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Full-Text Articles in Applied Mathematics
Logically Rectangular Finite Volume Methods With Adaptive Refinement On The Sphere, Marsha Berger, Donna Calhoun, Christiane Helzel, Randall Leveque
Logically Rectangular Finite Volume Methods With Adaptive Refinement On The Sphere, Marsha Berger, Donna Calhoun, Christiane Helzel, Randall Leveque
Donna Calhoun
The logically rectangular finite volume grids for two-dimensional partial differential equations on a sphere and for three-dimensional problems in a spherical shell introduced recently have nearly uniform cell size, avoiding severe Courant number restrictions. We present recent results with adaptive mesh refinement using the GEOCLAW software and demonstrate well-balanced methods that exactly maintain equilibrium solutions, such as shallow water equations for an ocean at rest over arbitrary bathymetry.
A Finite Volume Method For Solving Parabolic Equations On Logically Cartesian Curved Surface Meshes, Donna Calhoun, Christiane Helzel
A Finite Volume Method For Solving Parabolic Equations On Logically Cartesian Curved Surface Meshes, Donna Calhoun, Christiane Helzel
Donna Calhoun
We present a second-order, finite-volume scheme for the constant-coefficient diffusion equation on curved, parametric surfaces described via smooth or piecewise smooth mappings on logically Cartesian meshes. Our method does not require analytic metric terms, shows second-order accuracy, can be easily coupled to existing finite-volume solvers for logically Cartesian meshes and handles general mixed boundary conditions. We present numerical results demonstrating the accuracy of the scheme, and then use the scheme to solve advection-reaction-diffusion equations modeling biological pattern formation on surfaces.