# burgers_time_viscous

burgers_time_viscous, a MATLAB code which solves the time-dependent viscous Burgers equation using a finite difference discretization of the conservative form of the equation, and then carrying out a simple parabolic integration scheme.

The function u(x,t) is to be solved for in the equation:

du/dt + u * du/dx = nu * d^2u/dx^2
for 0 < nu, a <= x <= b, tmin <= t <= tmax with initial condition
u(x,tmin) = uinit(x);
and fixed Dirichlet conditions
u(a,t) = alpha, u(b,t) = beta

Problem data includes the spatial endpoints a and b, the Dirichlet boundary values u(a,t) = alpha, u(b,t) = beta, the time limits tmin and tmax, and the (positive) value of the viscosity nu.

The conservative form of the equation is

du/dt + 1/2 * d(u^2)/dx = nu * d^2u/dx^2
and this is the version we discretize.

### Languages:

burgers_time_viscous is available in a MATLAB version.

### Related Data and Programs:

burgers, a dataset directory which contains some solutions to the viscous Burgers equation.

burgers_solution, a MATLAB code which evaluates an exact solution of the time-dependent 1D viscous Burgers equation.

burgers_steady_viscous, a MATLAB code which solves the steady (time-independent) viscous Burgers equation using a finite difference discretization of the conservative form of the equation, and then applying Newton's method to solve the resulting nonlinear system.

fd1d_burgers_lax, a MATLAB code which applies the finite difference method and the Lax-Wendroff method to solve the non-viscous Burgers equation in one spatial dimension and time.

fd1d_burgers_leap, a MATLAB code which applies the finite difference method and the leapfrog approach to solve the non-viscous time-dependent Burgers equation in one spatial dimension.

### Reference:

1. Daniel Zwillinger,
Handbook of Differential Equations,
ISBN: 0127843965,
LC: QA371.Z88.

### Source Code:

• burgers_time_viscous.m, integrates a discretized form of the time dependent viscous Burgers equation.
• ic_expansion.m, an initial condition function for an expansion wave.
• ic_gaussian.m, an initial condition function for a Gaussian.
• ic_shock.m, an initial condition function for a shock wave.
• ic_spike.m, an initial condition function for a spike.
• ic_spline.m, an initial condition function for a spline through data.

Last revised on 04 December 2018.