Cost-effectiveness of fully implicit moving mesh adaptation: A practical investigation in 1D

The cost-effectiveness of moving mesh adaptation is studied in a number of 1D tests. We propose a method that is based on two established modern techniques. First, we use a moving mesh approach based on the classic equidistribution method. Second, we discretize the model equations for grid and physics using a conservative finite volume method and we solve the resulting equations with a preconditioned inexact Newton-Krylov method.Using these state of the art methods, we consider the question of whether a real improvement in performance can be achieved using adaptive grids. We consider rigorous metrics of the accuracy and cost of a numerical solution on uniform and adaptive grids. For a number of classic but challenging problems we demonstrate that indeed adaptive grids can lead to a great improvement in cost-effectiveness.

[1]  Shengtai Li,et al.  Stability of Moving Mesh Systems of Partial Differential Equations , 1998, SIAM J. Sci. Comput..

[2]  D. Mihalas,et al.  Foundations of Radiation Hydrodynamics , 1985 .

[3]  Mikhail Shashkov,et al.  The Error-Minimization-Based Strategy for Moving Mesh Methods , 2006 .

[4]  Weizhang Huang,et al.  Moving mesh partial differential equations (MMPDES) based on the equidistribution principle , 1994 .

[5]  J. Brackbill.  An adaptive grid with directional control , 1993 .

[6]  Per Lötstedt,et al.  Adaptive Error Control for Steady State Solutions of Inviscid Flow , 2001, SIAM J. Sci. Comput..

[7]  L. Petzold,et al.  Moving Mesh Methods with Upwinding Schemes for Time-Dependent PDEs , 1997 .

[8]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[9]  Homer F. Walker,et al.  Globally Convergent Inexact Newton Methods , 1994, SIAM J. Optim..

[10]  C. W. Hirt.  Heuristic stability theory for finite-difference equations☆ , 1968 .

[11]  J. Brackbill,et al.  Adaptive zoning for singular problems in two dimensions , 1982 .

[12]  B. P. Leonard,et al.  A stable and accurate convective modelling procedure based on quadratic upstream interpolation , 1990 .

[13]  A. M. Winslow.  Adaptive-mesh zoning by the equipotential method , 1981 .

[14]  C. Kelley.  Iterative Methods for Linear and Nonlinear Equations , 1987 .

[15]  R. Dembo,et al.  INEXACT NEWTON METHODS , 1982 .

[16]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[17]  M. Baines.  Moving finite elements , 1994 .

[18]  P. Gaskell,et al.  Curvature‐compensated convective transport: SMART, A new boundedness‐ preserving transport algorithm , 1988 .

[19]  Luis Chacón,et al.  A fully implicit, nonlinear adaptive grid strategy , 2006, J. Comput. Phys..

[20]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[21]  Giovanni Lapenta.  Variational grid adaptation based on the minimization of local truncation error: time-independent problems , 2004 .

[22]  Joseph M. Prusa,et al.  Dynamic Grid Adaptation Using the MPDATA Scheme , 2002 .

[23]  Vladimir D. Liseikin,et al.  Grid Generation Methods , 1999 .

[24]  Giovanni Lapenta,et al.  A recipe to detect the error in discretization schemes , 2004 .

[25]  M. Gehmeyr,et al.  Adaptive grid radiation hydrodynamics with TITAN , 1994 .

[26]  Robert B. Lowrie,et al.  A comparison of implicit time integration methods for nonlinear relaxation and diffusion , 2004 .

[27]  Robert D. Russell,et al.  Comparison of two-dimensional r -adaptive finite element methods using various error indicators , 2001 .

[28]  E. Dorfi,et al.  Simple adaptive grids for 1-d initial value problems , 1987 .

[29]  Michael L. Norman,et al.  Adaptive mesh techniques for fronts in star formation , 1984 .