The Solution of Burgers’ Equation for high Re’s

As the Reynolds number is ratcheted up to values of Re=1000, the equation becomes highly stiff. A 60-element MOL approach yields a solution that looks like the one shown below. The solution was plotted using the Rosenbrock method with the Shampine coefficients.

Severe discontinuities appear in the solution as one marches along the time domain. The number of elements along the spatial domain are clearly not adequate to capture the essence of the boundary value PDE. (How does one know what the “desired” solution is? In a case like this, one can cheat and refer to the literature to obtain it, as in the case of Blom et al’s paper).

Plotted below is the solution to the Burgers’ equation with Re=1000, but with 160 elements. The discontinuities appear to have quieted down with the increased number of elements.