A first-order system approach for diffusion equation. I: Second-order residual-distribution schemes

In this paper, we embark on a new strategy for computing the steady state solution of the diffusion equation. The new strategy is to solve an equivalent first-order hyperbolic system instead of the second-order diffusion equation, introducing solution gradients as additional unknowns. We show that schemes developed for the first-order system allow O(h) time step instead of O(h) and converge very rapidly toward the steady state. Moreover, this extremely fast convergence comes with the solution gradients (viscous stresses/heat fluxes for the Navier-Stokes equations) simultaneously computed with the same order of accuracy as the main variable. The proposed schemes are formulated as residual-distribution schemes (but can also be identified as finite-volume schemes), directly on unstructured grids. We present numerical results to demonstrate the tremendous gains offered by the new diffusion schemes, driving the rise of explicit schemes in the steady state computation for diffusion problems.