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Research Article
Characteristics of the differential quadrature method
and its improvement
Wang Fangzong, Liao Xiaobing, Xie Xiong
College of Electrical Engineering & New Energy, China Three Gorges University, Yichang 443002, Hubei Province, China
Correspondence should be addressed to Wang Fangzong; [email protected]
Abstract
The differential quadrature method has been widely used in scientific and engineering computation. However,
for the basic characteristics of time domain differential quadrature method, such as numerical stability and
calculation accuracy or order, it is still lack of systematic analysis conclusions. In this paper, according to the
principle of differential quadrature method, it has been derived and proved that the weighting coefficients matrix
of differential quadrature method meets the important V-transformation feature. Through the equivalence of the
differential quadrature method and the implic it Runge-Kutta method, it has been proved that the differential
quadrature method is A-stable and s-stage s-order method. On this basis, in order to further improve the accuracy
of the time domain differential quadrature method, a class of improved differential quadrature method of s-stage
2s-order have been proposed by using undetermined coefficients method and Padé approximations. The numerical
results show that the improved differential quadrature method is more precise than the traditional differential
quadrature method.
method has the capability of producing highly
accurate solutions with minimal computational effort
[3, 4] when the method is applied to problems with
globally smooth solutions. So far, the differential
1. Introduction
The differential quadrature method (DQM) was
first proposed by Bellman and his associates in the
early 1970s [1, 2], which is usually used for solving
ordinary and partial differential equations. As an
quadrature method has been widely applied to
boundary-value problems in many areas of
engineering and science, such as structural mechanics
[5-8], transport process [9], dynamic systems [10-12],
calculation of transmission line transient response [13,
14], etc. A comprehensive review of the chronological
development of the differential quadrature method can
be found in Reference [4]. Although the differential
quadrature method has been successfully applied in so
many fields, for the basic characteristics of the method,
analogous extension of the quadrature for integrals, it
can be essentially expressed as the values of the
derivatives at each grid point as weighted linear sums
approximately of the function values at all grid points
within the domain under consideration.
The differential quadrature method is conceptually
simple and the implementation is straightforward. It
has been recognized that the differential quadrature
1
such as numerical stability and calculation accuracy or
order, not much work about them has been done in
this area for the differential quadrature method.
According to Fung [15], using Lagrange interpolation
functions as test functions, the differential quadrature
in time domain was shown to be equivalent to the
recast implicit Runge-Kutta method [16-18], besides,
some low-order algorithms were discussed in detail.
accuracy with the defined three grid
Conclusions are then given in Section 7.
points.
2. Traditional differential quadrature method
Suppose function f (x) is sufficiently smooth in the
whole interval, there are (s+1) grid points with
coordinates
as ci , i  (0, s) .
The
first
order
derivative f 1 (ci ) at each grid point ci , i  (1, s) , is
However, the method used by Fung is not the
traditional sense of differential quadrature method, but
involved post-processing (i.e. numerical solution at
the end of grid points adopts polynomial
extrapolation).
In this paper, using general polynomial as test
functions [19], the weighting coefficients matrix of
differential quadrature method is proved to satisfy V-
approximated by a linear sum of all the function
values in the whole domain, that is
s
f 1 (ci )   g ij f (c j ) , i  (1, s)
(1)
j 0
Where f (ci ) represent function values at a grid
point ci , g ij is the weighting coefficients.
In order to compute the weighting coefficients g ij
transformation [17, 20]. The equivalent implicit
Runge-Kutta method is constructed through the
differential quadrature method. Hence, making use of
Butcher fundamental order theorem and the method of
linear stability analysis [17, 18], the basic
characteristics of the differential quadrature method
can be systematically analysed. Unfortunately, the
differential quadrature method is only a method of sstage s-order and A-stable. Consequently, the
differential quadrature method can’t yield higher
accurate solutions to the boundary-value problems
with fewer computational efforts. Based on above
deduction, the method of undetermined coefficients is
used to make the stability function of the equivalent
Runge-Kutta method become the diagonal Padé
approximations to the exponential function [17, 18].
Therefore, a class of improved differential quadrature
method of s-stage 2s-order is derived.
The manuscript is arranged as follows. In Section 2,
the weighting coefficients matrix of traditional
differential quadrature method using general
polynomial as test functions is briefly discussed. In
Section 3, the equivalent relationship between the
differential quadrature method and the Runge-Kutta
methods is deduced. In Section 4, the stability and
in Equation (1), the test functions can be chosen as
rk ( x)  x k , (k  0 ,1 , s)
(2)
Substituting Equation (2) into Equation (1) gives
s
k  0 , 0   g ij , i  (1, s)
(3)
j 0
s
 g ij c kj
j 0
 k  cik 1 , i , k  (1, s)
(4)
Equation (3) can be expanded into matrix form as
 g 11

 g 21
 

g
 s1
g 12  g 1s 1  g 10   0 
  
  
g 22  g 2 s 1  g 20   0 


           
  
  
g s 2  g ss 1  g s 0   0 
(5)
Let
 g10 
 g11



 g 21
 g 20 
G0  
, G 






g
g 
 s1
 s0 
g1s 
 1

 
 g2s 
 1
, e    (6)

 


 



 g ss 
 1
g12 
g 22

gs2
Using Equation (6), Equation (5) can be rewritten as
(7)
G0  Ge
From k  1, 2 , s , Equation (4) can be expanded as
 g i 0 c0  g i1c1  g i 2 c2    g is cs  1

2
2
2
2
 g i 0 c0  g i1c1  g i 2 c2    g is cs  2ci



 g c s  g c s  g c s    g c s  sc s 1
i1 1
i2 2
is s
i
 i0 0
accuracy characteristics of the differential quadrature
method are studied. A class of improved differential
quadrature method of s-stage 2s-order and A-stable is
proposed in Section 5. In Section 6, the transient
response of a double-degree-of-freedom system is
computed, which is given to verify the computational
(8)
Since initial grid point c 0 is usually defined as 0,
Equation (8) reduces to
2
 g i1 c1  g i 2 c 2    g is c s  1

2
2
2
 g i1 c1  g i 2 c 2    g is c s  2c i



s
s
 g c  g c    g c s  sc s 1
i2 2
is s
i
 i1 1
1
α s  [1 , 2 , , s ]T  V 1c s ,
s
(9)
c s  [c1s ,c2s , ,c ss ]T
Equations (13), i.e. G  VA s1 V 1 , is called the
From i  1, 2,s , Equation (9) can be also expanded
implicit expression of the weighting coefficients
matrix of the differential quadrature method, and is
also called V-transformation.
into matrix form as
 c1

c
G 2

c
 s
c12  c1s  1 2c1
 
c 22  c 2s  1 2c 2

     
c s2
c ss  1 2c s
1 c1

1 c 2

 
1 c
s

 sc1s 1 

 sc2s 1 


 
 scss 1 
 1



2



 
  

s 
 c ss 1 
s 1
1
s 1
2
 c
 c
When the grid points have been selected, the
weighting coefficients matrices G and G 0 are easy to
calculate with the above formula. Obviously, the
weighting coefficients of the differential quadrature
method depend on the test functions and distribution
of grid points, but are independent of some
specific problems. There are four typical grid points’
distributions: Legendre grid points, Chebyshev grid
points, Chebyshev-Gauss-Lobatto grid points and
Uniform grid points (also called Equally spaced grid
points) [18]. This paper will focus on the latter three
kinds of commonly used grid points, which are
defined as follows:
1) Chebyshev grid points
(10)
Vandermonde matrix V is defined as follows:
1 c1

1 c 2
V 
 
1 c
s

 c1s 1 

 c2s 1 

  
 c ss 1 
(11)
Making use of Equations (11), Equations (10) can be
expressed as
 c1

c
1
G V  2

c
 s
1 c1

1 c 2

 
1 c
s

c12
c 22

c s2
1
 c1s 
1

 c 2s 
2

  

 c ss 

0

2
s
c1  c1  1

c 22  c 2s 

   

c s2  c ss 








1

s
0  0



1

2



1

s
ck 
1
V G V  As
2) Chebyshev-Gauss-Lobatto grid points
ck 
(12)
1 

2 



 



 
1
0 
 s 
s 1

0
0
1
2

0
0
1
 k 
1  cos     , k  (0, s) .
2
 s 
3) Uniform grid points
k
ck  , k  (0, s) .
s
3. The equivalence of differe ntial quadrature
method and Runge-Kutta method
In order to analyse the numerical stability and order
of the differential quadrature method, the differential
quadrature method in time domain can be transformed
into equivalent implicit Runge-Kutta
method.
Consider the following ordinary differential equation:
(13)
Where As is defined as
0

1

As   0


 0

1
 2k  1  
1  cos 
   , k  (1, s  1) ,
2
 2s  2  
c 0  0 , cs  1 .
Finally, it can be inferred that
1
(15)


 dx
  f (t, x ) , 0  t  T
 dt
 x (t  0)  x 0

(14)
(16)
In the following, t n ,t n 1 represent respectively the
beginning and the end points at each step. h  t n1  t n
with
will be used to denote the step size. The time interval
3
[t n ,t n 1 ] will be normalized. i.e, c  (t  t n ) h ,
c1
c A 

b T cs
t  [t n ,t n1 ] , c  [0,1] . At the same time, Equation (16)
can be made in the standard normalized form
d
(17)
x  x(t n  ch)
x  hf (t n  ch , ~
x) , ~
dc
then, using s-stage differential quadrature method to
solve Equation (17) yeilds
x 
x1 ) 
~
 f (t n  c1 h , ~
 1


G     G 0 x n  h

(18)

~

 f (t  c h , ~

x
x
)
s
s 
 s
 n
4. Analysis of the basic characteristics of the
differential quadrature method
The stability and accuracy characteristics of the
equivalent Runge Kutta method will be investigated
next. From Equation (13) and (20), it can be inferred
that
(19)
1


c12  c1s  
1


2
s
c2  c2  
2




   
1
2
s 
cs
cs  

s

Equation (25) reduces to
(21)
Therefore, the weigthing coefficients matrix A also
satisfies V-transformation. It can be inferred from
Equation (19) that
s
 aij c kj 1 
s
~
x i  x n  h a ij f (t n  c j h，~
x j ) , i  (1, s)
j 1
(22)
Sinc cs  1 , t n  cs h  t n  h  t n1 , therefore,
~
x i (i  s) is the approximate solution at the end of the
step. Then, ~
x can be rewritten as the following form:
cik
, i  (1, s ) , k  (1, s )
k
(26)
from Equation (25), it can be inferred that
1
1
bTV  [1, ,, ]
2
s
s
(27)
Similarly, Equation (27) reduces to
s
~
x s  x n 1  x n  h a sj f (t n  c j h，~
xj)
s
(25)
On the other hand, since b j  a sj , j  (1, s ) and cs  1 ,
j 1
j 1
 c1s 1 

 c 2s 1 

  
 c ss 1 
 c1

 c2


c
 s
(20)
Clearly, making use of Equation (13) and (20) leads to
A  VA sV 1
a12  a1s  1 c1

a 22  a 2 s  1 c 2
      

a s 2  a ss  1 c s
 a11

 a 21
 

a
 s1
Let
G 1  A  [ a ij ] , i , j  (1 , s)
(24)
where b T  (b1 ,b2 , ,bs )  (a s1 , a s 2 , , a ss ) .
where ~
xi  x (t n  ci h), i  (1, s) . Since G0  Ge ,
Equation (18) reduces to
x 
x1 ) 
~
1
 f (t n  c1 h , ~
 1
 


1

    x n     hG 

~

1
 f (t  c h , ~

x
x
)
n
s
s 
 s
 

a11  a1s


a s1  a ss
b1  bs
s
1
 bi cik 1  k
(23)
, k  (1, s)
(28)
i 1
 x n  h b j f (t n  c j h，~
xj)
Obviously, it has been shown that the equivalent
Runge-Kutta method at least satisfies simplifying
j 1
where b j  a sj , j  (1, s ) . It can be seen that Equation
assumptions C(s) and B(s) from Equation (26) and
(28). Furthermore, it can be verified that the
equivalent Runge-Kutta method only satisfies
simplifying assumptions D(0). From Theorem 5.1 on
p.71 in Reference [17], it can be concluded that the
implicit Runge Kutta method or the corresponding
differential quadrature method is s-stage s-order.
The stability function R(z ) of the equivalent Runge-
(22) and (23) are the standard forms for an s-stage
Runge-Kutta method. Since ~
x s  x n1 , the equivalent
Runge-Kutta method is a reducible method. In fact,
the traditional differential quadrature method
generally doesn’t involve post-processing, so the
Runge-Kutta method converted from traditional
differential quadrature method will naturally become a
reducible method. The Runge-Kutta method can be
conveniently summarized in the Butcher tableau [18]
Kutta method or the corresponding differential
quadrature method is given by the formula[16-18]
as
4
R( z ) 
det(I  z (eb T  A))
det(I  zA)
Because As and As are a class of special matrices,
(29)
Equation(35) can be evaluated as
(k  1)!
( z ) s k 1
det(I  zA )
( s  1)!
k s
(36)
R( z ) 

1
(k  1)! s k 1
det(I  zAs )
1 k
z
( s  1)!
k s
Where, as usual, I is the identity matrix of dimension
s. Due to grid points’s asymmetric distribution, the
equivalent implicit Runge-Kutta methods is not a
symmetric method. As a result, there is an unique
adjoint method (also called reflected method) [16, 18],
which defines as
c
A
(b  ) T
1

s
Since cs  1 , resulting in c1  0 , and it implies 1  0 .
It can be verified that the stability function of
equivalent Runge-Kutta method is A-acceptability of
p-order ( p  s  1 ) rational approximation to
, satisfying
c   e  Pc
 
T
 PA P  eb  A
 
b  Pb
(30)
1



1 

s s
P 
 R



1



(31)
exponential
function
[21].
Therefore,
the
corresponding differential quadrature method is Astable.
In the following, the three-stage differential
quadrature method using Uniform grid points will be
given as an example. When s=3, c1 , c 2 and c 3 are
where P is
given by 1 3 , 2 3 ,1. It can be worked out that
matrices G 0 and G are given by
Futhermore, from Theorem 343B on p.221 in
Reference [18], if the original method satisfies the
simplifying assumptions C(s) and B(s), then the adjont
method also satisfies the same simplifying
assumptions. Hence, the adjont method enjoys V-
 
 3
  1

 
 2
1

G0 
,G   3
2

 
 9
  1

 
 2
transformation:
1 c1

1 c2

*
 
 1
A  V As (V ) , V  
 
1 c 
s

 (c1 ) s 1 

 (c2 ) s 1 
 (32)

 
 (c s ) s 1 
and the Butcher tableau
method would be
1
3
2
c A 3

bT 1
0
0

0
1
2

0



1 

2 

 



 
1
0 
 s 
s 1

(33)

1

A  1


1

with
1
β s  [1 ,  2 , , s ]T  (V  ) 1 (c  ) s
s
(c  ) s  [(c1 ) s ,(c2 ) s , ,(c s ) s ]T
(34)
det(I  z ( PA  P -1 )) det(I  zA )


det(I  z (VA sV 1 )) det(I  zAs )
det(I  z (V  As (V  ) 1 )) det(I  zAs )

det(I  zAs )
det(I  zAs )
1
1 
 0 0
9 
4 
1 0
9 

1
1  0
2

(37)
4
9
2

9

0
0
5
36
1
9
1
4
1
4
(38)
Α is
2 
1
27 
11
 1
27 
2 
1
3 
1
3
2
3
1
1

9
4
9

1

1
(39)
the stability function of the equivalent Runge-Kutta
method is
1
1 2
1 z 
z
3
27
R(z) 
(40)
2
11 2 1 3
1 z 
z 
z
3
54
27
Equation (29) can be reduced to
R( z ) 
1
3
2
3
1
 
2
3
1 

2
11 
9

2 
3
of equivalent Runge-Kutta
23
36
7
9
3
4
3
4
the V-transformation of matrix
where As is also defined as
0

1

A*s   0


 0

1  k
(35)
5
and the Butcher tableau of the adjoint method would
be
c
0
1
3
A
2
 T
(b )
3
0
1

36
1
9
1
4
0
2
9
4
9
0
5
36
1
4
3
4
0
Then, the stability function of this new Runge-Kutta
method becomes
1
(k  1)!
1 k
( z ) s k 1
~
det(I  zAs )
( s  1)!
k s
R( z ) 
~ 
1
(k  1)! s k 1
det(I  zAs )
1  k
z
( s  1)!
k s
(41)
(44)
From Equation (43) and (44), it can be inferred that
~
the last column elements in As determine the stability
the V-transformation of matrix Α* is

1

*
A  1


1

0
1
3
2
3

0  0

1 
1
9 


4
 0
9 
0
0
1
2

0 1

2 

1
27 


1
1
3 
0
1
3
2
3

0

1
9
4

9
1
function. To improve the order of new Runge-Kutta
method, undetermined coefficients γ can be selected
(42)
so that the stability function of new Runge-Kutta
method becomes the diagonal Padé approximations to
s
the exponential function (defined by e z ):
s
5.
The improved differential quadrature
method
(k  1)!
( z ) s k 1
(
s

1
)!
k s
R( z ) 
 ez
1
(k  1)! s k 1
1  k
z
( s  1)!
k s
1
1  k
Based on the above deduction, the traditional
differential quadrature method is a method of s-stage
s-order. Compared with the multi-stage high-order
Runge-Kutta method, for example, Gauss method (sstage 2s-order method), it has the disadvantage of
lower precision. As is well known that: if the stability
function of a numerical method is diagonal Padé
approximations to the exponential function, then this
method is the method of A-stable and s-stage 2s-order
[22]. Inspired by this idea, the stability function of
new Runge-Kutta method or new differential
quadrature method have been converted into diagonal
Padé approximation to the exponential function by
using undetermined coefficients method. From
conveniently obtained as
s  2 , γ 2   1 12 ,1 2 ;
s  3 , γ 3  1 60 ,  1 5 ,1 2  ;
s  4 , γ 4   1 280 ,1 14 ,  9 28 , 1 2 ;
……
， s  , so that coefficients
After getting γ s   1，
T
~
~
matrix A or G can also be easily computed through
using Equation (43). Therefore, a class of new RungeKutta
~
~
A  (G ) 1
0
0


0 


method
of
s-stage
2s-order
has
been
successfully constructed. In other words, a class of
improved differential quadrature method of s-stage 2s-
without changing b and V, a new Runge-Kutta
~
c A
method
is redefined as
bT
0
0
1
2

(45)
s
By comparing the coefficients on both sides of
Equation (45), undetermined coefficients γ can be
Equation (35), it can be seen that the stability function
of the equivalent Runge-Kutta method will be
~
determined by As and As . Suppose As  As  As ,
0

1

 V 0


 0

s
order has been derived. Besides, the adjoint method of
new Runge–Kutta method is also s-stage 2s-order.
Take the same as above, the improved differential
1 

2
quadrature method using Uniform grid points will be
given as an example. It can be worked out that the
new matrices G 0 and G are given by

~ 1
 V 1  VA
sV


 
1
 s 
s 1


(43)
6
u1 (0)  0 v1 (0)  0
,

u 2 (0)  0 v 2 (0)  0
17 
 17
 4 
10  



6
 2
 3 
~  4  ~  11
1 
G0 
G 
4
(46)
 2
 3 ,
6 
 75



33
 42


  12 
2 


 2
and the Butcher tableau of new Runge-Kutta method
would be
1 73
23 13

3 120
60 120
17 17
~ 2 97
c A 3 120  60 120
(47)

3
7
b T 1 27
40
20
40
3
1
0
4
4
~
The V-transformation of matrix Α is
 1
1
 3
~  2
A 1
 3

1 1

Then, the Butcher
be
c
the exact solution of this problem is
5
2

 x1  1  3 cos( 2t )  3 cos( 5t )

 x  3  5 cos( 2t )  4 cos( 5t )
 2
3
3
The following three Figures show the displacement
error trajectories comparison of traditional differential
quadrature method and improved differential
quadrature method with the same step size h=0.5s. In
these Figures, the exact solution at each step is used
for comparison. From Figure1-3, it can be evident that
improved differential quadrature method is two orders
of magnitude higher than traditional differential
quadrature method. The error of improved differential
-5
-4
quadrature method range between 10 and 10 , even
with a large step size h=0.5s.
In order to compare the calculation precision of the
traditional differential quadrature method and the
improved differential quadrature method better,
dynamic Equations (51) were computed for different
time steps. Table 1 shows the computational results at
t=60s. Comparing with the exact solution, the
numerical results using traditional differential
quadrature method are the same until the first decimal
place when s=5, h=1s; while the numerical results
using improved differential quadrature method are the
same until the third decimal place. When s=10, h=2s,
the numerical results using traditional differential
quadrature method are the same until the fifth decimal
place; while the numerical results using improved
differential quadrature method are the same until the
seventh decimal place (especially using Chebyshev
grid points and Chebysev-Gauss-Lobatto grid points,
the numerical results are almost the exact solution).
1
1
3
T
2
b
3
~
A*
3
40
13
120
17
120
1
4
3
20
17
60
23
60

0
3
40
7

120
17
120
3
4
(49)
~
The V-transformation of matrix Α* is

1

~* 
A  1


1

0
1
3
2
3

0  0

1 
1
9 
4 
 0
9 
0
0
1
2
1 
1
60 
1
 1
5 
1 
1
2 
0
1
3
2
3

0

1
9
4

9
1
(50)
6. Numerical examples
Consider a two-degree-of-freedom system governed
by
 2 0  x1   6  2  x1   0 

   
    
 0 1 x2    2 4  x2  10
(53)
The differential quadrature method can be used to find
the numerical solutions to transient response directly.
It can be seen the detailed calculation steps of solving
second-order differential equations in Reference [11].
1 
1  1 1 
 0 0
1

9 
60  3 9 
4 
1
2 4
(48)
1 0  1




9
5
3 9


1 1 
1  0
1 1 1 
2 2 


tableau of the adjoint method would
0
(52)
(51)
With initial condition
7
10
 x1 (t )
10
10
10
10
10
0
-1
traditional DQM
improved DQM
-2
-3
-4
-5
0.5
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5 10
7.5
8.5
9.5 10
t/s
(a) Error trajectories of x1 (t )
10
 x 2 (t )
10
10
10
10
10
0
-1
traditional DQM
improved DQM
-2
-3
-4
-5
0.5
1.5
2.5
3.5
4.5
5.5
6.5
t/s
(b) Error trajectories of x2 (t )
Fig 1. Error trajectories comparison of different DQM using Chebyshev grid (s=3, h=0.5s)
10
 x1 (t )
10
10
10
10
0
-1
traditional DQM
improved DQM
-2
-3
-4
0.5
1.5
2.5
3.5
4.5
5.5
6.5
t/s
(a) Error trajectories of x1 (t )
8
7.5
8.5
9.5 10
10
10
 x 2 (t )
10
10
10
10
0
-1
traditional DQM
improved DQM
-2
-3
-4
-5
0.5
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5 10
t/s
(b) Error trajectories of x2 (t )
Fig 2. Error trajectories comparison of different DQM using Chebyshev-Gauss-Lobatto grid (s=3, h=0.5s)
10
10
 x1 (t )
10
10
10
10
0
-1
traditional DQM
improved DQM
-2
-3
-4
-5
0.5
1.5
3
3.5
4.5
5.5
6.5
7.5
8.5
9.5 10
7.5
8.5
9.5 10
t/s
(a) Error trajectories of x1 (t )
10
10
 x 2 (t )
10
10
10
10
0
-1
traditional DQM
improved DQM
-2
-3
-4
-5
0.5
1.5
2.5
3.5
4.5
5.5
6.5
t/s
(b) Error trajectories of x2 (t )
Fig 3. Error trajectories comparison of different DQM using Uniform grid (s=3, h=0.5s)
9
Table 1 Computational results of the displacement (t=60s)
s=5
h=1s
s=10
h=2s
t=60s
x1 (t )
x2 (t )
Exact solution
2.264386552
5.469005157
Traditional DQM-Chebyshev
2.253975383
5.491424542
Improved DQM-Chebyshev
2.264406270
5.468965760
2.276613635
5.439926064
2.264406270
5.468965760
Traditional DQM-Uniform
2.342513616
5.278336836
Improved DQM-Uniform
2.264406270
5.468965760
Traditional DQM-Chebyshev
2.264383332
5.469011768
Improved DQM-Chebyshev
2.264386552
5.469005157
2.264399479
5.468978613
2.264386552
5.469005157
Traditional DQM-Uniform
2.265087814
5.467565205
Improved DQM-Uniform
2.264386561
5.469005173
Traditional DQM-ChebyshevGauss-Lobatto
Improved DQM-ChebyshevGauss-Lobatto
Traditional DQM-ChebyshevGauss-Lobatto
Improved DQM-ChebyshevGauss-Lobatto
7. Conclusion
(3) differential quadrature method can also gives
more accurate solutions. This is the main
mechanism that differential quadrature method
has been successfully applied in many fields.
In this paper, the linear stability and the order of
differential quadrature method in time domain are
systematically studied in detail and a class of new
differential quadrature method of s-stage 2s-order is
proposed. From the above analysis and derivation, the
following conclusions can be made:
(4) Finally, by making the stability function of
equivalent Runge-Kutta method become the
diagonal Padé approximations to the exponential
function, a class of improved differential
quadrature method of s-stage 2s-order and Astable is proposed. Therrfore, the improved
differential quadrature method can be extended
to multi-degree-of-freedom time domain dynamic
systems, which can produce higher accurate at
lower computational cost.
(1) Based on general polynomial as test functions,
the weighting coefficients matrix of the
differential quadrature method satisfies the Vtransformation. It plays an extremely important
role in the analysis of basic characteristics of
differential quadrature method
and
its
improvement.
(2) The traditional differential quadrature method can
be converted into equivalent Runge-Kutta method
of A-stable and s-stage s-order. Therefore,
compared with the commonly used single-stage
low-order numerical integral methods, even with
Conflict of Interests
The authors declare that there is no conflict of
interests regarding the publication of this paper.
Acknowledgements
a small number of grid points, the traditional
The authors gratefully acknowledge the support from
the National Natural Science Foundation of China
10
IEEE M icrowave and Guided Wave Letters, vol.9, no.4,
pp.145-147, 1999.
(NSFC) through its grant 51377098.
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11