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An Implicit Compact Fourth-Order Algorithm for Solving the Shallow-Water Equations in Conservation-Law Form

I. M. NavonNational Research Institute for Mathematical Sciences of the CSIR, Pretoria, South Africa

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H. A. RiphagenNational Research Institute for Mathematical Sciences of the CSIR, Pretoria, South Africa

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Abstract

An alternating-direction implicit finite-difference scheme is developed for solving the nonlinear shallow-water equations in conservation-law form.

The algorithm is second-order time accurate, while fourth-order compact differencing is implemented in a spatially factored form. The application of the higher order compact Padé differencing scheme requires only the solution of either block-tridiagonal or cyclic block-tridiagonal coefficient matrices, and thus permits the use of economical block-tridiagonal algorithms. The integral invariants of the shallow-water equations, i.e., mass, total energy and enstrophy, are well conserved during the numerical integration, ensuring that a realistic nonlinear structure is obtained.

Largely in an experimental way, two methods are investigated for determining stable approximations for the extraneous boundary conditions required by the fourth-order method. In both methods, third-order uncentered differences at the boundaries are utilized, and both preserve the overall fourth-order convergence rate of the more accurate interior approximation.

A fourth-order dissipative term was added to the equations to overcome the increased aliasing due to the fourth-order method. Alternatively, Wallington and Shapiro low-pass filters were applied.

The numerical integration of the shallow-water equations is performed in a channel corresponding to a middle-latitude band. A linearized version of this method is shown to be unconditionally stable.

Abstract

An alternating-direction implicit finite-difference scheme is developed for solving the nonlinear shallow-water equations in conservation-law form.

The algorithm is second-order time accurate, while fourth-order compact differencing is implemented in a spatially factored form. The application of the higher order compact Padé differencing scheme requires only the solution of either block-tridiagonal or cyclic block-tridiagonal coefficient matrices, and thus permits the use of economical block-tridiagonal algorithms. The integral invariants of the shallow-water equations, i.e., mass, total energy and enstrophy, are well conserved during the numerical integration, ensuring that a realistic nonlinear structure is obtained.

Largely in an experimental way, two methods are investigated for determining stable approximations for the extraneous boundary conditions required by the fourth-order method. In both methods, third-order uncentered differences at the boundaries are utilized, and both preserve the overall fourth-order convergence rate of the more accurate interior approximation.

A fourth-order dissipative term was added to the equations to overcome the increased aliasing due to the fourth-order method. Alternatively, Wallington and Shapiro low-pass filters were applied.

The numerical integration of the shallow-water equations is performed in a channel corresponding to a middle-latitude band. A linearized version of this method is shown to be unconditionally stable.

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