Together with my collaborator Lorenzo Pareschi, we have just submitted a new preprint entitled Residual equilibrium schemes for time dependent partial differential equations. This work extends the note about steady state preserving spectral methods to a large class of dissipative PDE’s. Here is the abstract:

Many applications involve partial differential equations which admits nontrivial steady state solutions. The design of schemes which are able to describe correctly these equilibrium states may be challenging for numerical methods, in particular for high order ones. In this paper, inspired by micro-macro decomposition methods for kinetic equations, we present a class of schemes which are capable to preserve the steady state solution and achieve high order accuracy for a class of time dependent partial differential equations including nonlinear diffusion equations and kinetic equations. Extension to systems of conservation laws with source terms are also discussed.