# SIMULATION OF NONLINEAR DIFFUSION ON A SPHERE

### Yuri N. Skiba, Denis M. Filatov

#### Abstract

A new numerical technique for the simulation of nonlinear diffusion processes on a sphere is developed. The core of our approach is to split the original equation's operator, thus reducing the two-dimensional problem to two one-dimentional problems. Further, we apply two different coordinate grids to cover the entire sphere for solving the split 1D problems. This allows avoiding the question of imposing adequate boundary conditions near the poles, which is always a serious problem when modelling on a sphere. Yet, therefore we can employ finite difference schemes of any even approximation order in space. The developed approach is cheap to implement from the computational point of view. Numerical experiments prove the suggested technique, simulating several diffusion phenomena with high accuracy.

#### References

- Bear, J. (1988). Dynamics of Fluids in Porous Media. Dover Publications, New York.
- 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. J. Comput. Phys., 202:577-601.
- Lacey, A. A., Ockendon, J. R., and Tayler, A. B. (1982). “waiting-time” solutions of a nonlinear diffusion equation. SIAM J. Appl. Math., 42:1252-1264.
- Marchuk, G. I. (1982). Methods of Computational Mathematics. Springer-Verlag, Berlin.
- Peletier, L. A. (1981). The porous media equation. In Ammam, H. and Bazley, N., editors, Applications of Nonlinear Analysis in the Physical Sciences, pages 229- 241. Pitman, Boston.
- Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (2007). Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge.
- Seshadri, R. and Na, T. Y. (1985). Group Invariance in Engineering Boundary Value Problems. Springer-Verlag, New York.
- Skiba, Y. N. and Filatov, D. M. (2011). On an efficient splitting-based method for solving the diffusion equation on a sphere. Numer. Meth. Part. Diff. Eq., 27:doi: 10.1002/num.20622.
- Wu, Z., Zhao, J., Yin, J., and Li, H. (2001). Nonlinear Diffusion Equations. World Scientific Publishing, Singapore.

#### Paper Citation

#### in Harvard Style

N. Skiba Y. and M. Filatov D. (2011). **SIMULATION OF NONLINEAR DIFFUSION ON A SPHERE** . In *Proceedings of 1st International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,* ISBN 978-989-8425-78-2, pages 24-30. DOI: 10.5220/0003574500240030

#### in Bibtex Style

@conference{simultech11,

author={Yuri N. Skiba and Denis M. Filatov},

title={SIMULATION OF NONLINEAR DIFFUSION ON A SPHERE},

booktitle={Proceedings of 1st International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,},

year={2011},

pages={24-30},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0003574500240030},

isbn={978-989-8425-78-2},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of 1st International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,

TI - SIMULATION OF NONLINEAR DIFFUSION ON A SPHERE

SN - 978-989-8425-78-2

AU - N. Skiba Y.

AU - M. Filatov D.

PY - 2011

SP - 24

EP - 30

DO - 10.5220/0003574500240030