Preconditioned methods for solving the incompressible low speed compressible equations
Journal of Computational Physics
SIAM Journal on Scientific Computing
Cell vertex algorithms for the compressible Navier-Stokes equations
Journal of Computational Physics
Numerical schemes for hyperbolic conservation laws with stiff relaxation terms
Journal of Computational Physics
The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
SIAM Journal on Numerical Analysis
A multigrid tutorial: second edition
A multigrid tutorial: second edition
Flux difference splitting and the balancing of source terms and flux gradients
Journal of Computational Physics
Toward the ultimate conservative scheme: following the quest
Journal of Computational Physics
Central Schemes for Balance Laws of Relaxation Type
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Essentially non-oscillatory Residual Distribution schemes for hyperbolic problems
Journal of Computational Physics
Journal of Computational Physics
Residual distribution schemes for advection and advection-diffusion problems on quadrilateral cells
Journal of Computational Physics
Third-order-accurate fluctuation splitting schemes for unsteady hyperbolic problems
Journal of Computational Physics
A first-order system approach for diffusion equation. II: Unification of advection and diffusion
Journal of Computational Physics
Divergence formulation of source term
Journal of Computational Physics
First-, second-, and third-order finite-volume schemes for diffusion
Journal of Computational Physics
Hi-index | 31.46 |
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^2) 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.