equations (36.e) and (36.f), the following
difference equation is obtained:
(35.c)
ut  v
v in 1  2 v in  v in 1
 v in V i  n
h2
(40.a)
t

w tt 
v in 1  v in 1
2 h
w in 1  2 w in  w in 1
 w inV i  n
h2
(42.a)
(42.b)
Leapfrog method: We will use half-steps for nonlinear
wave equation. In fact, we will use the same grid
system as in the three-variable wave equation that is,
our lattice will be constructed. So if we consider the
values of u at the lattice points  i, n  , the difference
vtt  vxx  vV (u )
equation will be:
(36.d)
wt  vx
(36.e)
wtt  wxx  wV (u )
(36.f)
u
n  1
i
 u
n
i
 u
 u
n 1
i
 2  ( v in  0 . 5 )
(43.a)
and also the values of v at the lattice points  i, n  5 ,
the difference equation will be:
Considering equations (36.a) and (36.b), the following
difference equation is obtained:
u
n 1
i
t


2
u
(37)
tt
win 0.5  win 0.5
(43.b)
 Vi n0.5 )
h
and then the values of w at the lattice points  i, n  5  ,
the difference equation will be:
vin  0.5  vin  0.5  2 (
where,
3861 Res. J. Appl. Sci. Eng. Technol., 4(19): 3858-3864, 2012
w in01.5  w in01.5  2 (
v in 0 .5  v in 0 .5
)
h
(43.c)
Iterative Crank-Nicholson method: We will consider
each term in the second term on the right-hand side of
the FTCS (Forward Time, Centered Space) method and
replaced it by the average of that value and the value of
the function at the next time step. Thus, our difference
equations are:
u in  1  u
v in  1  v in 


n
i
 w in11  w in11
2
(
(44.a)
( v in  1  v in )
2
2h
 V in 1 
w in 1  w in 1
 V in )
2h
(44.b)
w in  1  w in 
 v in11  v in11
2
(
2

v in 1  v in 1
)
2
(44.c)
Example: A specific of Eq. (34) can be considered as:
V (u )  1  cos( u )
Fig. 2: Crank-Nicholson on sine-gordon equation:
(45)
u ( x,0)  0  u i0  0
which results in the sine-Gordon equation:
utt  u xx  sin(u )
u t ( x ,0 )  0 
(46)
To simulate the periodic boundary condition as follows:
u ( 0 , t )  u (1, t )
(47)
(48.a)
u t ( x ,0 )  0
(48.b)
Proposition: For some function, u (x. t) and v(x, t), the
partial differential equation u  c 2 u is equivalent to
tt
 win11  win11
vin1  vin  (
2
2h
( v in 1  v in )
 sin(uin1) 
w in 1  w in 
2
(
2
vt  cu x
(51.a)
ut  cv x
(51.b)
(49.a)
Proof: If u  cv and v  cu for functions u , v, then:
t
x
t
x
win1  win1
 sin(uin ))
2h
(49.b)
 v in11  v in11
xx
the system of equations:
We use the Crank-Nicholson method (44) to
discretize (46) and to solve the system of equations:
2
(50.b)
TWO-VARIABLE SYSTEM
u ( x ,0 )  0


( u i0  u i1 )  0  u i1  u i0
Note that, the black points are the known data that
can be used to evaluate and therefore, the values of
ui1 , vi1 , wi1 for i  1, 2 ,..., n can be computed (Fig. 2).
and the initial conditions are given:
u in 1  u in 1 
1
(50.a)
v n  v in1 (49.c)
 i 1
)
2
utt  cvxt  cvtx  c 2uxx
(52)
Now assume u  c 2 u for some function u . There
tt
xx
exists a family of functions v such that vt  cu x . Then
each v has the form:
To predict the values two initial conditions are
required which they are:
v ( x, t )   cu x dt  f ( x )
3862 Res. J. Appl. Sci. Eng. Technol., 4(19): 3858-3864, 2012
where,
for some function f .
Now we choose f so that cv  u at t  0 . Therefore, we
x
t
need only to show that cv  u always, that is
x
vt 
uin1  2uin  uin1
Vin
2
h
vtt 
vin1  vin1 n n
 vi Vi
2h
t
. Since:

( cv x  u t )  0
t

cv x  cv xt  cv tx  c 2 u xx
t
Leapfrog method: In the method, we have a second
derivative. So we should evaluate both u and v on
whole steps, thus our difference equations are:
(54)
uin 1  uin 1  2 (vin )
Thus the wave equation and the system are equivalent.
Nonlinear system: The nonlinear wave Eq. (34) is
considered. It is defined two-variable system and can be
written as:
ut  v
(55.a)
vt  uxx  V (u )
(55.b)
Lax-Wendroff method : Lax-Wendrof approximates u
by the second-order Taylor series, where the derivatives
are replaced by two (56) equivalent difference equation:
ut  v
(56.a)
utt  uxx  V (u)
(56.b)
(56.c)
vtt  vxx  vV (u)
(56.d)
2
u tt
u in  1  u in 

2
( v in  1  v in )
2
(
 Vi  n 1 
u
n
i 1
h2
 2 u in  u in1
 Vi  n )
h2
V (u )  1  cos(u )

2
v tt
(63)
which results in the sine-Gordon equation:
(64)
To simulate the periodic boundary condition as:
u ( 0 , t )  u (1, t )
(58.b)
Considering equations (56.c) and (56.d), the
following difference equation is obtained:
v in  1  v in   v t 
(62.b)
(57)
(58.a)
u in1  2uin  u in1
 Vin
2
h
(62.a)
 u in11  2 u in 1  u in11
utt  u xx  sin(u)
u t  v in
(61.b)
Iterative Crank-Nicholson method: We will consider
each term in the second term on the right-hand side of
the FTCS method (Forward Time, Centered Space) and
replace it by the average of that value and the value of
the function at the next time step. Thus, our difference
equations are:
where,
u tt 
uin1  2uin  uin1
 Vin )
2
h
(61.a)
Example: A specific of equation (34) can be
considered as:
Considering equations (56.a) and (56.b), the following
difference equation is obtained:

vin 1  vin 1  2 (
vin 1  vin 
vt  uxx  V (u )
u in  1  u in   u t 
(60.b)
(53)
We have:

(cv x  u t )  c 2 u xx  u tt  0
t
(60.a)
(65)
and initial conditions are given:
(59)
3863 u ( x ,0 )  0
(66)
u t ( x ,0 )  v
(67)
Res. J. Appl. Sci. Eng. Technol., 4(19): 3858-3864, 2012
Fig. 3: Leapfrog for sine-gordon
We use the Leapfrog method (62) to discretize
(65), so solve the system of equations:
u in 1  u in 1  2 ( vin )
vin 1  vin 1  2 (
u in1  2u in  u in1
 sin( u in ))
h2
(68.a)
(68.b)
Note that, the black points are the known date that
can be used to evaluate. Therefore, the values of u1 , v1
i
i
for i  1,2,..., n can be computed (Fig. 3).
CONCLUSION
In this study, we have used Lax-Wendroff,
Leapfrog and Iterated-Crank-Nicholson methods to
solve numerically certain generalized nonlinear wave
equation in terms of two-variable and three-variable
systems. In addition, provided with the knowledge of
using the various numerical methods and imposing the
various boundary conditions, we consider a nonlinear
system without general solution.
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