High-order finite difference methods on the cubed sphere: Application to passive scalar advection

Document Type : Research Article

Authors

Department of Space Physics, Institute of Geophysics, University of Tehran, Tehran, Iran.

Abstract

The present work is part of our ongoing effort to make use of the cubed sphere for global atmospheric modeling applications. To this end, high-order numerical solutions of advection equation on the gnomonic cubed sphere grid are presented. Arbitrary-order one-dimensional interpolation based on spherical distance of the grid points have been used at the halo regions to bind the faces of the cubed sphere together. Although, the implementation of the model in this work is such that it makes using finite difference operators of various types and orders straightforward, we only discuss the results of 2nd-, 4th-, and 6th-order centered explicit operators. For time integration, the classical 4th-order Runge-Kutta method and the leapfrog method with a first-degree polynomial regression time filter with 2nd-order accuracy have been used. These methods have been compared by solving the advection equation using solid body rotations of  and  cosine bells. The long-range variant of this test is used to analyze the stability of the schemes. The methods and test cases have been chosen in such a way that facilitates the assessment of different components of the numerical scheme and factors affecting their accuracy.
Results of the advection test cases, demonstrate that although low differentiability class of the advected field or low order of accuracy of the time integration scheme, could limit the actual order of accuracy of the solution, conservation properties of the model could, nevertheless, be improved significantly by the use of high-order finite difference operators. All of the methods used in this work need some scale-selective artificial dissipation for stability which has been supplied in the form of high-order explicit filters. Single-parameter fractional filters (i.e. filters that remove only a fraction of the shortest waves) are used in the long-range integrations to determine the required amount of artificial dissipation for each method.
Results demonstrate that lower-order methods and lower resolutions require more artificial dissipation for stability. Since there is no systematic cascade of variance to smaller scales in the solid body rotation test-case, application of filter does not reduce the order of accuracy of the normalized variance. Also, the minimum required dissipation which has been calculated here might not be sufficient in cases where there is such a cascade. Time traces of 2-norm of error display a linear increase with time with no jumps or transient increases. Although such transient increases are present in time traces of normalized mean and variance, which might signify grid imprinting, this does not pose a problem since these non-conservation errors converge with or faster than the expected order. On the whole, the results demonstrate the superiority of the high-order methods in terms of accuracy and performance. In long-range integrations, high-order methods exhibit convergence of errors at low resolutions even after 100 revolutions of the cosine bells.

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References

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Journal of the Earth and Space Physics
Volume 51, Issue 2
online publication date: 20 September 2025
September 2025
Pages 353-375

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