Convergence of Linearized and Adjoint Approximations for Discontinuous Solutions of Conservation Laws. Part 1: Linearized Approximations and Linearized Output Functionals

This paper analyzes the convergence of discrete approximations to the linearized equations arising from an unsteady one-dimensional hyperbolic equation with a convex flux function. A simple modified Lax–Friedrichs discretization is used on a uniform grid, and a key point is that the numerical smoothing increases the number of points across the nonlinear discontinuity as the grid is refined. It is proved that this gives pointwise convergence almost everywhere for the solution to the linearized discrete equations with smooth initial data, and also convergence in the discrete approximation of linearized output functionals. In Part 2 [M. Giles and S. Ulbrich, SIAM J. Numer. Anal., 48 (2010), pp. 905–921] we extend the results to Dirac initial data for the linearized equation and will prove the pointwise convergence almost everywhere for the solution of the adjoint discrete equations. In addition, we present numerical results illustrating the asymptotic behavior which is analyzed.