Finite Volume Approximation of One-Dimensional Stiff Convection-Diffusion Equations

In this work, we present a novel method to approximate stiff problems using a finite volume (FV) discretization. The stiffness is caused by the existence of a small parameter in the equation which introduces a boundary layer. The proposed semi-analytic method consists in adding in the finite volume space the boundary layer corrector which encompasses the singularities of the problem. We verify the stability and convergence of our finite volume schemes which take into account the boundary layer structures. A major feature of the proposed scheme is that it produces an efficient stable second order scheme to be compared with the usual stable upwind schemes of order one or the usual costly second order schemes demanding fine meshes.

[1]  Wiktor Eckhaus,et al.  Boundary Layers in Linear Elliptic Singular Perturbation Problems , 1972 .

[2]  Roger Temam,et al.  Boundary layers in smooth curvilinear domains: Parabolic problems , 2009 .

[3]  Guido Kanschat,et al.  A locally conservative LDG method for the incompressible Navier-Stokes equations , 2004, Math. Comput..

[4]  Roger Temam,et al.  On Parabolic Boundary Layers for Convection–Diffusion Equations in a Channel: Analysis and Numerical Applications , 2006, J. Sci. Comput..

[5]  Christoph Schwab,et al.  The p and hp versions of the finite element method for problems with boundary layers , 1996, Math. Comput..

[6]  E. M. de Jager,et al.  Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type , 1966 .

[7]  R. Bruce Kellogg,et al.  Layers and corner singularities in singularly perturbed elliptic problems , 2008 .

[8]  J. Lions.  Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal , 1973 .

[9]  Shagi-Di Shih,et al.  Asymptotic anaylsis of a singular perturbation problem , 1987 .

[10]  Jens Markus Melenk,et al.  hp-Finite Element Methods for Singular Perturbations , 2002 .

[11]  Chang-Yeol Jung,et al.  Numerical approximation of two‐dimensional convection‐diffusion equations with boundary layers , 2005 .

[12]  Weeratunge Malalasekera,et al.  An introduction to computational fluid dynamics - the finite volume method , 2007 .

[13]  Chang-Yeol Jung,et al.  Finite elements scheme in enriched subspaces for singularly perturbed reaction-diffusion problems on a square domain , 2008, Asymptot. Anal..

[14]  Robert E O'Malley.  Singular perturbation analysis for ordinary differential equations , 1977 .

[15]  Lutz Tobiska,et al.  Numerical Methods for Singularly Perturbed Differential Equations , 1996 .

[16]  R. Eymard,et al.  Finite Volume Methods , 2019, Computational Methods for Fluid Dynamics.

[17]  Gung-Min Gie,et al.  Singular perturbation problems in a general smooth domain , 2009, Asymptot. Anal..

[18]  Bernardo Cockburn,et al.  Convergence of the finite volume method for multidimensional conservation laws , 1995 .

[19]  Roger Temam,et al.  COMPARISON OF FINITE VOLUME AND FINITE DIFFERENCE METHODS AND APPLICATION , 2006 .

[20]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[21]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

[22]  Roger Temam,et al.  Asymptotic analysis for singularly perturbed convection-diffusion equations with a turning point , 2007 .

[23]  Roger Temam,et al.  Numerical approximation of one-dimensional stationary diffusion equations with boundary layers , 2002 .