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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)
Department of Mathematics, Hong Kong Baptist University
International Workshop on Scientific Computing
On the Occasion of Prof Cui Jun-zhi’s 70th Birthday
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).Spectral postprocessing (Tang and X. Xu/Fudan)
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.Spectral postprocessing
We introduce the linear coordinate transformation
and the transformations
Then problem (1.1)-(1.2) becomes
Y’(x) = G(Y; s), 1 < s 1;
Y(1) = y0.
Let be the Chebyshev-Gauss-Labbato points:
We project G to the polynomial space PN:
where Fj is the j-th Lagrange interpolation polynomial associated with the Chebyshev-Gauss-Labbato points.
Since Fj PN, it can be expanded by the Chebyshev basis functions:
Assume it is satisfied in the collocation points , i.e.,
we finally obtain the following numerical scheme
It is noticed that
y’ = y + cos(x+1)ex+1, x (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.Example 1
As an application, we apply the spectral postprocessing technique for the Hamiltonian system:
with the initial valuep(t0) = p0, q(t0) = q0,
Feng Kang, Difference schemes for Hamiltonian formalism an symplectic geometry, J. Comput. Math., 4 1986, pp. 279-289.
Integrating (1.5) leads to a system of integral equation
Assume (1.6) holds at the Legendre or Chebyshev collocation points:
tkj = (tk + 1) + j, 0 j N.
We can discretize the integral terms in (1.7) using Gauss quadrature together with the Lagrange interpolation:
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 1010, the symplectic scheme without postprocessing requires about 5 times more 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.
Legendre spectral method is proposed and analyzed for Volterra type integral equations:
where the kernel k and the source term g are given.
Let be the zeros of Legendre polynomials of degree Ns+1, i.e., LNs+1(x). Then the spectral collocation points are
We collocate (1.8) at the above points:
Using the linear transform
Consider Eq. (1.8) with
Example 4: errors vs Ns and iterative steps.
Theorem 1Let u be the exact solution of the Volterra equation (1.9) and assume that
where uj is given by spectral collocation method and Fj(x) is the j-th Lagrange basis function associated with the Gauss-points If u Hm(I), then for m 1,
provided that N is sufficiently large.
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 Gauss-Lobatto points Then
Lemma 3.3 Assume that Fj(x) is the N-th Lagrange interpolation polynomials associated with the Gauss, or Gauss-Radau, or Gauss-Lobatto points. ThenThe convergence analysis (Proof ingredients)
[Yanping Chenand Tang]
Ishtiaq Ali (CAS)
Hermann Brunner (Newfoundland/HKBU)
u(x) = a(x)u(qx),0 < x T,
u(0) = y0,
where 0 < q < 1 is a given constant …
y(t) = b(t)y(qt + q1), -1 < t 1,
y(-1) = y0.
(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.
provided that N is sufficiently large
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 isy(t) = sin(t).
(a): q = 0.5 and (b): q = 0.99.
Consider the time fractional diffusion equation of the form
subject to the following initial and boundary conditions:
u(x,0) = g(x), x ,
u(0,t) = u(L,t)=0, 0 t T,
where is the order of the time fractional derivative. is defined as the Caputo fractional derivatives of order given by
where S is an elastic tensor related to the extra-stress tensor of the fluid by is the rate of deformation tensor.
The extra-stress tensor is given by an adequate constitutive equation,
where the memory function is
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.
[H. Brunner/Newfoundland and HKBU and Tang]