AN ENERGY-CONSERVING DIFFERENCE SCHEME FOR THE STORM SURGE EQUATIONS1

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Abstract
A system of finite difference equations for storm surge prediction has been constructed, using forward time differences. The scheme was tested for special simple geometrical configurations, and it was found to be stable without introducing smoothing operators. The variation with time of the total energy was, in each case, the test of stability. The small-scale oscillation of the energy with time (characteristic of forward difference schemes) was studied in detail. A method of reducing this effect is suggested. A completely implicit finite difference scheme is discussed from the point of view of stability and convergence. It is shown how the requirement of a convergent iterative process actually introduces a severe restriction on the ratio Δt/Δs, thus canceling the advantages of the otherwise unconditionally stable implicit schemes. Abstract A system of finite difference equations for storm surge prediction has been constructed, using forward time differences. The scheme was tested for special simple geometrical configurations, and it was found to be stable without introducing smoothing operators. The variation with time of the total energy was, in each case, the test of stability. The small-scale oscillation of the energy with time (characteristic of forward difference schemes) was studied in detail. A method of reducing this effect is suggested. A completely implicit finite difference scheme is discussed from the point of view of stability and convergence. It is shown how the requirement of a convergent iterative process actually introduces a severe restriction on the ratio Δt/Δs, thus canceling the advantages of the otherwise unconditionally stable implicit schemes.

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