Uniform convergence analysis of finite difference approximations for advection–reaction–diffusion problem on adaptive grids

References

• Axelsson , O. and Carey , G. F. 1985 . On the numerical solution of two-point singularly perturbed boundary value problems . Comput. Methods Appl. Mech. Eng , 50 : 217 – 229 .
   Web of Science ®Google Scholar
• Bai , Z.-Z. and Guo , X.-P. 2010 . On Newton–HSS methods for systems of nonlinear equations with positive-definite Jacobian matrices . J. Comput. Math , 28 : 235 – 260 .
   Web of Science ®Google Scholar
• Beckett , G. M. and Mackenzie , J. A. 2000 . Convergence analysis of finite difference approximations on equidistributed grids to a singularly perturbed boundary value problem . Appl. Numer. Math , 35 : 87 – 109 .
   Web of Science ®Google Scholar
• Budd , C. J. , Huang , W.-Z. and Russell , R. D. 2009 . Adaptivity with moving grids . Acta Numer , : 1 – 131 .
   Google Scholar
• Chen , Y.-P. 2003 . Uniform pointwise convergence for a singularly perturbed problem using arc-length equidistribution . J. Comput. Appl. Math , 159 : 25 – 34 .
   Web of Science ®Google Scholar
• Chen , Y.-P. 2006 . Uniform convergence analysis of finite difference approximations for singular perturbation problems on an adaptive grid . Adv. Comput. Math , 24 : 197 – 212 .
   Web of Science ®Google Scholar
• Clavero , C. , Gracia , J. L. and Lisbona , F. J. 2010 . An almost third order finite difference scheme for singularly perturbed reaction–diffusion systems . J. Comput. Appl. Math , 234 : 2501 – 2515 .
   Web of Science ®Google Scholar
• Delzanno , G. L. and Finn , J. M. 2011 . The fluid dynamic approach to equidistribution methods for grid adaptation . Comput. Phys. Comm , 182 : 330 – 346 .
   Web of Science ®Google Scholar
• Kellogg , R. B. and Tsan , A. 1978 . Analysis of some difference approximations for a singular perturbation problem without turning points . Math. Comput , 32 : 1025 – 1039 .
   Web of Science ®Google Scholar
• Kopteva , N. 2001 . Maximum norm a posteriori error estimates for a one-dimensional convection–diffusion problem . SIAM J. Numer. Anal , 39 : 423 – 441 .
   Web of Science ®Google Scholar
• Kopteva , N. and Stynes , M. 2001 . A robust adaptive method for quasi-linear one-dimensional convection–diffusion problem . SIAM J. Numer. Anal , 39 : 1446 – 1467 .
   Web of Science ®Google Scholar
• Larsson , S. and Thomée , V. 2003 . Partial Differential Equations with Numerical Methods , Berlin : Springer .
   Google Scholar
• Li , R. , Tang , T. and Zhang , P. 2001 . Moving mesh methods in multiple dimensions based on harmonic maps . J. Comput. Phys , 170 : 562 – 588 .
   Web of Science ®Google Scholar
• Linss , T. 2001 . Uniform pointwise convergence of finite difference schemes using grid equidistribution . Computing , 66 : 27 – 39 .
   Web of Science ®Google Scholar
• Mackenzie , J. A. 1999 . Uniform convergence analysis of an upwind finite-difference approximation of a convection–diffusion boundary value problem on an adaptive grid . IMA J. Numer. Anal , 19 : 233 – 249 .
   Web of Science ®Google Scholar
• Mackenzie , J. A. and Mekwi , W. R. 2007 . An analysis of stability and convergence of a finite-difference discretization of a model parabolic PDE in 1D using a moving mesh . IMA J. Numer. Anal , 27 : 507 – 528 .
   Web of Science ®Google Scholar
• Miller , J. J.H. , O'Riordan , E. and Shishkin , G. I. 1996 . Fitted Numerical Methods for Singular Perturbation Problems , Singapore : World Scientific .
   Google Scholar
• O'Malley , R. E. 1974 . Introduction to Singular Perturbations , New York : Academic .
   Google Scholar
• Qiu , Y. and Sloan , D. M. 1999 . Analysis of difference approximations to a singularly perturbed two-point boundary value problem on an adaptively generated grid . J. Comput. Appl. Math , 101 : 1 – 25 .
   Web of Science ®Google Scholar
• Qiu , Y. , Sloan , D. M. and Tang , T. 2000 . Numerical solution of a singularly perturbed two-point boundary value problem using equidistribution: Analysis of convergence . J. Comput. Appl. Math , 116 : 121 – 143 .
   Web of Science ®Google Scholar
• Roos , H.-G. , Stynes , M. and Tobiska , L. 1996 . Numerical Methods for Singularly Perturbed Differential Equations , Berlin : Springer .
   Google Scholar
• Smart , D. R. 1974 . Fixed-Point Theorems , Cambridge : University of Cambridge .
   Google Scholar
• Tan , Z. , Lim , K. M. and Khoo , B. C. 2007 . An adaptive mesh redistribution method for the incompressible mixture flows using phase-field model . J. Comput. Phys , 225 : 1137 – 1158 .
   Web of Science ®Google Scholar
• Tang , T. 2005 . Moving mesh methods for computational fluid dynamics . Contemp. Math , 383 : 141 – 173 .
   Google Scholar