Back
 JAMP  Vol.6 No.6 , June 2018
Analysis on Sixth-Order Compact Approximations with Richardson Extrapolation for 2D Poisson Equation
Abstract: By using Richardson extrapolation and fourth-order compact finite difference scheme on different scale grids, a sixth-order solution is computed on the coarse grid. Other three techniques are applied to obtain a sixth-order solution on the fine grid, and thus give out three kinds of Richardson extrapolation-based sixth order compact computation methods. By carefully analyzing the truncation errors respectively on 2D Poisson equation, we compare the accuracy of these three sixth order methods theoretically. Numerical results for two test problems are discussed.
Cite this paper: Dai, R. and Lin, P. (2018) Analysis on Sixth-Order Compact Approximations with Richardson Extrapolation for 2D Poisson Equation. Journal of Applied Mathematics and Physics, 6, 1139-1159. doi: 10.4236/jamp.2018.66097.
References

[1]   Wang, Z.J., et al. (2013) High-Order CFD Methods: Current Status and Perspective. International Journal for Numerical Methods in Fluids, 72, 811-845.
https://doi.org/10.1002/fld.3767

[2]   Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A. (1998) Spectral Methods in Fluid Dynamics. Springer, New York.

[3]   Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A. (2006) Spectral Methods: Fundamentals in Single Domains. Springer, New York.

[4]   Bueno-Orovio, A., Pérez-Garcia, V.M. and Fenton, F.H. (2006) Spectral Methods for Partial Differential Equations in Irregular Domains: The Spectral Smoothed Boundary Method. SIAM Journal on Scientific Computing, 28, 886-900.
https://doi.org/10.1137/040607575

[5]   Saad, Y. (1996) Iterative Methods for Sparse Linear Systems. PWS Publishing, New York.

[6]   Kalita, J.C., Datal, D.C. and Dass, A.K. (2002) A Class of Higher Order Compact Schemes for the Unsteady Two-Dimensional Convection Diffusion Equation with Variable Convection Coefficients. International Journal for Numerical Methods in Fluids, 38, 1111-1131.
https://doi.org/10.1002/fld.263

[7]   Spotz, W.F. and Carey, G.F. (1995) High-Order Compact Scheme for the Steady Stream-Function Vorticity Equations. International Journal for Numerical Methods in Engineering, 38, 3497-3512.
https://doi.org/10.1002/nme.1620382008

[8]   Zhang, J. (2002) Multigrid Method and Fourth Order Compact Difference Scheme for 2D Poisson Equation with Unequal Mesh-Size Discretization. Journal of Computational Physics, 179, 170-179.
https://doi.org/10.1006/jcph.2002.7049

[9]   Zhang, J., Geng, Z. and Dai, R. (2012) Analysis on Two Approaches for High Order Accuracy Finite Difference Computation. Applied Mathematics Letters, 25, 2081-2085.
https://doi.org/10.1016/j.aml.2012.05.003

[10]   Spotz, W.F. and Carey, G.F. (1996) A High-Order Compact Formulation for the 3D Poisson Equation. Numerical Methods for Partial Differential Equations, 12, 235-243.
https://doi.org/10.1002/(SICI)1098-2426(199603)12:2<235::AID-NUM6>3.0.CO;2-R

[11]   Lele, S.K. (1992) Compact Finite Difference Schemes with Spectral-Like Resolution. Journal of Computational Physics, 103, 16-42.
https://doi.org/10.1016/0021-9991(92)90324-R

[12]   Sutmann, G. and Steffen, B. (2006) High-Order Compact Solvers for the Three-Dimensional Poisson Equation. Journal of Computational and Applied Mathematics, 187, 142-170.
https://doi.org/10.1016/j.cam.2005.03.041

[13]   Chu, P.C. and Fan, C. (1998) A Three-Point Combined Compact Difference Scheme. Journal of Computational Physics, 140, 370-399.
https://doi.org/10.1006/jcph.1998.5899

[14]   Zhang, J. and Zhao, J. (2005) Truncation Error and Oscillation Property of the Combined Compact Difference Scheme. Applied Mathematics and Computation, 161, 241-251.
https://doi.org/10.1016/j.amc.2003.12.023

[15]   Nabavi, M., Kamran Siddiqui, M.H. and Dargahia, J. (2007) A New 9-Point Sixth-Order Accurate Compact Finite-Difference Method for the Helmholtz Equation. Journal of Sound and Vibration, 307, 972-982.
https://doi.org/10.1016/j.jsv.2007.06.070

[16]   Kyei, Y., Roop, J.P. and Tang, G. (2010) A Family of Sixth-Order Compact Finite-Difference Schemes for the Three-Dimensional Poisson Equation. Advances in Numerical Analysis, 2010, Article ID: 352174.

[17]   Richardson, L.F. (1910) The Approximate Arithmetical Solution by Finite Differences of Physical Problems Involving Differential Equations, with an Application to the Stresses in a Masonry Dam Trans. Proceedings of the Royal Society of London. Series A, 210, 307-357.

[18]   Cheney, W. and Kincard, E. (1999) Numerical Mathematics and Computing. 4th Edition, Cole Publishing, Pacific Grove.

[19]   Roache, P.J. and Knupp, P.M. (1993) Completed Richardson Extrapolation. Communications in Numerical Methods in Engineering, 9, 365-374.
https://doi.org/10.1002/cnm.1640090502

[20]   Burg, C. and Erwin, T. (2009) Application of Richardson Extrapolation to the Numerical Solution of Partial Differential Equations. Numerical Methods for Partial Differential Equations, 25, 810-832.
https://doi.org/10.1002/num.20375

[21]   Sun, H. and Zhang, J. (2004) A High Order Finite Difference Discretization Strategy Based on Extrapolation for Convection Diffusion Equations. Numerical Methods for Partial Differential Equations, 20, 18-32.
https://doi.org/10.1002/num.10075

[22]   Wang, Y. and Zhang, J. (2009) Sixth Order Compact Scheme Combined with Multigrid Method and Extrapolation Technique for 2D Poisson Equation. Journal of Computational Physics, 228, 137-146.
https://doi.org/10.1016/j.jcp.2008.09.002

[23]   Dai, R., Zhang, J. and Wang, Y. (2014) Multiple Coarse Grid Acceleration for Multiscale Multigrid Computation. Journal of Computational and Applied Mathematics, 269, 75-85.
https://doi.org/10.1016/j.cam.2014.03.021

[24]   Dai, R., Wang, Y. and Zhang, J. (2013) Fast and High Accuracy Multiscale Multigrid Method with Multiple Coarse Grid Updating Strategy for 3D Convection Diffusion Equation. Computers & Mathematics with Applications, 66, 542-559.
https://doi.org/10.1016/j.camwa.2013.06.008

[25]   Gibou, F. and Fedkiw, R. (2005) A Fourth Order Accurate Discretization for the Laplace and Heat Equations on Arbitrary Domains, with Applications to the Stefan Problem. Journal of Computational Physics, 202, 577-601.
https://doi.org/10.1016/j.jcp.2004.07.018

[26]   Lin, Q. and Xie, H. (2013) Extrapolation of the Finite Element Method on General Meshes. International Journal of Numerical Analysis and Modeling, 10, 139-153.

 
 
Top