Error Estimate and the Geometric Corrector for the Upwind Finite Volume Method Applied to the Linear Advection Equation

This paper deals with the upwind finite volume method applied to the linear advection equation on a bounded domain and with natural boundary conditions. We introduce what we call the geometric corrector, which is a sequence associated with every finite volume mesh in $\mathbf{R}^{nd}$ and every nonvanishing vector $\mathbf{a}$ of $\mathbf{R}^{nd}$. First we show that if the continuous solution is regular enough and if the norm of this corrector is bounded by the mesh size, then an order one error estimate for the finite volume scheme occurs. Afterwards we prove that this norm is indeed bounded by the mesh size in several cases, including the one where an arbitrary coarse conformal triangular mesh is uniformly refined in two dimensions. Computing numerically exactly this corrector allows us to state that this result might be extended under conditions to more general cases, such as the one with independent refined meshes.

[1]  E. Süli,et al.  A convergence analysis of Yee's scheme on nonuniform grids , 1994 .

[2]  T. A. Manteuffel,et al.  Numerical solution of partial differential equations on irregular grids , 1987 .

[3]  Todd E. Peterson,et al.  A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation , 1991 .

[4]  D. Kröner.  Numerical Schemes for Conservation Laws , 1997 .

[5]  Andrew B. White,et al.  Supra-convergent schemes on irregular grids , 1986 .

[6]  Bernardo Cockburn,et al.  A priori error estimates for numerical methods for scalar conservation laws. Part I: The general approach , 1996, Math. Comput..

[7]  N. N. Kuznetsov.  Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation , 1976 .

[8]  Bernardo Cockburn,et al.  A priori error estimates for numerical methods for scalar conservation laws. Part II : flux-splitting monotone schemes on irregular Cartesian grids , 1997, Math. Comput..

[9]  Joe D. Hoffman,et al.  Relationship between the truncation errors of centered finite-difference approximations on uniform and nonuniform meshes , 1982 .

[10]  Randall J. LeVeque,et al.  H-Box Methods for the Approximation of Hyperbolic Conservation Laws on Irregular Grids , 2003, SIAM J. Numer. Anal..

[11]  C. Bardos.  Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport , 1970 .

[12]  Andrew B. White,et al.  Some Supraconvergent Schemes for Hyperbolic Equations on Irregular Grids , 1989 .

[13]  Philippe Villedieu,et al.  Convergence of an explicit finite volume scheme for first order symmetric systems , 2003, Numerische Mathematik.

[14]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[15]  Eli Turkel,et al.  Accuracy of schemes with nonuniform meshes for compressible fluid flows , 1986 .

[16]  A. A. Samarskii,et al.  Homogeneous difference schemes on non-uniform nets☆ , 1963 .

[17]  J. Pike,et al.  Grid adaptive algorithms for the solution of the euler equations on irregular grids , 1987 .

[18]  Andrew B. White,et al.  A supraconvergent scheme for nonlinear hyperbolic systems , 1989 .

[19]  Pingwen Zhang,et al.  Optimal L 1 -Rate of Convergence for The Viscosity Method and Monotone Scheme to Piecewise Constant Solutions with Shocks , 1997 .

[20]  Thomas A. Manteuffel,et al.  On the efficient numerical solution of systems of second order boundary value problems , 1986 .

[21]  Thomas A. Manteuffel,et al.  A calculus of difference schemes for the solution of boundary-value problems on irregular meshes , 1992 .

[22]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[23]  Bruno Després,et al.  Lax theorem and finite volume schemes , 2003, Math. Comput..

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

[25]  Thomas A. Manteuffel,et al.  The numerical solution of second-order boundary value problems on nonuniform meshes , 1986 .

[26]  Randall J. LeVeque,et al.  Finite volume methods for irregular one-dimensional grids , 1994 .

[27]  J. M. Sanz-Serna,et al.  A finite difference formula for the discretization of d^3/dx^3 on nonuniform grids , 1991 .

[28]  Bruno Després,et al.  An Explicit A Priori Estimate for a Finite Volume Approximation of Linear Advection on Non-Cartesian Grids , 2004, SIAM J. Numer. Anal..

[29]  Bernardo Cockburn,et al.  A Priori Error Estimates for Numerical Methods for Scalar Conservation Laws Part III: Multidimensional Flux-Splitting Monotone Schemes on Non-Cartesian Grids , 1998 .