Spectral methods for
initial value problems
and integral equations
Tang Tao
Department of Mathematics, Hong Kong Baptist University
International Workshop on Scientific Computing
On the Occasion of Prof Cui Jun-zhi’s 70th Birthday
Outline of the talk
Motivations (accuracy in time)
 Spectral postprocessing (efficiency)
 Singular kernels
 Delay-differential equations
 Extensions


Joint with Cheng Jin, Xu Xiang (Fudan)
2
Spectral postprocessing (Tang and X. Xu/Fudan)
We begin by considering a simple ordinary differential equation with given
initial value:
y’(x) = g(y; x), 0 < x  T,
(1.1)
y(0) = y0.
(1.2)




Can we obtain exponential rate of convergence for (1.1)-(1.2)?
For BVPs, the answer is positive and well known.
For the IVP (1.1)-(1.2), spectral methods are not attractive since (1.1)(1.2) is a local problem
A global method requires larger storage and computational time (need
to solve a linear system for large T or a nonlinear system in case that g
in (1.1) is nonlinear).
3
Spectral postprocessing



Purpose: a spectral postprocessing technique
which uses lower order methods to provide
starting values.
A few Gauss-Seidal type iterations for a well
designed spectral method.
Aim: to recover the exponential rate of
convergence with little extra computational
resource.
4
Formulas …
We introduce the linear coordinate transformation
T  t0
T  t0
x
s
,
 1  s  1,
2
2
and the transformations
T  t0 
T  t0 
 T  t0
 T  t0
Y ( s)  y
s
G (Y ; s )  g  Y ;
s
,
.
2 
2
2 
 2

Then problem (1.1)-(1.2) becomes
Y’(x) = G(Y; s),
1 < s  1;
Y(1) = y0.
N
Let {s j } j 0 be the Chebyshev-Gauss-Labbato points:
 j 
s j  cos ,
0  j  N.
N
 
We project G to the polynomial space PN:
N
G(Y ; s)   G(Y j ; s j ) Fj ( s),
j 0
where Fj is the j-th Lagrange interpolation polynomial associated with the
Chebyshev-Gauss-Labbato points.
5
Formulas …
Since Fj  PN, it can be expanded by the Chebyshev basis functions:
N
F j ( s )    mjTm ( s ).
m 0
N
Assume it is satisfied in the collocation points {si }i 0 , i.e.,
N
F j ( si )    mjTm ( si ),
0  i  N,
m 0
which gives
2
 jm 
 mj  ~ ~ cos
,
Ncm c j
 N 
we finally obtain the following numerical scheme
N
Y j  y0   ijG(Y j ; s j ),
j 0
where
1
ij  ~
Nc j
(1.3)
1
1
2(1) m 
 jm  1
cos
Tm1 ( si ) 
Tm1 ( si )  2


.
~
m 1
m 1 
 N  m  1
m 1 cm
N
It is noticed that Tm1 ( si )  cos(i(m  1) / N ).
6
Legendre collocation (Lobatto III)
N
Let {x j } j 0 be the Legendre-Gauss-Labatto points, we obtain the following
numerical scheme
N
Yi  y0   w jiG(Y j , x j ),
where
(1.4)
j 0
1 N Lm ( x j ) 1
Lm1 ( xi )  Lm1 ( xi ).
w ji 

N  1 m  0 LN ( x j ) 2 m  1
7
Example 1
Consider a simple example
y’ = y + cos(x+1)ex+1, x  (1,1],
y(1)=1.
The exact solution of the is y=(1+sin(x+1))exp(x+1).


First use explicit Euler method to solve the
problem (with a fixed mesh size h=0.1).
Then we use the spectral postprocessing
formulas to update the solutions using the
Gauss-Seidal type iterations.
8
(a)
(b)
(c)
Example 1: errors vs Ns for
spectral postprocessing method
(1.4), with (a): Euler, (b): RK2,
and (c): RK4 solutions as the
initial data.
9
Spectral postprocessing for Hamiltonian systems
As an application, we apply the spectral postprocessing
technique for the Hamiltonian system:
dp
  q H ( p, q ),
dt
dq
  p H ( p, q),
dt
t0  t  T ,
(1.5)
with the initial value p(t0) = p0, q(t0) = q0,
Feng Kang, Difference schemes for Hamiltonian formalism an symplectic
geometry, J. Comput. Math., 4 1986, pp. 279-289.
4th-order explicit Runge-Kutta
 4th-order explicit symplectic method

10
Spectral postprocessing for Hamiltonian systems
Integrating (1.5) leads to a system of integral equation
t
t
p(t )  pk    q H ( p, q)ds,
q(t )  qk    p H ( p, q)ds.
tk
tk
(1.6)
Assume (1.6) holds at the Legendre or Chebyshev collocation points:
t kj
t kj
p(t kj )  pk    q H ( p, q)ds,
q(t kj )  qk    p H ( p, q)ds. (1.7)
tk
tk
where
tkj = (tk + 1) + j, 0  j  N.
We can discretize the integral terms in (1.7) using Gauss quadrature
together with the Lagrange interpolation:
p(t kj )  pk 
q(t kj )  qk 
t kj  t k
2
tkj  t k
2
N

l 0
N

l 0
q
H ( I N p( skjl ), I N q( skjl ) wl ,
p
H ( I N p( skjl ), I N q( skjl ) wl .
11
Example 2
Consider the Hamiltonian problem (1.5) with
1 2
H ( p, q)  ( p  q 2 ),
2
p(t0 )  sin( t0 ), q(t0 )  cos(t0 ).
This system has an exact solution (p, q) = (sint, cost).
We take T=1000 in our computations.



Table 1(a) presents the maximum error in t[0,1000] using
both the RK4 method and the symplectic method.
Table 2(b) shows the performance of the postprocessing
with initial data in [tk, t2+2] generated by using RK4 t=0.1).
To reach the same accuracy of about 1010, the symplectic
scheme without postprocessing requires about 5 times
more CPU time.
12
(a)
RK4
Symplectic
Max. Error CPU time Max. Error CPU time
t = 101
blow up

9.20e-03
0.16s
t = 102
1.81e-0
1.60s
1.32e-06
1.52s
t = 103
2.43e-2
12.02s
9.16e-11
10.60s
(b)
iter step=3
Max. Error CPU time
iter step=6
Max. Error CPU time
N=8
1.74e-2
1.796s
5.49e-07
1.828s
N = 10
1.74e-2
1.813s
5.49e-07
1.843s
N = 12
1.74e-2
1.828s
5.49e-07
1.859s
(c)
iter step=3
Max. Error
CPU time
N=8
2.23e-5
1.797s
7.07e-10
1.843s
N = 10
2.17e-5
1.812s
6.83e-10
1.862s
N = 12
2.14e-5
1.860s
6.83e-10
1.906s
iter step=6
Max. Error CPU time
Example 2.
(a): the maximum errors
obtained by RK4 and the
symplectic method;
(b): spectral postprocessing
results using the RK4 (t =
0.1) as the initial data in
each sub-interval [tk, tk+2];
(c): same as (b), except that
RK4 is replaced by the
symplectic method. Here N
denotes the number of
spectral collocation points
used.
13
(a)
(b)
Example 2: errors vs Ns and iterative steps with (a): RK4
results and (b): symplectic results as the initial data.
14
Spectral postprocessing for Volterra integral equations
Legendre spectral method is proposed and analyzed for Volterra type
integral equations:
x
u( x)   k ( x, s, u(s))ds  g ( x),
a
x [a, b]
(1.8)
where the kernel k and the source term g are given.
Let { i }i s0 be the zeros of Legendre polynomials of degree Ns+1, i.e.,
LNs+1(x). Then the spectral collocation points are
ba
ba
xis 
i 
.
2
2
We collocate (1.8) at the above points:
N
xis
u ( x )  g ( x )   k ( xis , s, u ( s)) ds  g ( x),
s
i
s
i
a
Using the linear transform
xis  a
xis  a
xa
xa
s( ) 

, si ( ) 

2
2
2
2
we have
x s  a Ns
u( xis )  g ( xis ) 
i
2

0  i  Ns .
1    1

s
k
x
 i , si (k ), u(si (k )) wk .
k
15
Example 4
Consider Eq. (1.8) with
2 tan( u )
k ( x, s , u ) 
, a  1, b  1,
2
2
1 x  s
g ( x)  arctan( x)  ln( 1  2 x 2 )  ln( 2  x 2 ).
Example 4: errors vs Ns and
iterative steps.
16
The convergence analysis
[Tang, Xu, Cheng/Fudan Univ]
x
u( x)   K ( x, s)u(s)ds  g ( x),
x [1, 1].
1
(1.9)
Theorem 1 Let u be the exact solution of the Volterra
equation (1.9) and assume that
N
U ( x)   u j Fj ( x),
j 0
where uj is given by spectral collocation method and Fj(x) is
the j-th Lagrange basis function associated with the Gausspoints {x j }Nj0 . If u  Hm(I), then for m  1,
~
u  U L ( I )  CN 1/ 2m max K ( xi , s( xi ,  )) ~
u L ( I )  CN  m | u |H~ ( I ) ,

1i  N
H m ,n ( I )
2
m ,n
provided that N is sufficiently large.
17
The convergence analysis (Proof ingredients)
Lemma 3.1 Assume that a (N+1)-point Gauss, or Gauss-Radau, or GaussLobatto quadrature formula relative to the Legendre weight is used to
integrate the product u, where u  Hm(I), with I:=(1, 1) for some m  1 and
  PN. Then there exists a constant C independent of N such that
1

1
u ( x) ( x)dx  (u,  ) N  CN m | u |H~ m ,N ( I ) ||  || L2 ( I ) ,
Lemma 3.2 Assume that u  Hm(I) and denote INu its interpolation
polynomial associated with the (N+1)-point Gauss, or Gauss-Radau, or
N
Gauss-Lobatto points {x j } j 0 . Then
u  INu
2
L (I )
 CN  m | u |H~ m ,N ( I ) .
Lemma 3.3 Assume that Fj(x) is the N-th Lagrange interpolation
polynomials associated with the Gauss, or Gauss-Radau, or GaussLobatto points. Then
N
23 / 2 1/ 2
max  | F j ( x) |
N .
x( 1,1)
j 0

18
Methods and convergence analysis for
t
u(t )   (t  s)  k (t , s)u(s)ds  g (t ),
0
0  t  T.
[Yanping Chen and Tang]
(a)
(b)
Chebyshev spectral for \alpha=0.5
Jacobi-spectral for general \alpha
19
Spectral methods for
pantograph-type DDEs
Ishtiaq Ali (CAS)
Hermann Brunner (Newfoundland/HKBU)
Tao Tang

Consider the delay differential equation:
u(x) = a(x)u(qx), 0 < x  T,
u(0) = y0,
where 0 < q < 1 is a given constant …

Using a simple transformation, the above problem
becomes
y(t) = b(t)y(qt + q1), -1 < t  1,
y(-1) = y0.
21

Difficulties in using finite-difference type methods
(a). u(qx) – un-matching of the grid points so interpolations
are needed – difficult to obtain high order methods
(b). Difficult in obtaining stable numerical methods (analysis
has been available for q=0.5 only)
(c). Difficult when q close to 0 or 1.
22
1 qt j  q1  v  q1 
 y (v)dv,
 y (t j )  y0  
b
q 1
 q 
j  1.
Projecting the above integrand to N , we have
N
 v  q1 
 t k  q1 
 y (v)   b
 y (t k ) Fk (v),
b
q 
k 0 
 q 
Expand Fk (v) in terms of the Legendre polynomial s :
N
Fk (v)   ckm Lm (v).
m 0
N
 Y j  y0   bk Yk wk , j ,
1 j  N,
k 0
 wk , j
1

2
qN ( N  1)LN ( xk )
N
L
m 0
m
( xk )Lm 1 qt j  q1   Lm 1 qt j  q1 .
23
Theorem: If the function b is sufficiently smooth (which also implies
that the solution is smooth), then
Yy
L ( I )
 CN
 m 1/ 2
b yq   q1  H~
 CN 1/ 2m b H~
y
m ,N ( I )
m ,N ( I )
,
L2 ( I )
provided that N is sufficiently large
24

Consider the general pantograph equation
 y(t )  a(t ) y(t )  b(t ) y(qt )  c(t ) y(qt )  g (t ),

 y0  0.
t  (1,1]
with a(t) = sin(t), b(t) = cos(qt),
c(t) = -sin(qt), g(t) = cos(t) – sin2(t).
The exact solution of the problem is y(t) = sin(t).
25
Figure: L errors for general pantograph equation with neutral term.
(a): q = 0.5 and (b): q = 0.99.
26
Spectral methods for fractional diffusion equation
(Huang/Xu/Tang)
Consider the time fractional diffusion equation of the form
 u ( x, t )  2u ( x, t )

 f ( x, t ), x  , 0  t  T

2
t
x
subject to the following initial and boundary conditions:
u(x,0) = g(x), x  ,
u(0,t) = u(L,t)=0, 0  t  T,
  u ( x, t )
t 
where  is the order of the time fractional derivative.
is
defined as the Caputo fractional derivatives of order  given
by
t u ( x, s )
 u ( x, t )
1
ds

, 0    1.



0
t
(1   )
s (t  s)
27
Basic equations for Viscoelastic flows
  v  0,
 v

   ( v v)  p  0 2 v    S   0 g,
 t

0 
where S is an elastic tensor related to the extra-stress tensor
T







S
where



v

(

v
)
of the fluid by
is the rate of
0
deformation tensor.
The extra-stress tensor is given by an adequate constitutive
equation,
t
 (t )   M (t  t ) H ( I1 , I 2 )Bt (t )dt 

where the memory function is
m1
M (t  t )  
m 1
am
m

e
t t 
m
28
(a)
(b)
Predicted streamlines for the flow through a 4:1 planar
contraction for Re=1 using the finite volume code of Alves et al.
(a) Newtonian; (b) UCM model with We=4.
29
Happy Birthday, Professor Cui!
30
Methods and error analysis for delay equations
qt
u(t )  a(t )u(qt )   (t  s)  b(t , s)u(s)ds,
0
0  t  T.
[H. Brunner/Newfoundland and HKBU and Tang]
31