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Gyroscopic polynomials
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DOI：10.1016/j.jcp.2023.112268.png)
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Abstract
En 中文
Gyroscopic alignment gives rise to highly spatially anisotropic columnar structures that in combination with complex domain boundaries pose challenges for efficient numerical dis-cretizations and computations. We define gyroscopic polynomials to be three-dimensional polynomials expressed in a coordinate system that conforms to rotational alignment. We remap the original domain with radius-dependent boundaries onto a right cylindrical or annular domain to create the computational domain in this coordinate system. We find the volume element expressed in gyroscopic coordinates leads naturally to a hierarchy of orthonormal bases. We build the bases out of Jacobi polynomials in the vertical and gen-eralized Jacobi polynomials in the radial. Because these coordinates explicitly conform to flow structures found in rapidly rotating systems the bases represent fields with a rel-atively small number of modes. We develop the operator structure for one-dimensional semi-classical orthogonal polynomials as a building block for differential operators in the full three-dimensional cylindrical and annular domains. The differentiation operators of generalized Jacobi polynomials generate a sparse linear system for discretization of differ-ential operators acting on the gyroscopic bases. This enables efficient simulation of systems with strong gyroscopic alignment. & COPY; 2023 Elsevier Inc. All rights reserved.
Keywords:
Stretched cylinder geometry
Stretched annulus geometry
Coordinate singularities
Spectral methods
Generalized Jacobi polynomials
Sparse operators
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