COBY: Standard 1-D Chebyshev collocation method

Purpose

• The package "Coby" provides the common Chebyshe collocation method for solving the 1-D elliptic equation with the general boundary condition.

Specifications

• Name: Coby.
• Author: Feng Chen.
• Finishing date: 10/13/2012.
• Languages: MATLAB.

Simple Example

• Equation: \begin{equation} \begin{aligned} & \alpha u - u'' = f, \quad x \in (-1,1), \\ & a_{\pm} u(\pm 1)+b_{\pm} u'(\pm 1)=c_{\pm}. \end{aligned} \end{equation}
• Parameters: \begin{equation} \alpha = 3, \quad a_{\pm} =1, \quad b_{\pm} = 0. \end{equation}
• Exact solution and input functions: \begin{equation} u(x) = \sin(\pi x), \end{equation} then $f(x)$ and $c_{\pm}$ are calculated accordingly.

Quick Start

• Compiling and running:

  matlab Coby_Driver_Dirichlet
  

• Output:

• CPU: Intel(R) Xeon(R) CPU X5550 @2.67GHz.
• OS: CentOS release 6.4 (Final).
• Release: MATLAB R2012a.

References

• Lloyd N. Trefethen. Spectral Methods in MATLAB (Software, Environments, Tools), SIAM, (2001).

Code Highlights

% initialize the first-order collocation matrix
D = cheb(N); 

% second-order collocation matrix
D2 = D*D;

% construct the linear system
A = alp*diag(ones(N+1,1))-D2;

% modify the first and last row for boundary conditions
A(1,:) = bp*D(1,:)+ap*[1,zeros(1,N)];
A(N+1,:) = bm*D(N+1,:)+am*[zeros(1,N), 1];

% modify right hand side for boundary conditions
b = [cp;f(2:N);cm];

% solve for the solution
u = A\b;