We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and a splitting scheme of the fourth-order governing equations in terms of two second-order elliptic operators. We show our methods have high-order convergence rates for the metric and other geometric quantities of the manifold , for the truncation errors of exterior calculus operators , and for the solution errors of the Stokes problem for hydrodynamic flows on curved surfaces. Our approaches also may be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.
We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators . The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.
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a splitting scheme of
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We show our methods have
Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit
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solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature,
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, for the truncation errors of
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, and for the solution errors of the Stokes problem for hydrodynamic flows on curved surfaces. Our approaches also may
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