Adjoints of Nonoscillatory Advection Schemes

Nonoscillatory advection schemes contain switches, so that the derivative of the numerical solution at any time step with respect to that at the previous time step may be discontinuous. In consequence, sensitivities calculated using the adjoint of the numerical scheme may be discontinuous or ambiguous. This discontinuity is not a property of the continuous advection equation; it is an artefact of the numerical schemes used to solve it. The problem is demonstrated in some simple one-dimensional test cases. We derive a result showing that there is no possibility of smoothing the switches in nonoscillatory advection schemes to remove the discontinuities while retaining an obvious and desirable scaling property. We discuss some alternative approaches to deriving the adjoint schemes needed for sensitivity calculations.