A maximum-principle preserving finite element method for scalar conservation equations

This paper introduces a first-order viscosity method for the explicit approximation of scalar conservation equations with Lipschitz fluxes using continuous finite elements on arbitrary grids in any space dimension. Provided the lumped mass matrix is positive definite, the method is shown to satisfy the local maximum principle under a usual CFL condition. The method is independent of the cell type; for instance, the mesh can be a combination of tetrahedra, hexahedra, and prisms in three space dimensions.

[1]  Eitan Tadmor,et al.  Nonoscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws , 1998, SIAM J. Sci. Comput..

[2]  Alexandre Ern,et al.  Nonlinear diffusion and discrete maximum principle for stabilized Galerkin approximations of the convection-diffusion-reaction equation , 2002 .

[3]  Xiangxiong Zhang,et al.  Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments , 2011, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[4]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[5]  S. Zalesak.  Introduction to “Flux-Corrected Transport. I. SHASTA, A Fluid Transport Algorithm That Works” , 1997 .

[6]  Ludmil T. Zikatanov,et al.  A monotone finite element scheme for convection-diffusion equations , 1999, Math. Comput..

[7]  Jean-Luc Guermond,et al.  Entropy-based nonlinear viscosity for Fourier approximations of conservation laws , 2008 .

[8]  Thomas J. R. Hughes,et al.  A Petrov-Galerkin finite element method for convection-dominated flows: An accurate upwinding technique for satisfying the maximum principle☆ , 1985 .

[9]  J. Nédélec,et al.  First order quasilinear equations with boundary conditions , 1979 .

[10]  M. Christon.  The influence of the mass matrix on the dispersive nature of the semi-discrete, second-order wave equation , 1999 .

[11]  J. Boris,et al.  Flux-Corrected Transport , 1997 .

[12]  Yong Yang,et al.  A Second-Order Maximum Principle Preserving Lagrange Finite Element Technique for Nonlinear Scalar Conservation Equations , 2014, SIAM J. Numer. Anal..

[13]  Xu-Dong Liu,et al.  A maximum principle satisfying modification of triangle based adaptive stencils for the solution of scalar hyperbolic conservation laws , 1993 .

[14]  Mikhail Shashkov,et al.  A tensor artificial viscosity using a mimetic finite difference algorithm , 2001 .

[15]  Tayfun E. Tezduyar,et al.  Stabilization Parameters in SUPG and PSPG Formulations , 2003 .

[16]  R. Sani,et al.  Incompressible Flow and the Finite Element Method, Volume 1, Advection-Diffusion and Isothermal Laminar Flow , 1998 .

[17]  Tayfun E. Tezduyar,et al.  Finite element stabilization parameters computed from element matrices and vectors , 2000 .

[18]  Jay P. Boris,et al.  Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works , 1973 .

[19]  H. Holden,et al.  Splitting methods for partial differential equations with rough solutions : analysis and MATLAB programs , 2010 .

[20]  Dmitri Kuzmin,et al.  On the design of general-purpose flux limiters for finite element schemes. I. Scalar convection , 2006, J. Comput. Phys..

[21]  Jean-Luc Guermond,et al.  Entropy viscosity method for nonlinear conservation laws , 2011, J. Comput. Phys..

[22]  Xiangxiong Zhang,et al.  Maximum-Principle-Satisfying and Positivity-Preserving High Order Discontinuous Galerkin Schemes for Conservation Laws on Triangular Meshes , 2011, Journal of Scientific Computing.

[23]  Ramon Codina,et al.  A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation , 1993 .

[24]  Yifan Zhang,et al.  Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection-diffusion equations on triangular meshes , 2013, J. Comput. Phys..

[25]  Xiangxiong Zhang,et al.  On maximum-principle-satisfying high order schemes for scalar conservation laws , 2010, J. Comput. Phys..

[26]  Huijiang Zhao.  FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES WITH SINGULAR INITIAL DATA Lp(p , 1996 .

[27]  Thomas Eugene Voth,et al.  Generalized Fourier analyses of the advection–diffusion equation—Part I: one‐dimensional domains , 2004 .

[28]  M. Tabata.  A Theoretical and Computational Study of Upwind-Type Finite Element Methods , 1986 .

[29]  J. Szmelter.  Incompressible flow and the finite element method , 2001 .

[30]  Bojan Popov,et al.  Adaptive Semidiscrete Central-Upwind Schemes for Nonconvex Hyperbolic Conservation Laws , 2007, SIAM J. Sci. Comput..

[31]  Richard Sanders,et al.  A third-order accurate variation nonexpansive difference scheme for single nonlinear conservation laws , 1988 .

[32]  Tzanio V. Kolev,et al.  High-Order Curvilinear Finite Element Methods for Lagrangian Hydrodynamics , 2012, SIAM J. Sci. Comput..

[33]  Jean-Luc Guermond,et al.  A correction technique for the dispersive effects of mass lumping for transport problems , 2013 .

[34]  S. Zalesak.  Fully multidimensional flux-corrected transport algorithms for fluids , 1979 .

[35]  Alexandre Ern,et al.  Stabilized Galerkin approximation of convection-diffusion-reaction equations: discrete maximum principle and convergence , 2005, Math. Comput..

[36]  P. Lax.  Weak solutions of nonlinear hyperbolic equations and their numerical computation , 1954 .