Monoslope and multislope MUSCL methods for unstructured meshes

We present new MUSCL techniques associated with cell-centered finite volume method on triangular meshes. The first reconstruction consists in calculating one vectorial slope per control volume by a minimization procedure with respect to a prescribed stability condition. The second technique we propose is based on the computation of three scalar slopes per triangle (one per edge) still respecting some stability condition. The resulting algorithm provides a very simple scheme which is extensible to higher dimensional problems. Numerical approximations have been performed to obtain the convergence order for the advection scalar problem whereas we treat a nonlinear vectorial example, namely the Euler system, to show the capacity of the new MUSCL technique to deal with more complex situations.

[1]  Jean-Marc Moschetta,et al.  Regular Article: Positivity of Flux Vector Splitting Schemes , 1999 .

[2]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[3]  D. Kröner,et al.  Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions , 1995 .

[4]  Serge Piperno,et al.  Criteria for the design of limiters yielding efficient high resolution TVD schemes , 1998 .

[5]  Thierry Buffard.  Analyse de quelques methodes de volumes finis non structures pour la resolution des equations d'euler , 1993 .

[6]  K. P.,et al.  HIGH RESOLUTION SCHEMES USING FLUX LIMITERS FOR HYPERBOLIC CONSERVATION LAWS * , 2012 .

[7]  Timothy J. Barth,et al.  The design and application of upwind schemes on unstructured meshes , 1989 .

[8]  Thomas J. R. Hughes,et al.  Encyclopedia of computational mechanics , 2004 .

[9]  Jean-Antoine Désidéri,et al.  Compressible flow solvers using unstructured grids , 1992 .

[10]  P. Woodward,et al.  The numerical simulation of two-dimensional fluid flow with strong shocks , 1984 .

[11]  Matthew E. Hubbard,et al.  Regular Article: Multidimensional Slope Limiters for MUSCL-Type Finite Volume Schemes on Unstructured Grids , 1999 .

[12]  S. Spekreijse.  Multigrid solution of monotone second-order discretizations of hyperbolic conservation laws , 1986 .

[13]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[14]  P. Colella.  Multidimensional upwind methods for hyperbolic conservation laws , 1990 .

[15]  T. Barth,et al.  Finite Volume Methods: Foundation and Analysis , 2004 .

[16]  A. Jameson.  Artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence in transonic and hypersonic flows , 1993 .

[17]  P. Chévrier,et al.  A Van Leer finite volume scheme for the Euler equations on unstructured meshes , 1993 .

[18]  S. Osher,et al.  Triangle based adaptive stencils for the solution of hyperbolic conservation laws , 1992 .

[19]  Chi-Wang Shu,et al.  On positivity preserving finite volume schemes for Euler equations , 1996 .

[20]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[21]  T. Barth,et al.  Error estimation and adaptive discretization methods in computational fluid dynamics , 2003 .

[22]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[23]  D. Kuzmin,et al.  High-resolution FEM-TVD schemes based on a fully multidimensional flux limiter , 2004 .

[24]  A. Jameson,et al.  Finite volume solution of the two-dimensional Euler equations on a regular triangular mesh , 1985 .

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

[26]  E. Bertolazzi,et al.  A CELL-CENTERED SECOND-ORDER ACCURATE FINITE VOLUME METHOD FOR CONVECTION–DIFFUSION PROBLEMS ON UNSTRUCTURED MESHES , 2004 .

[27]  Paul-Henry Cournède,et al.  Positivity statements for a mixed-element-volume scheme on fixed and moving grids , 2006 .

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

[29]  B. V. Leer,et al.  Towards the Ultimate Conservative Difference Scheme , 1997 .