The Burgers’ Equation

My colleague Mikhail had recently introduced me to the Burger’s equation, and had posed the question of how the Rosenbrock techniques fare at solving the Burger’s equation for large values of Reynolds number.
The Burgers’ equation is a partial differential equation that is used to describe shock phenomenon in gas flows, first presented in J.M.Burgers’ 1948 paper. The general form of the Burgers’ equation is 

 

This is a 1-dimensional PDE with the Reynolds number Re controlling the stiffness of the equation. As indicated earlier, PDE’s can be converted into ODE’s using the Method of Lines. The MOL representation for the Burgers’ equation thus becomes:

With a known set of initial and boundary conditions, the MOL representation can be put to use to solve the PDEs.