We describe our current efforts to develop finite volume
schemes for solving PDEs on logically Cartesian locally adapted
surfaces meshes. Our methods require an underlying smooth or
piecewise smooth grid transformation from a Cartesian computational
space to 3d surface meshes, but does not rely on analytic metric terms
to obtain second order accuracy. Our hyperbolic solvers are based on
Clawpack (R. J. LeVeque) and the parabolic solvers are based on a
diamond-cell approach (Y. Coudi\`ere, T. Gallou\"et, R. Herbin et
al). If time permits, I will also discuss Discrete Duality Finite
Volume methods for solving elliptic PDEs on surfaces.
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To do local adaption and time subcycling in regions requiring high
spatial resolution, we are developing ForestClaw, a hybrid adaptive
mesh refinement (AMR) code in which non-overlapping fixed-size
Cartesian grids are stored as leaves in a forest of quad- or
oct-trees. The tree-based code p4est (C. Burstedde) manages the
multi-block connectivity and is highly scalable in realistic
applications.
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I will present results from reaction-diffusion systems on surface
meshes, and test problems from the atmospheric sciences community.