Discrete adjoint approximations with shocks

Abstract

This paper is concerned with the formulation and discretisation of adjoint equations when there are shocks in the underlying solution to the original nonlinear hyperbolic p.d.e. For the model problem of a scalar unsteady one-dimensional p.d.e. with a convex flux function, it is shown that the analytic formulation of the adjoint equations requires the imposition of an interior boundary condition along any shock. A 'discrete adjoint' discretisation is defined by requiring the adjoint equations to give the same value for the linearised functional as a linearisation of the original nonlinear discretisation. It is demonstrated that convergence requires increasing numerical smoothing of any shocks. Without this, any consistent discretisation of the adjoint equations without the inclusion of the shock boundary condition may yield incorrect values for the adjoint solution.