Approximate inertial manifold-base finite-difference operators and quasi-steady solutions of parabolic PDES with application to sediment transport

Here we discuss the approximate inertial manifold-based finite-difference schemes of Margolin and Jones [1] as a way to derive discrete analogs to elementary, quasi-steady models. Quasi-steady models provide simplified, often closed form, steady state solutions for a number of parabolic PDEs by introducing approximate temporal behavior. Using these approximations, we show that a nontrivial, physically meaningful approximate steady state may exist even for a linear operator. In this article, we discuss the particular example of sediment transport governing quasi-steady hill-slope and alluvial fan profile evolution by a 1D unsteady diffusion based model.