Spectral methods for initial value problems and integral equations

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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

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)
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)
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 …

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.

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

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
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

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 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:

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

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.

(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

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

Consider Eq. (1.8) with

Example 4: errors vs Ns and iterative steps.

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.

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)
Methods and convergenceanalysis for

[Yanping Chenand Tang]

• Chebyshev spectral for \alpha=0.5
• Jacobi-spectral for general \alpha

### 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.

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.

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

provided that N is sufficiently large

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).

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

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

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

(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.

Methods and error analysis for delay equations

[H. Brunner/Newfoundland and HKBU and Tang]