A 4th-order and single-cell-based advection scheme on unstructured grids using multi-moments

Abstract This paper presents a 2D conservative scheme on triangular meshes for advection transport problem. Different from the conventional finite volume method (FVM), the present scheme uses and carries forward in time not only the cell-integrated average but also the point values at the vertices and Gaussian points at the boundary as the model variables for each triangular element. Treating all these quantities, which are generically called the ‘moments’ of the physical variable herein, as the prognostic variables enables us to construct more accurate spatial discretization with less computational stencils. Moreover, the resulting numerical scheme appears much more robust to various triangular unstructured meshes. A 4th-order accuracy is achieved by using a cubic polynomial constructed over a single triangular mesh element. The accuracy and the robustness of the proposed method are evaluated by numerical experiments with comparisons to other existing schemes.

[1]  Nobuatsu Tanaka Development of a highly accurate interpolation method for mesh‐free flow simulations I. Integration of gridless, particle and CIP methods , 1999 .

[2]  T. Yabe,et al.  Completely conservative and oscillationless semi-Lagrangian schemes for advection transportation , 2001 .

[3]  T. Yabe,et al.  The constrained interpolation profile method for multiphase analysis , 2001 .

[4]  Robin Sibson,et al.  Locally Equiangular Triangulations , 1978, Comput. J..

[5]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems , 1989 .

[6]  Chi-Wang Shu,et al.  The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V , 1998 .

[7]  Jean-Raymond Abrial,et al.  On B , 1998, B.

[8]  B. M. Fulk MATH , 1992 .

[9]  Takashi Yabe,et al.  A universal solver for hyperbolic equations by cubic-polynomial interpolation. II, Two- and three-dimensional solvers , 1991 .

[10]  Z. Wang,et al.  Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids , 2002 .

[11]  Takashi Yabe,et al.  A universal solver for hyperbolic equations by cubic-polynomial interpolation I. One-dimensional solver , 1991 .

[12]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[13]  T. Yabe,et al.  An Exactly Conservative Semi-Lagrangian Scheme (CIP–CSL) in One Dimension , 2001 .

[14]  T. Yabe,et al.  Conservative and oscillation-less atmospheric transport schemes based on rational functions , 2002 .

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

[16]  Zhi J. Wang,et al.  Evaluation of discontinuous Galerkin and spectral volume methods for scalar and system conservation laws on unstructured grids , 2004 .

[17]  Scott W. Sloan,et al.  An implementation of Watson's algorithm for computing 2-dimensional delaunay triangulations , 1984 .

[18]  F. Xiao,et al.  Numerical simulations of free-interface fluids by a multi-integrated moment method , 2005 .

[19]  Mengping Zhang,et al.  An analysis of and a comparison between the discontinuous Galerkin and the spectral finite volume methods , 2005 .

[20]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .

[21]  Remo Guidieri Res , 1995, RES: Anthropology and Aesthetics.

[22]  Takashi Yabe,et al.  Constructing exactly conservative scheme in a non-conservative form , 2000 .

[23]  O. Friedrich,et al.  Weighted Essentially Non-Oscillatory Schemes for the Interpolation of Mean Values on Unstructured Grids , 1998 .

[24]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[25]  Zhi J. Wang,et al.  Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids. Basic Formulation , 2002 .

[26]  Rémi Abgrall,et al.  On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation , 1994 .

[27]  Chaowei Hu,et al.  No . 98-32 Weighted Essentially Non-Oscillatory Schemes on Triangular Meshes , 1998 .

[28]  Chi-Wang Shu TVB uniformly high-order schemes for conservation laws , 1987 .

[29]  Chi-Wang Shu Total-variation-diminishing time discretizations , 1988 .

[30]  A. Staniforth,et al.  Semi-Lagrangian integration schemes for atmospheric models - A review , 1991 .

[31]  Feng Xiao,et al.  Unified formulation for compressible and incompressible flows by using multi-integrated moments I: one-dimensional inviscid compressible flow , 2004 .