An adaptive staggered grid finite difference method for modeling geodynamic Stokes flows with strongly variable viscosity

Abstract

Here we describe a new staggered grid formulation for discretizing incompressible Stokes flow which has been specifically designed for use on adaptive quadtree‐type meshes. The key to our new adaptive staggered grid (ASG) stencil is in the form of the stress‐conservative finite difference constraints which are enforced at the “hanging” velocity nodes between resolution transitions within the mesh. The new ASG discretization maintains a compact stencil, thus preserving the sparsity within the matrix which both minimizes the computational cost and enables the discrete system to be efficiently solved via sparse direct factorizations or iterative methods. We demonstrate numerically that the ASG stencil (1) is stable and does not produce spurious pressure oscillations across regions of grid refinement, which intersect discontinuous viscosity structures, and (2) possesses the same order of accuracy as the classical nonadaptive staggered grid discretization. Several pragmatic error indicators that are used to drive adaptivity are introduced in order to demonstrate the superior performance of the ASG stencil over traditional nonadaptive grid approaches. Furthermore, to demonstrate the potential of this new methodology, we present geodynamic examples of both lithospheric and planetary scales models.