BARIS: Fourier method for the Burgers' equation

Purpose

• The package "BariS" solves the 1-D viscous Burgers' equation with periodic boundary condition. The spatial discretization is a Fourier psuedospectral method. Three methods were proposed to calculate the nonlinear term: (i) evaluating Fourier coefficients exactly with the convolution; (ii) padding zeros in the $k$-space; and (iii) filtering with the two thirds rule. For the temporal discretization, a second-order symmetric Strang splitting was used. The linear diffusion term was integrated exactly, while the nonlinear term by a third-order Runge-Kutta scheme.

Specifications

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

Simple Example

• Equation: \begin{equation} \begin{aligned} & u_t + uu_x = \epsilon u_{xx}, \quad x\in (0,2\pi), \\ & u \text{ is periodic}. \end{aligned} \end{equation}
• Parameters: \begin{equation} T=2, \quad \epsilon=0.001, \quad N_x = 1024, \quad \delta t = 0.005. \end{equation}
• Initial condition: \begin{equation} u_0(x) = \sin(x). \end{equation}

Quick Start

• Compiling and running:

  matlab Bari_Driver_Serial
  

• Graphics:
  
• 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).
• Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, Th.A. Spectral Methods: Fundamentals in Single Domains (Scientific Computation), Springer, (2007).

Code Highlights

%%% calculate the nonlinear term with convolution 

% re-order the modes
uu = [ui(lcut+1:N); ui(1:lcut)]/N;

% convolution
uc = conv(uu,uu);

% re-order to matlab convention
m = length(uc);
ucc = [uc((m+1)/2:m); uc(1:(m-1)/2)]*2*N;

% first-order derivative
du = rfft_grad_1d(ucc, 1);

% cut off the high modes
rhs = [du(1:lcut); du(lcut+N:m)]/2;
rhs = -rhs/2;