The Cartesian method for solving partial differential equations in spherical geometry

Abstract Cartesian coordinates are used to solve the nonlinear shallow-water equations on the sphere. The two-dimensional equations, in spherical coordinates, are first embedded in a three-dimensional system in a manner that preserves solutions of the two-dimensional system. That is, solutions of the three-dimensional system, with appropriate initial conditions, also solve the two-dimensional system on the surface of the sphere. The higher dimensional system is then transformed to Cartesian coordinates. Computations are limited to the surface of the sphere by projecting the equations, gradients, and solution onto the surface. The projected gradients are approximated by a weighted sum of function values on a neighboring stencil. The weights are determined by collocation using the spherical harmonics in trivariate polynomial form. That is, the weights are computed from the requirement that the projected gradients are near exact for a small set of spherical harmonics. The method is applicable to any distribution of points and two test cases are implemented on an icosahedral geodesic grid. The method is both vectorizable and parallelizable.

[1]  Norman A. Phillips,et al.  NUMERICAL INTEGRATION OF THE PRIMITIVE EQUATIONS ON THE HEMISPHERE , 1959 .

[2]  P. Swarztrauber,et al.  A standard test set for numerical approximations to the shallow water equations in spherical geometry , 1992 .

[3]  David L. Williamson,et al.  Integration of the barotropic vorticity equation on a spherical geodesic grid , 1968 .

[4]  C. Lawson,et al.  Solving least squares problems , 1976, Classics in applied mathematics.

[5]  R. Salmon.  New equations for nearly geostrophic flow , 1985, Journal of Fluid Mechanics.

[6]  R. Heikes,et al.  Numerical Integration of the Shallow-Water Equations on a Twisted Icosahedral Grid , 1995 .

[7]  G. L. Browning,et al.  A comparison of three numerical methods for solving differential equations on the sphere , 1989 .

[8]  P. Swarztrauber.  The Vector Harmonic Transform Method for Solving Partial Differential Equations in Spherical Geometry , 1993 .

[9]  Yoshinobu Masuda,et al.  An Integration Scheme of the Primitive Equation Model with an Icosahedral-Hexagonal Grid System and its Application to the Shallow Water Equations , 1986 .

[10]  G. Starius,et al.  Composite mesh difference methods for elliptic boundary value problems , 1977 .

[11]  F. Mesinger,et al.  A global shallow‐water model using an expanded spherical cube: Gnomonic versus conformal coordinates , 1996 .

[12]  M. Taylor.  The Spectral Element Method for the Shallow Water Equations on the Sphere , 1997 .

[13]  Richard Asselin,et al.  Frequency Filter for Time Integrations , 1972 .

[14]  G. Starius,et al.  On composite mesh difference methods for hyperbolic differential equations , 1980 .

[15]  N. A. Phillips,et al.  A Map Projection System Suitable for Large-scale Numerical Weather Prediction , 1957 .

[16]  Charles L. Lawson,et al.  Solving least squares problems , 1976, Classics in applied mathematics.

[17]  Paul N. Swarztrauber,et al.  The Approximation of Vector Functions and Their Derivatives on the Sphere , 1981 .

[18]  David A. Randall,et al.  Numerical Integration of the Shallow-Water Equations on a Twisted Icosahedral Grid. Part II. A Detailed Description of the Grid and an Analysis of Numerical Accuracy , 1995 .

[19]  Paul N. Swarztrauber.  Spectral Transform Methods for Solving the Shallow-Water Equations on the Sphere , 1996 .

[20]  John L. McGregor.  Semi-Lagrangian Advection on Conformal-Cubic Grids , 1996 .

[21]  André Robert,et al.  The Integration of a Low Order Spectral Form of the Primitive Meteorological Equations , 1966 .

[22]  P. Paolucci,et al.  The “Cubed Sphere” , 1996 .

[23]  Akio Arakawa,et al.  Integration of the Nondivergent Barotropic Vorticity Equation with AN Icosahedral-Hexagonal Grid for the SPHERE1 , 1968 .

[24]  William Bourke,et al.  An Efficient, One-Level, Primitive-Equation Spectral Model , 1972 .

[25]  David L. Williamson,et al.  INTEGRATION OF THE PRIMITIVE BAROTROPIC MODEL OVER A SPHERICAL GEODESIC GRID , 1970 .

[26]  Jack Dongarra,et al.  LINPACK Users' Guide , 1987 .