Playing with Burgers's equation

International audienceThe 1D Burgers equation is used as a toy model to mimick the resulting behaviour of numerical schemes when replacing a conservation law by a form which is equivalent for smooth solutions, such as the total energy by the internal energy balance in the Euler equations. If the initial Burgers equation is replaced by a balance equation for one of its entropies (the square of the unknown) and discretized by a standard scheme, the numerical solution converges, as expected, to a function which is not a weak solution to the initial problem. However, if we first add to Burgers' equation a diffusion term scaled by a small positive parameter ǫ before deriving the entropy balance (this yields a non conservative diffusion term in the resulting equation), and then choose ǫ and the discretization parameters adequately and let them tend to zero, we observe that we recover a convergence to the correct solution