Dispersive regularizations and numerical discretizations for the inviscid Burgers equation

We study centred second order in time and space discretizations of the inviscid Burgers equation. Although this equation in its continuum formulation supports non-smooth shock wave solutions, the discrete equation generically supports smooth solitary wave solutions. Using backward error analysis we derive the modified equation associated with the numerical scheme. We identify three different equations, the Korteweg-de Vries (KdV) equation, the Camassa-Holm (CH) equation and the b = 0 member of the b-family. Solutions of the first two equations are solitary waves and do not converge to the shock solutions of the Burgers equation. The third equation however supports solutions which strongly approximate weak solutions of the Burgers equation. We corroborate our analytical results with numerical simulations.