NONLINEAR INSTABILITY OF WAVETRAIN UNDER INFLUENCES OF SHEAR CURRENT WITH VARYING VORTICITY AND AIR PRESSURE

Under the assumption of weak shear current with varying vorticity in water and weak air pressure the Zakharov theory is extended to include the effects of vorticity and air pressure on the modulation of water waves. This new equation is used to examine the influence of current and wind on the Benjamin-Feir sideband instability and long-time evolution of wavetrain. As strength of the current increases the bandwidth is found broadened, and the maximum growth rate of sidebands decreased. Periodic solution of sidebands in the presence of current is indicated, which means that shear current does not affect the downshift of wave spectrum peak. Energy input by imposing the air pressure leads to the enhancement of the lower sideband, which is in agreement with the finding of Hara and Mei (1991).

Efficient and Accurate Numerical Methods Using the Accelerated Spectral Deferred Correction for Solving Fractional Differential Equations

We develop an efficient and accurate spectral deferred correction(SDC)method for fractional differential equations(FDEs)by extending the algorithm in[14]for classical ordinary differential equations(ODEs).Specifically,we discretize the resulted Picard integral equation by the SDC method and accelerate the convergence of the SDC iteration by using the generalized minimal residual algorithm(GMRES).We first derive the correction matrix of the SDC method for FDEs and analyze the convergence region of the SDC method.We then present several numerical examples for stiff and non-stiff FDEs including fractional linear and nonlinear ODEs as well as fractional phase field models,demonstrating that the accelerated SDC method is much more efficient than the original SDC method,especially for stiff problems.Furthermore,we resolve the issue of low accuracy arising from the singularity of the solutions by using a geometric mesh,leading to highly accurate solutions compared to uniform mesh solutions at almost the same computational cost.Moreover,for long-time integration of FDEs,using the geometric mesh leads to great computational savings as the total number of degrees of freedom required is relatively small.