Linear and Nonlinear Splitting Schemes Conserving Total Energy and Mass in the Shallow Water Model

Abstract

Two linear and one nonlinear implicit unconditionally stable finite-difference schemes of the second-order approximation in all variables are given for a shallow-water model including the rotation and topography of the earth. The schemes are based on splitting the model equation into two one-dimensional subsystems. Each of the subsystems conserves the mass and total energy in both differential and discrete (in time and space) forms. One of the linear schemes contains a smoothing procedure not violating the conservation laws and suppressing spurious oscillations caused by the application of central-difference approximations of spatial derivatives. The unique solvability of the linear schemes and convergence of iterations used to find their solutions are proved.