Unconditional stability of alternating difference schemes with variable time steplengthes for dispersive equation

In this paper, the difference method with intrinsic parallelism for dispersive equation is studied. The general alternating difference schemes with variable time steplengthes are constructed and proved to be unconditionally stable. Some concrete parallel difference schemes are the special cases of the general schemes, such as the alternating group explicit scheme with variable time steplengthes, the alternating segment explicit-implicit scheme with variable time steplengthes, the alternating segment Crank-Nicolson scheme with variable time steplengthes, and so on. The numerical results are given to show the effectiveness of the present method. They show that the variable time steplengthes schemes are more accurate and can be obtained with less computational effort than the equal time steplengthes schemes.