Solving mild-slope equation by explicit scheme

An explicit scheme of Saul'yev for parabolic equations has been used to solve the time-dependent parabolic equation of the mild-slope equation. Theoretical analysis has been presented for stability and consistency of the explicit scheme applied to the mild-slope equation. The explicit scheme is simpler than most existing numerical schemes for solving the mild-slope equation, including elliptical, hyperbolic and parabolic (time-dependent) types and it is faster than the generalised conjugate gradient (GCG) scheme. It has about the same convergent speed as the alternating direction implicit (ADI) scheme for the parabolic equation, but the scheme is simpler. Experimental data obtained by Berkhoff et al. have been used to test the model. A comparison of present model results, for the case of an elliptic shoal, against the results of the GCG model and ADI model has been made and good agreement has been achieved. A nested grid method has also been combined with the present explicit scheme for solving the mild-slope equation. The simplicity of the explicit scheme makes the nested grid method very attractive for problems of wave propagation from deep to shallow water, where in deep water a coarse mesh can be used and in the shallow water region the mesh can be refined to achieve greater accuracy.