Couplex Benchmark Computations Obtained with the Software Toolbox UG

This paper describes the numerical results for the COUPLEX benchmark obtained with the simulation software UG using vertex centered finite volume and higher order discontinuous Galerkin schemes. Multigrid solvers on unstructured grids, local mesh refinement and parallel computation are employed to yield very accurate solutions. Since the full range of results required in the benchmarks is too large to be displayed in this paper we focus on the comparison of discretization schemes, assessment of numerical errors and the presentation of parallel computations.

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

[2]  M. Wheeler,et al.  A characteristics-mixed finite element method for advection-dominated transport problems , 1995 .

[3]  Gerry E. Schneider,et al.  A SKEWED, POSITIVE INFLUENCE COEFFICIENT UPWINDING PROCEDURE FOR CONTROL-VOLUME-BASED FINITE-ELEMENT CONVECTION-DIFFUSION COMPUTATION , 1986 .

[4]  Bernardo Cockburn,et al.  The local discontinuous Galerkin method for contaminant transport , 2000 .

[5]  B. Rivière,et al.  Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I , 1999 .

[6]  Hong Wang,et al.  An ELLAM Scheme for Advection-Diffusion Equations in Two Dimensions , 1999, SIAM J. Sci. Comput..

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

[8]  I. Babuska,et al.  A DiscontinuoushpFinite Element Method for Diffusion Problems , 1998 .

[9]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[10]  H. Rentz-Reichert,et al.  UG – A flexible software toolbox for solving partial differential equations , 1997 .

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

[12]  T. F. Russell,et al.  An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation , 1990 .

[13]  Rainer Helmig,et al.  Efficient fully-coupled solution techniques for two-phase flow in porous media: Parallel multigrid solution and large scale computations , 1999 .

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

[15]  Bernardo Cockburn,et al.  The Runge-Kutta local projection P1-discontinuous-Galerkin finite element method for scalar conservation laws , 1988 .

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

[17]  Bernardo Cockburn.  Discontinuous Galerkin methods , 2003 .

[18]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[19]  Jürgen Bey,et al.  Finite-Volumen- und Mehrgitter-Verfahren für elliptische Randwertprobleme , 1998 .

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

[21]  Mary F. Wheeler,et al.  A Priori Error Estimates for Finite Element Methods Based on Discontinuous Approximation Spaces for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[22]  B. Rivière,et al.  Superconvergence and H(div) projection for discontinuous Galerkin methods , 2003 .

[23]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

[24]  Stefan Lang,et al.  High Level Software Tools for Unstructured Adaptive Grids on Massively Parallel Systems , 1999, PPSC.

[25]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[26]  R. Alexander.  Diagonally implicit runge-kutta methods for stiff odes , 1977 .

[27]  Rolf Rannacher.  Accurate Time Discretization Schemes for Computing Nonstationary Incompressible Fluid Flow , 1994 .

[28]  Richard E. Ewing,et al.  Numerical Simulation of Multiphase Flow in Fractured Porous Media , 2000 .

[29]  Klaus Birken,et al.  Ein Modell zur effizienten Parallelisierung von Algorithmen auf komplexen, dynamischen Datenstrukturen , 1998 .

[30]  Rainer Helmig,et al.  Numerical simulation of non-isothermal multiphase multicomponent processes in porous media.: 1. An efficient solution technique , 2002 .

[31]  Chi-Wang Shu,et al.  The Runge-Kutta local projection $P^1$-discontinuous-Galerkin finite element method for scalar conservation laws , 1988, ESAIM: Mathematical Modelling and Numerical Analysis.

[32]  C. Dawson.  Godunov-mixed methods for advective flow problems in one space dimension , 1991 .

[33]  T. F. Russell,et al.  NUMERICAL METHODS FOR CONVECTION-DOMINATED DIFFUSION PROBLEMS BASED ON COMBINING THE METHOD OF CHARACTERISTICS WITH FINITE ELEMENT OR FINITE DIFFERENCE PROCEDURES* , 1982 .

[34]  B. Rivière,et al.  Part II. Discontinuous Galerkin method applied to a single phase flow in porous media , 2000 .

[35]  M. Wheeler.  An Elliptic Collocation-Finite Element Method with Interior Penalties , 1978 .