Numerical modeling of three dimensional self-gravitating Stokes flow problem with free surface

Abstract

A solution code of the Stokes flow system in three dimensions under a self-gravitating field was developed with a free surface in order to simulate the long-time scale evolution of the solid earth system on the massively parallel-vector super computer Earth Simulator 2 (ES2). The largely deformable surface of a planetary body is represented by a sharp boundary of scalar color function, which is advected by the CIP-CLSR method. The self-gravitating field is calculated by the Poisson equation of the gravity potential. A low viscous material (sticky air) surrounding the planet mimics its free surface motion. In order to solve the ill conditioned Stokes problem of the finite difference discretization on a staggered grid, I employ the iterative Stokes flow solver, robust to large viscosity jumps, using a strong Schur complement preconditioner and mixed precision arithmetic utilizing the double-double method. In this paper, I present an overview of the numerical algorithms and the current parallel programming strategy for the efficient performance on ES2. In addition, preliminary results of the core formation simulation are demonstrated in three dimensions as an example of the target Stokes flow system of this simulation code.

A high‐order multi‐moment constrained finite‐volume global shallow‐water model on the Yin‐Yang grid

“…This indicates the suitability of the Yin‐Yang grid to short‐ or medium‐range simulations for global geophysical fluid dynamics. Global models for different purposes and applications have been so far developed (Li et al , ; Qaddouri et al , ; Baba et al , ; Furuchi, ; Qaddouri, ; Qaddouri and Lee, ) based on the Yin‐Yang grid, which show the potential of the Yin‐Yang grid to be a simple and quasi‐uniform spherical grid for practical purposes. In this article we report a new shallow‐water model on the Yin‐Yang grid by using a new high‐order numerical method.…”

Implicit solution of the material transport in Stokes flow simulation: Toward thermal convection simulation surrounded by free surface

“…The sticky-air material is defined as a fluid with low viscosity and zero density which is evolved according to the governing equations given by Eqs. (1), (2), (4), (5).…”

“…[1][2][3][4][5]). Simulating such systems over million-year time scales enables numerical investigation of the interaction between the surface geometry (topography) and interior dynamics of planets, which is of fundamental importance * Corresponding author.…”