# Global Linear Stability of the Two-Dimensional Shallow-Water Equations: An Application of the Distributive Theorem of Roots for Polynomials on the Unit Circle

Jia Wang Ocean Prediction Experimental Laboratory, Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, Florida

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## Abstract

This paper deals with the numerical stability of the linearized shallow-water dynamic and thermodynamic system using centered spatial differencing and leapfrog time differencing. The nonlinear version of the equations is commonly used in both 2D and 3D (split technique) numerical models. To establish the criteria, we employ the theorem of the root distributive theory of a polynomial proposed by Cheng (1966). The Fourier analysis or von Neumann method is applied to the linearized system to obtain a characteristic equation that is a sixth-order polynomial with complex coefficients. Thus, a series of necessary and sufficient criteria (but only necessary conditions for the corresponding nonlinear equations) are obtained by applying Cheng's theorem within the unit circle. It is suggested that the global stability should be determined by this set of criteria rather than the Courant–Friedrichs–Lewy (CFL) criterion alone. Each of the conditions has physical meaning: for instance, h + ζ > 0, |f| Δt < 1, and 0 < Δtβ^′ < 1, etc., must be satisfied as well, which helps define the model domain and the relation between damping coefficients and integration time step, where h is the undisturbed water depth, ζ the free surface elevation, f the Coriolis parameter, β^′ the sum of bottom friction coefficient and horizontal viscosity, and Δt the integrating time step. The full solution and the physical implications are given in the paper. Since Cheng's theorem was published in Chinese only and is of considerably theoretical and practical value in numerical stability analysis, the translation of the theorem is in appendix A.

## Abstract

This paper deals with the numerical stability of the linearized shallow-water dynamic and thermodynamic system using centered spatial differencing and leapfrog time differencing. The nonlinear version of the equations is commonly used in both 2D and 3D (split technique) numerical models. To establish the criteria, we employ the theorem of the root distributive theory of a polynomial proposed by Cheng (1966). The Fourier analysis or von Neumann method is applied to the linearized system to obtain a characteristic equation that is a sixth-order polynomial with complex coefficients. Thus, a series of necessary and sufficient criteria (but only necessary conditions for the corresponding nonlinear equations) are obtained by applying Cheng's theorem within the unit circle. It is suggested that the global stability should be determined by this set of criteria rather than the Courant–Friedrichs–Lewy (CFL) criterion alone. Each of the conditions has physical meaning: for instance, h + ζ > 0, |f| Δt < 1, and 0 < Δtβ^′ < 1, etc., must be satisfied as well, which helps define the model domain and the relation between damping coefficients and integration time step, where h is the undisturbed water depth, ζ the free surface elevation, f the Coriolis parameter, β^′ the sum of bottom friction coefficient and horizontal viscosity, and Δt the integrating time step. The full solution and the physical implications are given in the paper. Since Cheng's theorem was published in Chinese only and is of considerably theoretical and practical value in numerical stability analysis, the translation of the theorem is in appendix A.

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