Limitations of Richardson Extrapolation and Some Possible Remedies

The origin of oscillatory convergence in finite difference methods is investigated. Fairly simple implicit schemes are used to solve the steady one-dimensional convection-diffusion equation with variable coefficients, and possible scenarios are shown that exhibit the oscillatory convergence. Also, a manufactured solution to difference equations is formulated that exhibits desired oscillatory behavior in grid convergence, with a varying formal order of accuracy. This model-error equation is used to statistically assess the performance of several methods of extrapolation. Alternative extrapolation schemes, such as the deferred extrapolation to limit technique, to calculate the coefficients in the Taylor series expansion of the error function are also considered. A new method is proposed that is based on the extrapolation of approximate error, and is shown to be a viable alternative to other methods. This paper elucidates the problem of oscillatory convergence, and brings a new perspective to the problem of estimating discretization error by optimizing the information from a minimum number of calculations.

[1]  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 .

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

[3]  George M. Phillips,et al.  Theory and applications of numerical analysis , 1976, The Mathematical Gazette.

[4]  H. Keller.  Numerical Solution of Two Point Boundary Value Problems , 1976 .

[5]  Application of Richardson extrapolation to some simple turbulent flow calculations. , 1993 .

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

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

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

[9]  Hugh W. Coleman,et al.  Comprehensive Approach to Verification and Validation of CFD Simulations—Part 2: Application for Rans Simulation of a Cargo/Container Ship , 2001 .

[10]  Hugh W. Coleman,et al.  Comprehensive Approach to Verification and Validation of CFD Simulations—Part 1: Methodology and Procedures , 2001 .

[11]  J. Cadafalch,et al.  Verification of Finite Volume Computations on Steady-State Fluid Flow and Heat Transfer , 2002 .

[12]  Ismail Celik,et al.  Assessment of numerical uncertainty for the calculations of turbulent flow over a backward‐facing step , 2005 .

[13]  J. S. Turner,et al.  QUANTIFICATION OF UNCERTAINTY IN COMPUTATIONAL FLUID DYNAMICS , 2006 .