**Nikiforov**

Vladimir O.

D.Sc., Prof.

Vladimir O.

D.Sc., Prof.

doi: 10.17586/2226-1494-2016-16-1-161-167

doi: 10.17586/2226-1494-2016-16-1-161-167

# BENCHMARK SOLUTIONS FOR STOKES EQUATIONS WITH VARIABLE VISCOSITY IN CYLINDRICAL AND SPHERICAL COORDINATES

**Read the full article**';

**Article in**Russian

**For citation:**Makeev I.V., Popov I.Yu., Blinova I.V. Benchmark solutions for Stokes equations with variable viscosity in cylindrical and spherical coordinates for algorithm testing.

*Scientific and Technical Journal of Information Technologies, Mechanics and Optics*, 2016, vol. 16, no. 1, pp. 161–167.

**Abstract**

Stokes flows in cylindrical and spherical geometry are considered. Such flows are rather natural for geophysics. We derive some exact particular solutions of Stokes and continuity equations for particular dependence of viscosity and density on cylindrical coordinates. These solutions correspond to axisymmetric flows for the case when viscosity is a function of radius. We suggest exact particular solutions of Stokes and continuity equations with variable viscosity and density in spherical coordinates for the case of spherically symmetric viscosity and density distributions. We demonstrate how these solutions can be used for creation of test problems suitable for benchmarking numerical algorithms. Examples of such benchmarking are presented. The advantage of this benchmarking approach is the ability to test numerical algorithms for variable viscosity and density gradients. We suggest numerical scheme of multigrid algorithm for solving Stokes and continuity equations with variable viscosity in a spherical coordinate system. Calculations are performed on a sequence of orthogonal staggered grids. The quality of the numerical scheme was verified by comparing the numerical solution with the analytical solution of the test problem.

**Keywords:**Stokes equations, variable viscosity, Gauss-Seidel method, benchmark solutions, multigrid methods

**References**

1. Gerya T. Introduction to Numerical Geodynamic Modeling. Cambridge, Cambridge University Press, 2010, 358 p.

2. Ismail-Zadeh A., Tackley P. Computational Methods for Geodynamics. Cambridge, Cambridge University Press, 2010, 313 p.

3. Deubelbeiss Y., Kaus B.J. Comparison of Eulerian and Lagrangian numerical techniques for the Stokes equations in the presence of strongly varying viscosity.* Physics of Earth Planetary Interiors*, 2008, vol. 171, no. 1–4, pp. 92–111. doi: 10.1016/j.pepi.2008.06.023

4. Duretz T., May D.A., Gerya T.V., Tackley P.J. Discretization errors and free surface stabilization in the finite-difference and marker-in-cell method for applied geodynamics: a numerical study. Geochemistry, Geophysics, Geosystems, 2008, vol. 12, no. 7, art. Q07004. doi: 10.1029/2011GC003567

5. Blankenbach B., Busse F. et al. A benchmark comparison for mantle convection code.* Geophysical Journal International*, 1989, vol. 98, no. 1, pp. 23–38.

6. Busse F.H., Christensen U., Clever R., Cserepes L., Gable C., Giannandrea E., Duillou L., Houseman G. et al. 3D convection at infinite Prandtl number in Cartesian geometry – a benchmark comparison. *Geophysical and Astrophysical Fluid Dynamics*, 1993, vol. 75, pp. 39–59. doi: 10.1080/03091929408203646

7. Popov A.I., Lobanov I.S., Popov I.Yu., Gerya T.V. Benchmark solutions for nanoflows. *Nanosystems: Physics, Chemistry, Mathematics*, 2014, vol. 5, pp. 391–399.

8. Popov I.Yu., Lobanov I.S., Popov S.I., Popov A.I., Gerya T.V. Practical analytical solutions for benchmarking of 2-D and 3-D geodynamic Stokes problems with variable viscosity. *Solid Earth*, 2014, vol. 5, no. 1, pp. 461–476. doi: 10.5194/se-5-461-2014

9. Tosl N., Stein C., Noack L., Huttig C., Maierova P., Samuel H., Davies D.R., Wilson C.R., Kramer S.C., Thieulot C., Glerum A., Fraters M., Spakman W., Rozel A., Tackey P.J. A community benchmark for viscoplastic thermal convection in a 2-D square box. *Geochemistry, Geophysics, Geosystems*, 2015, vol. 16, no. 7, pp. 2175–2196. doi: 10.1002/2015GC005807.

10. Popov A.I., Gerya T.V., Lobanov I.S., Popov I.Yu. On the Stokes flow computation algorithm based on Woodbury formula. *Nanosystems: Physics, Chemistry, Mathematics*, 2015, vol. 6, pp. 140–145. doi: 10.17586/2220-8054-2015-6-1-140-145

11. Tackley P. Modelling compressible mantle convection with large viscosity contrasts in a three-dimensional spherical shell using the yin-yang grid. *Physics of the Earth and Planetary Interiors*, 2008, vol. 171, no. 1–4, pp. 7–18. doi: 10.1016/j.pepi.2008.08.005

12. van Keken P.E., Currie C., King S.D., Behn M., Cagnioncle A., He J., Katz R.F., Lin S.-C., Parmentier E.M., Spiegelman M., Wang K. A community benchmark for subduction zone modeling. *Physics of the Earth and Planetary Interiors*, 2008, vol. 171, pp. 187–197. doi: 10.1016/j.pepi.2008.04.015

13. Brenner S.C. A nonconforming multigrid method for the stationary stokes equations. *Mathematics of Computation*, 1990, vol. 55, pp. 411–437. doi: 10.1090/S0025-5718-1990-1035927-5

14. Kang K., Kwak D., Yon Y. The analysis of multigrid method for nonconforming method for the stationary stokes equations.* Bulletin of the Korean Mathematical Society*, 1996, vol. 33, no. 3, pp. 343–357.

15. John V., Tobiska L. A coupled multigrid method for nonconforming finite element discretizations of the 2D-stokes equation. *Computing*, 2000, vol. 64, no. 4, pp. 307–332.

16. Wang M., Chen L. Multigrid methods for the stokes equations using distributive Gauss-Seidel relaxations based on the least squares commutator. *Journal of Scientific Computing*, 2013, vol. 56, no. 2, pp. 409–431. doi: 10.1007/s10915-013-9684-1