The tangent linear model for semi-Lagrangian schemes: linearizing the process of interpolation

The tangent linear model may be used in diverse applications such as Kalman filtering, variational assimilation using the adjoint method, sensitivity studies or predictability studies. A “correct” tangent linear variation contains all of the linear part of the nonlinear variation. This concept is used to show that simply differentiating a nonlinear model’s code does not necessarily lead to a tangent linear model which is correct in all circumstances. The example of linearizing interpolation schemes is used. For infinitesimal variations, the linear variation is correct if and only if the first derivative of an interpolator is continuous. Even if the tangent linear variation is occasionally incorrect, the size of the error can be determined and may in fact be quite tolerable. Therefore, there should be no fundamental difficulty in linearizing semi-Lagrangian schemes if care is taken in choosing an appropriate interpolation scheme.