Adjoints of nonoscillatory advection schemes

Nonoscillatory advection schemes contain switches, so that the derivative o f the numerical solution at any time step with respect to that at the previ ous time step may be discontinuous. In consequence, sensitivities calculate d using the adjoint of the numerical scheme may be discontinuous or ambiguo us. This discontinuity is not a property of the continuous advection equati on; it is an artefact of the numerical schemes used to solve it. The proble m is demonstrated in some simple one-dimensional test cases. We derive a re sult showing that there is no possibility of smoothing the switches in nono scillatory advection schemes to remove the discontinuities while retaining an obvious and desirable scaling property. We discuss some alternative appr oaches to deriving the adjoint schemes needed for sensitivity calculations. (C) 2001 Academic Press.