Highly accurate numerical solutions with repeated Richardson extrapolation for 2D laplace equation

Abstract A theoretical basis is presented for the repeated Richardson extrapolation (RRE) to reduce and estimate the discretization error of numerical solutions for heat conduction. An example application is described for the 2D Laplace equation using the finite difference method, a domain discretized with uniform grids, second-order accurate approximations, several variables of interest, Dirichlet boundary conditions, grids with up to 8,193 × 8,193 nodes, a multigrid method, single, double and quadruple precisions and up to twelve Richardson extrapolations. It was found that: (1) RRE significantly reduces the discretization error (for example, from 2.25E-07 to 3.19E-32 with nine extrapolations and a 1,025 × 1,025 grid, yielding an order of accuracy of 19.1); (2) the Richardson error estimator works for numerical results obtained with RRE; (3) a higher reduction of the discretization error with RRE is achieved by using higher calculation precision, a larger number of extrapolations, a larger number of grids and correct error orders; and (4) to obtain a given value error, much less CPU time and RAM memory are required for the solution with RRE than without it.

[1]  G. D. Davis.  Natural convection of air in a square cavity: A bench mark numerical solution , 1983 .

[2]  E. Erturk,et al.  Numerical solutions of 2‐D steady incompressible driven cavity flow at high Reynolds numbers , 2004, ArXiv.

[3]  Rahul Kumar,et al.  One-sided finite-difference approximations suitable for use with Richardson extrapolation , 2006, J. Comput. Phys..

[4]  P. Roache.  Perspective: A Method for Uniform Reporting of Grid Refinement Studies , 1994 .

[5]  Gerry E. Schneider,et al.  A MODIFIED STRONGLY IMPLICIT PROCEDURE FOR THE NUMERICAL SOLUTION OF FIELD PROBLEMS , 1981 .

[6]  David Levin,et al.  Practical extrapolation methods: theory and applications , 2005, Math. Comput..

[7]  Patrick J. Roache,et al.  Verification and Validation in Computational Science and Engineering , 1998 .

[8]  M. Garbey,et al.  A least square extrapolation method for improving solution accuracy of PDE computations , 2003 .

[9]  Richard H. Pletcher,et al.  Computational Fluid Mechanics and Heat Transfer , 1984 .

[10]  Clarence O. E. Burg,et al.  Application of Richardson extrapolation to the numerical solution of partial differential equations , 2009 .

[11]  Fred J. Vermolen,et al.  Numerical Methods in Scientific Computing , 2006 .

[12]  Ismail Celik,et al.  Numerical Experiments on Application of Richardson Extrapolation With Nonuniform Grids , 1997 .

[13]  Yin Wang,et al.  Sixth order compact scheme combined with multigrid method and extrapolation technique for 2D poisson equation , 2009, J. Comput. Phys..

[14]  Comments on the Richardson Extrapolation , 1986 .

[15]  L. Richardson.  The Approximate Arithmetical Solution by Finite Differences of Physical Problems Involving Differential Equations, with an Application to the Stresses in a Masonry Dam , 1911 .

[16]  D. C. Joyce.  Survey of Extrapolation Processes in Numerical Analysis , 1971 .

[17]  H. B. Keller,et al.  Driven cavity flows by efficient numerical techniques , 1983 .

[18]  Ismail Celik,et al.  Calculation of Numerical Uncertainty Using Richardson Extrapolation: Application to Some Simple Turbulent Flow Calculations , 1995 .

[19]  Round-off errors in Richardson's extrapolation method , 1988 .

[20]  P. Knupp,et al.  Completed Richardson extrapolation , 1993 .

[21]  Dennis G. Zill,et al.  Advanced Engineering Mathematics , 2021, Technometrics.

[22]  R. Pletcher,et al.  Computational Fluid Mechanics and Heat Transfer. By D. A ANDERSON, J. C. TANNEHILL and R. H. PLETCHER. Hemisphere, 1984. 599 pp. $39.95. , 1986, Journal of Fluid Mechanics.

[23]  V. Denny,et al.  On the convergence of numerical solutions for 2-D flows in a cavity at large Re , 1979 .

[24]  L. Richardson,et al.  The Deferred Approach to the Limit. Part I. Single Lattice. Part II. Interpenetrating Lattices , 1927 .

[25]  Ayodeji O. Demuren,et al.  Estimating Uncertainty in Computations of Two-Dimensional Separated Flows , 1994 .

[26]  Marc Garbey,et al.  EVALUATION OF RICHARDSON EXTRAPOLATION IN COMPUTATIONAL FLUID DYNAMICS , 2002 .

[27]  Yongbin Ge,et al.  A high order finite difference method with Richardsonextrapolation for 3D convection diffusion equation , 2010, Appl. Math. Comput..

[28]  C. H. Marchi,et al.  The lid-driven square cavity flow: numerical solution with a 1024 x 1024 grid , 2009 .

[29]  Hiroyuki Ozoe,et al.  Extrapolation of finite-difference calculations of laminar natural convection in enclosures to zero grid size , 1981 .

[30]  I. Celik,et al.  Limitations of Richardson Extrapolation and Some Possible Remedies , 2005 .

[31]  Timothy G. Trucano,et al.  Verification and validation. , 2005 .

[32]  Hai-wei Sun,et al.  A high‐order finite difference discretization strategy based on extrapolation for convection diffusion equations , 2004 .

[33]  C. H. Marchi,et al.  UNIDIMENSIONAL NUMERICAL SOLUTION ERROR ESTIMATION FOR CONVERGENT APPARENT ORDER , 2002 .