A first-order system approach for diffusion equation. II: Unification of advection and diffusion

In this paper, we unify advection and diffusion into a single hyperbolic system by extending the first-order system approach introduced for the diffusion equation [J. Comput. Phys., 227 (2007) 315-352] to the advection-diffusion equation. Specifically, we construct a unified hyperbolic advection-diffusion system by expressing the diffusion term as a first-order hyperbolic system and simply adding the advection term to it. Naturally then, we develop upwind schemes for this entire system; there is thus no need to develop two different schemes, i.e., advection and diffusion schemes. We show that numerical schemes constructed in this way can be automatically uniformly accurate, allow O(h) time step, and compute the solution gradients (viscous stresses/heat fluxes for the Navier-Stokes equations) simultaneously to the same order of accuracy as the main variable, for all Reynolds numbers. We present numerical results for boundary-layer type problems on non-uniform grids in one dimension and irregular triangular grids in two dimensions to demonstrate various remarkable advantages of the proposed approach. In particular, we show that the schemes solving the first-order advection-diffusion system give a tremendous speed-up in CPU time over traditional scalar schemes despite the additional cost of carrying extra variables and solving equations for them. We conclude the paper with discussions on further developments to come.

[1]  Laszlo Fuchs,et al.  Compact Third-Order Multidimensional Upwind Scheme for Navier–Stokes Simulations , 2002 .

[2]  Gabriel Nagy,et al.  The behavior of hyperbolic heat equations' solutions near their parabolic limits , 1994 .

[3]  Hiroaki Nishikawa,et al.  A first-order system approach for diffusion equation. I: Second-order residual-distribution schemes , 2007, J. Comput. Phys..

[4]  A. Jameson.  Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings , 1991 .

[5]  Fermín Navarrina,et al.  A discontinuous Galerkin method for a hyperbolic model for convection–diffusion problems in CFD , 2007 .

[6]  Hiroaki Nishikawa.  Towards Future Navier-Stokes Schemes: Uniform Accuracy, O(h) Time Step, and Accurate Viscous/Heat Fluxes , 2009 .

[7]  Herman Deconinck,et al.  Positive matrix distribution schemes for hyperbolic systems, with application to the Euler equations , 1996 .

[8]  Hiroaki Nishikawa,et al.  On High-Order Fluctuation-Splitting Schemes for Navier-Stokes Equations , 2009 .

[9]  Philip L. Roe,et al.  High-Order Fluctuation-Splitting Schemes for Advection-Diffusion Equations , 2009 .

[10]  M. Ricchiutoa,et al.  On uniformly high-order accurate residual distribution schemes for advection – diffusion , 2008 .

[11]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[12]  Clinton P. T. Groth,et al.  Computational Fluid Dynamics 2004 , 2006 .

[13]  C. D. Levermore,et al.  Numerical Schemes for Hyperbolic Conservation Laws with Stiff Relaxation Terms , 1996 .

[14]  Pietro De Palma,et al.  Third-order-accurate fluctuation splitting schemes for unsteady hyperbolic problems , 2007, J. Comput. Phys..

[15]  T. Taylor,et al.  Computational methods for fluid flow , 1982 .

[16]  H. Nishikawa,et al.  Higher-Order Discretization of Diffusion Terms in Residual-Distribution Methods , 2005 .

[17]  Boris Diskin,et al.  Comparison of Node-Centered and Cell-Centered Unstructured Finite-Volume Discretizations , 2013 .

[18]  Rémi Abgrall,et al.  Application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes , 2007, J. Comput. Phys..

[19]  Chi-Wang Shu,et al.  High order residual distribution conservative finite difference WENO schemes for convection-diffusion steady state problems on non-smooth meshes , 2007, J. Comput. Phys..

[20]  Jiri Blazek,et al.  Accurate and Efficient Discretization of Navier-Stokes Equations on Mixed Grids , 2000 .

[21]  Rémi Abgrall,et al.  Toward the ultimate conservative scheme: following the quest , 2001 .

[22]  Philip L. Roe,et al.  Characteristic‐based schemes for dispersive waves I. The method of characteristics for smooth solutions , 1993 .

[23]  Jim E. Morel,et al.  Methods for hyperbolic systems with stiff relaxation , 2002 .

[24]  Philip L. Roe,et al.  A multidimensional generalization of Roe's flux difference splitter for the euler equations , 1993 .

[25]  R. Pletcher,et al.  Computational Fluid Mechanics and Heat Transfer. By D. A ANDERSON, J. C. TANNEHILL and R. H. PLETCHER. Hemisphere, 1984. 599 pp. $39.95. , 1986, Journal of Fluid Mechanics.

[26]  L. Franca,et al.  Stabilized Finite Element Methods , 1993 .

[27]  Chi-Wang Shu,et al.  High order residual distribution conservative finite difference WENO schemes for steady state problems on non-smooth meshes , 2006, J. Comput. Phys..

[28]  Vittorio Romano,et al.  Central Schemes for Balance Laws of Relaxation Type , 2000, SIAM J. Numer. Anal..

[29]  Philip L. Roe,et al.  Grids and Solutions from Residual Minimisation , 2001 .

[30]  Mohit Arora,et al.  Explicit characteristic-based high-resolution algorithms for hyperbolic conservation laws with stiff source terms. , 1996 .

[31]  Richard H. Pletcher,et al.  Computational Fluid Mechanics and Heat Transfer , 1984 .

[32]  Erik Dick,et al.  Computational Fluid Dynamics 2006 , 2009 .

[33]  Philip L. Roe,et al.  Towards High-Order Fluctuation-Splitting Schemes for Navier-Stokes Equations , 2005 .

[34]  D. Arnold.  An Interior Penalty Finite Element Method with Discontinuous Elements , 1982 .

[35]  Bram van Leer,et al.  In Search Of An Optimal Local Navier-Stokes Preconditioner , 2003 .

[36]  N. Ron-Ho,et al.  A Multiple-Grid Scheme for Solving the Euler Equations , 1982 .

[37]  H. Deconinck,et al.  A parallel, implicit, multi-dimensional upwind, residual distribution method for the Navier-Stokes equations on unstructured grids , 1999 .

[38]  Philip L. Roe,et al.  A Third-Order Fluctuation Splitting Scheme That Preserves Potential Flow , 2001 .

[39]  Laszlo Fuchs,et al.  Compact third-order multidimensional upwind discretization for steady and unsteady flow simulations , 2005 .

[40]  T. Hughes,et al.  Stabilized finite element methods. I: Application to the advective-diffusive model , 1992 .