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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)

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spectral methods for initial value problems and integral equations

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
Outline of the talk
  • Motivations (accuracy in time)
  • Spectral postprocessing (efficiency)
  • Singular kernels
  • Delay-differential equations
  • Extensions
  • Joint with Cheng Jin, Xu Xiang (Fudan)
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).

Spectral postprocessing (Tang and X. Xu/Fudan)
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.

Spectral postprocessing
formulas
Formulas …

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.

formulas6
Formulas …

Since Fj  PN, it can be expanded by the Chebyshev basis functions:

Assume it is satisfied in the collocation points , i.e.,

which gives

we finally obtain the following numerical scheme

(1.3)

where

It is noticed that

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.

Example 1
slide9
Example 1: errors vs Ns for spectral postprocessing method (1.4), with (a): Euler, (b): RK2, and (c): RK4 solutions as the initial data.

(a)

(b)

(c)

spectral postprocessing for hamiltonian systems
Spectral postprocessing for Hamiltonian systems

As an application, we apply the spectral postprocessing technique for the Hamiltonian system:

(1.5)

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.

  • 4th-order explicit Runge-Kutta
  • 4th-order explicit symplectic method
spectral postprocessing for hamiltonian systems11
Spectral postprocessing for Hamiltonian systems

Integrating (1.5) leads to a system of integral equation

Assume (1.6) holds at the Legendre or Chebyshev collocation points:

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:

example 2
Consider the Hamiltonian problem (1.5) with

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.

Example 2
slide13

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.

slide14

(a)

Example 2: errors vs Ns and iterative steps with (a): RK4 results and (b): symplectic results as the initial data.

(b)

spectral postprocessing for volterra integral equations
Spectral postprocessing for Volterra integral equations

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

we have

example 4
Example 4

Consider Eq. (1.8) with

Example 4: errors vs Ns and iterative steps.

the convergence analysis tang xu cheng fudan univ
The convergence analysis [Tang, Xu, Cheng/Fudan Univ]

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.

the convergence analysis proof ingredients
Lemma 3.1 Assume that a (N+1)-point Gauss, or Gauss-Radau, or Gauss-Lobatto 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

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. Then

The convergence analysis (Proof ingredients)
slide19
Methods and convergenceanalysis for

[Yanping Chenand Tang]

  • Chebyshev spectral for \alpha=0.5
  • Jacobi-spectral for general \alpha
s pectral method s for pantograph type ddes

Spectral methods for pantograph-type DDEs

Ishtiaq Ali (CAS)

Hermann Brunner (Newfoundland/HKBU)

Tao Tang

slide21
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.

slide22
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.

slide24
Theorem: If the function b is sufficiently smooth (which also implies that the solution is smooth), then

provided that N is sufficiently large

slide25
Consider the general pantograph equation

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).

slide26
Figure:L errors for general pantograph equation with neutral term.

(a): q = 0.5 and (b): q = 0.99.

spectral methods for fractional diffusion equation huang xu tang
Spectral methods for fractional diffusion equation(Huang/Xu/Tang)

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

basic equations for viscoelastic flows
Basic equations for Viscoelastic flows

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

slide29

(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.

slide31
Methods and error analysis for delay equations

[H. Brunner/Newfoundland and HKBU and Tang]