Dispersive regularisations for the inviscid Burgers equation

Abstract

We study centred second order in time and space finite difference methods of the inviscid Burgers equation, deriving a more general numerical discretisation scheme, than the one introduced in G.A. Gottwald, 2007, for this equation. In particular, using backward error analysis we derive the modified equation associated with the numerical scheme. We also automatise the search for particular schemes, allowing us to study a whole class of numerical discretisations and tune the parameters to obtain a wide range of explicit and implicit numerical schemes of interest. We determine conditions for our discretisation scheme so that it approximates the b=0 member of the b-family, whose solutions converge strongly in the zero-alpha limit to weak solutions of the inviscid Burgers equation. We then investigate the meaning of alpha^2 in this equation. We find that the smoothing parameter alpha^2 has no physical meaning, therefore it cannot be exclusively considered as a length scale over which smoothing takes place, nor can it be assumed that the sharp shock fronts are completely related to the value of alpha^2. These findings are in accordance with G.A. Gottwald, 2007. We then stabilise our schemes by using spatial averaging, but since this adds some artificial viscosity, we search for the maximum number of time steps over which we can perform the averaging, so as to add the least artificial viscosity possible.