HR: 0800h
AN: DI21A-0337    [Abstracts]
TI: A Comparison of Finite Difference Formulations for the Stokes Equations in the Presence of Strongly Varying Viscosity
AU: * Deubelbeiss, Y
EM: yolanda.deubelbeiss@erdw.ethz.ch
AF: AU: * Deubelbeiss, Y
EM: yolanda.deubelbeiss@erdw.ethz.ch
AF: AU: Kaus, B J
EM: kaus@erdw.ethz.ch
AF: AU: Kaus, B J
EM: kaus@erdw.ethz.ch
AF: AB: Numerical modeling of geodynamic processes typically requires the solution of the Stokes equations for creeping, highly viscous flows. Since material properties such as effective viscosity of rocks can vary many orders of magnitudes over small spatial scales, the Stokes solver needs to be robust even in the case of highly variable viscosity. Currently, a number of different techniques (e.g. finite element, finite difference and spectral methods) are in use by different authors. Benchmark studies indicate that the accuracy of the velocity solution is satisfying for most methods. The accuracy of deviatoric stresses and pressures, however, is typically less than that of velocities. In the case of highly varying viscosity, some methods even result in oscillating pressures. Over recent years there has been an increased demand for accurate pressures. E.g. melt migration through compacting, two- phase flow materials requires solving equations for the fluid and the solid matrix. Shear-localization in partially molten rocks couples moving fluid within a deforming solid. Therefore it is necessary to have accurate knowledge of pressures, which feed back to the solution. It becomes increasingly important to understand the accuracy of numerical methods for Stokes flow in the presence of large variations in material properties. The objective of this study is therefore to evaluate the accuracy of the pressure solution for a number of numerical techniques. Thereby, we make use of a 2-D analytical solution for the stress distribution inside and around a viscous inclusion in a matrix of different viscosity subjected to pure-shear boundary conditions. Furthermore, numerical simulations have been compared with the analytical solution for density-driven flow (Rayleigh-Taylor instabilities). Results are presented for a staggered grid velocity- pressure finite difference method, a stream function finite difference method and a rotated staggered grid velocity- pressure finite difference method. The staggered grid and stream function formulations require viscosities to be defined both at center and at corner points of control volumes, while the rotated staggered grid finite difference method only requires viscosity defined at center points. We demonstrate that the manner in which viscosities are defined at these locations is of extreme importance for the accuracy of the overall solution. The problem is investigated by studying a simple physical quasi 1-D model with a contact of two media representing the contact between an inclusion embedded in a matrix (2-D case). Analytically and numerically, it is demonstrated that viscosity interpolation using harmonic averaging yields the best results. 2-D numerical results for the above mentioned setups show that for different interpolation methods the errors can vary one order of magnitude. Accuracy of velocity solutions are more than half an order of magnitude better than pressure solutions. The Rayleigh-Taylor instability test, on the other hand, has a weaker sensitivity to viscosity interpolation methods. Results are mainly dependent on the manner in which density is interpolated, which is the driving force in this system. Differences between the three numerical schemes for both setups are secondary compared to the effect of the viscosity interpolation. The best averaging method, for the setups studied here, is a geometric- harmonic averaging of viscosity and an arithmetic averaging of density.
DE: 8020 Mechanics, theory, and modeling
DE: 8100 TECTONOPHYSICS
DE: 8120 Dynamics of lithosphere and mantle: general (1213)
DE: 8125 Evolution of the Earth (0325)
DE: 8160 Rheology: general (1236, 8032)
SC: Study of the Earth's Deep Interior [DI]
MN: 2007 Fall Meeting