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

Abstract

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 Pade 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 o...