High order multi-moment constrained finite volume method. Part I: Basic formulation

A new high-order finite volume method based on local reconstruction is presented in this paper. The method, so-called the multi-moment constrained finite volume (MCV) method, uses the point values defined within single cell at equally spaced points as the model variables (or unknowns). The time evolution equations used to update the unknowns are derived from a set of constraint conditions imposed on multi kinds of moments, i.e. the cell-averaged value and the point-wise value of the state variable and its derivatives. The finite volume constraint on the cell-average guarantees the numerical conservativeness of the method. Most constraint conditions are imposed on the cell boundaries, where the numerical flux and its derivatives are solved as general Riemann problems. A multi-moment constrained Lagrange interpolation reconstruction for the demanded order of accuracy is constructed over single cell and converts the evolution equations of the moments to those of the unknowns. The presented method provides a general framework to construct efficient schemes of high orders. The basic formulations for hyperbolic conservation laws in 1- and 2D structured grids are detailed with the numerical results of widely used benchmark tests.

[1]  Pierre Sagaut,et al.  A Class of Explicit ENO Filters with Application to Unsteady Flows , 2001 .

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

[3]  Eleuterio F. Toro,et al.  Derivative Riemann solvers for systems of conservation laws and ADER methods , 2006, J. Comput. Phys..

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

[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]  Zhi J. Wang,et al.  Spectral (finite) volume method for conservation laws on unstructured grids IV: extension to two-dimensional systems , 2004 .

[8]  G. Sod.  A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws , 1978 .

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

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

[11]  A KoprivaDavid,et al.  A Conservative Staggered-Grid Chebyshev Multidomain Method for Compressible Flows , 1996 .

[12]  Feng Xiao,et al.  CIP/multi-moment finite volume method with arbitrary order of accuracy , 2007 .

[13]  Feng Xiao,et al.  Unified formulation for compressible and incompressible flows by using multi-integrated moments II: Multi-dimensional version for compressible and incompressible flows , 2006, J. Comput. Phys..

[14]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

[15]  Kenji Takizawa,et al.  Conservative form of interpolated differential operator scheme for compressible and incompressible fluid dynamics , 2008, J. Comput. Phys..

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

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

[18]  R. Akoh,et al.  A CIP/multi-moment finite volume method for shallow water equations with source terms , 2008 .

[19]  Jianxian Qiu,et al.  Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one-dimensional case , 2004 .

[20]  Antony Jameson,et al.  Spectral Difference Method for Unstructured Grids II: Extension to the Euler Equations , 2007, J. Sci. Comput..

[21]  Feng Xiao,et al.  CIP/multi-moment finite volume method for Euler equations: A semi-Lagrangian characteristic formulation , 2007, J. Comput. Phys..

[22]  Karkenahalli Srinivas,et al.  Computational Fluid Dynamics 2002 , 2003 .

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

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

[25]  C. Angelopoulos.  High resolution schemes for hyperbolic conservation laws , 1992 .

[26]  Feng Xiao,et al.  An efficient method for capturing free boundaries in multi‐fluid simulations , 2003 .

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

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

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

[30]  F. Xiao,et al.  A Multimoment Finite-Volume Shallow-Water Model On The Yin-Yang Overset Spherical Grid , 2008 .

[31]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

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

[33]  Yong-Tao Zhang,et al.  Resolution of high order WENO schemes for complicated flow structures , 2003 .

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

[35]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

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

[37]  Wang Chi-Shu,et al.  Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws , 1997 .

[38]  Feng Xiao,et al.  A 4th-order and single-cell-based advection scheme on unstructured grids using multi-moments , 2005, Comput. Phys. Commun..

[39]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[40]  Marcel Vinokur,et al.  Spectral difference method for unstructured grids I: Basic formulation , 2006, J. Comput. Phys..

[41]  Feng Xiao,et al.  Shallow water model on cubed-sphere by multi-moment finite volume method , 2008, J. Comput. Phys..

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

[43]  Chris Lacor,et al.  On the connection between the spectral volume and the spectral difference method , 2007, J. Comput. Phys..

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

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

[46]  Feng Xiao,et al.  Profile-modifiable Conservative Transport Schemes and a Simple Multi-integrated Moment Formulation for Hydrodynamics , 2003 .

[47]  John H. Kolias,et al.  A CONSERVATIVE STAGGERED-GRID CHEBYSHEV MULTIDOMAIN METHOD FOR COMPRESSIBLE FLOWS , 1995 .

[48]  Eleuterio F. Toro,et al.  ADER: Arbitrary High Order Godunov Approach , 2002, J. Sci. Comput..

[49]  Zhi Jian Wang,et al.  Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids III: One Dimensional Systems and Partition Optimization , 2004, J. Sci. Comput..