Shall4 — an implicit compact fourth-order FORTRAN program for solving the shallow-water equations in conservation-law form

Abstract

A FORTRAN IV computer program is documented implementing a compact fourth-order accurate finite difference scheme in a spatially factored form, for solving the nonlinear shallow-water equations on a limited domain. In contrast to the usual fourth-order schemes this compact fourth-order scheme requires the solution of only either block-tridiagonal or cyclic block-tridiagonal coefficient matrices. Moreover this compact fourth-order scheme is related to the finite-element method and has a smaller truncation error than the usual fourth-order schemes. The integral invariants of the shallow-water equations are calculated at each time-step and were determined to be well conserved during the numerical integration, ensuring that a realistic nonlinear structure is obtained. A Schumann-Wallington low-pass filtering procedure was incorporated in the program to overcome the increased aliasing due to the higher accuracy method. A third-order boundary condition is imposed, preserving the overall fourth-order convergence rate of the interior approximation.