Most of numerical methods for diffusion equations, refer to vertex unknowns directly or indirectly, and their accuracy is ultimately determined by the approximation to vertex unknowns. Based on the “twin-fitting” method, a simple and high accurate treatment for the vertex unknowns is developed and is applied to the nine-point scheme for diffusion problem. Numerical experiments show that the resulting nine-point scheme is high accurate for diffusion problems with discontinuous diffusion coefficients on distorted meshes.
KershawD.S., Differencing of the diffusion equation in Lagrangian hydrodynamic codes, Journal of Computational Physics39 (1991), 375–395.
2.
LiD.Y. and ChenG.N., An introduction to difference methods of parabolic equations, Science Press (in Chinese), 1995.
3.
HermelineF., Approximation of 2-D and 3-D diffusion operators with variable full tensor coefficients on arbitrary meshes, Computer Methods in Applied Mechanics and Engineering196 (2007), 2497–2526.
4.
AavatsmarkI., An introduction to multipoint flux approximations for quadrilateral grids, Computational Geosciences6 (2002), 405–432.
5.
DroniouJ. and PotierC.L., Construction and convergence study of schemes preserving the elliptic local maximum principle, SIAM Journal on Numerical Analysis49(2) (2011), 459–490.
6.
MorelJ.E., DendyJ.E., HallM. L and WhiteS.W., A cell-centered Lagrangian-mesh diffusion differencing scheme, Journal of Computational Physics103 (1992), 286–299.
7.
WuJ.M. and GaoZ.M., A nine-point scheme with explicit weights for diffusion equation on distorted meshes, Applied Numerical Mathematics61 (2011), 844–867.
8.
ShashkovM.J., Conservative finite-difference methods on general grids, CRC Press, Boca Raton, FL, 1996.
9.
SongS.H. and WangS.H., High accuracy numerical method for heatconduction equation with discontinuous diffusion coefficient, Acta Mathematicae Applicatae Sinica (Chinese Series)34(2) (2011), 229–239.
10.
SongS.H. and WangS.H., A Simple and High Accurate Algorithm for Diffusion Problem, Acta Mathematicae Applicatae Sinica (Chinese Series)37(2) (2014), 367–378.
11.
ZhukovV.T. and FeodoritovaO.B., Difference schemes for the heat condunction equation based on local least-squares approximations, Preprint No, 97, IPMathem. Akad. Nauk S. S. S. R., Moscow, 1989.
12.
LiuX.Z., WangR.L., LinZ. and YuY.L., Computational Methods and Program for Langrange Radiation Hydrodynamics Based On Unstructured Polygonal Meshes, Inertial Confinement Fusion Annual Report (in Chinese)2 (2014), 92–100.
13.
ShengZ.Q. and YuanG.W., The finite volume scheme preserving extrenum principle for diffusion equations on polygonal meshes, Journal of Computational Physics230 (2011), 2588–2604.
