Explicit construction of solutions to the Burgers equation with discontinuous initial-boundary conditions

A solution of the initial-boundary value problem on the strip (0,1) × [0, 1] for scalar conservation laws with strictly convex flux can be obtained by considering gradients of the unique solution V to an associated Hamilton-Jacobi equation (with appropriately defined initial and boundary conditions). It was shown in Frankowska (2010) that V can be expressed as the minimum of three value functions arising in calculus of variations problems that, in turn, can be obtained from the Lax formulae. Moreover the traces of the gradients V_x satisfy generalized boundary conditions (as in LeFloch (1988)). In this work we illustrate this approach in the case of the Burgers equation and provide numerical approximation of its solutions.