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ABSTRACT
The aim of this study is to extend the use of repeated Richardson extrapolation to one-dimensional (1D) and two-dimensional (2D) fields in computational fluid dynamics (CFD). The following two methods are tested: completed Richardson extrapolation (CRE), a method that has been used previously in the literature, and full Richardson extrapolation (FRE), a new method developed in this study. The Poisson’s, advection–diffusion, Laplace’s, and Burgers’ equations are solved using the finite difference method. The CRE and FRE methods were found to significantly reduce the discretization error of the numerical solutions for all nodes of the grid.
Nomenclature
| C | = | Richardson correction |
| CDS-2 | = | second-order central differencing scheme |
| CDS-4 | = | fourth-order central differencing scheme |
| E | = | discretization error in the numerical solution |
| g | = | number of a grid |
| G | = | number of grids |
| h | = | distance between two consecutive nodes in each grid |
| k | = | weighting factor |
| L1, L2 and Li | = | L1-norm, L2-norm, and |
| m | = | number of Richardson extrapolations |
| N | = | total number of nodes in grid |
| Ng | = | total number of nodes in grid g |
| p | = | pressure (Pa) |
| P, W, E | = | spatial position of the node in the grid |
| pE | = | effective order |
| Pe | = | Peclet number |
| pf | = | order of accuracy of numerical solution |
| pm | = | true orders |
| p0 | = | theoretical order of accuracy |
| r | = | grid refinement ratio |
| Re | = | Reynolds number |
| u, v | = | dependent variables |
| UDS-1 | = | first-order upwind differencing scheme |
| x, y | = | spatial coordinates |
Nomenclature
| C | = | Richardson correction |
| CDS-2 | = | second-order central differencing scheme |
| CDS-4 | = | fourth-order central differencing scheme |
| E | = | discretization error in the numerical solution |
| g | = | number of a grid |
| G | = | number of grids |
| h | = | distance between two consecutive nodes in each grid |
| k | = | weighting factor |
| L1, L2 and Li | = | L1-norm, L2-norm, and |
| m | = | number of Richardson extrapolations |
| N | = | total number of nodes in grid |
| Ng | = | total number of nodes in grid g |
| p | = | pressure (Pa) |
| P, W, E | = | spatial position of the node in the grid |
| pE | = | effective order |
| Pe | = | Peclet number |
| pf | = | order of accuracy of numerical solution |
| pm | = | true orders |
| p0 | = | theoretical order of accuracy |
| r | = | grid refinement ratio |
| Re | = | Reynolds number |
| u, v | = | dependent variables |
| UDS-1 | = | first-order upwind differencing scheme |
| x, y | = | spatial coordinates |