The program considers the composite form of a numerical integration method for solving differential equations and constructs its linear multistep form. A set of tasks is performed concerning the dispersion analysis of the numerical method in a 1D context. Dispersion analysis is used to assess how well a numerical method replicates the propagation of waves compared to the exact solution.

First, the linear multistep form of the method is obtained. This form is used to calculate future states in a numerical simulation using several previous states.

Next, dispersion curves have been plotted. These curves show how the numerical wave speed differs from the true wave speed against a normalized wave number. This helps in visualizing the accuracy of the numerical method for wave propagation.

Next, certain values of the normalized wave number for given ratios of time step to wave period (( \Delta t / T )) and for different Courant-Friedrichs-Lewy (CFL) numbers are calculated under a specified condition for ( \beta ) parameters.

Finally, CFL values that optimize the numerical method performance for various ( \beta_1 ) values have been obtained. The optimal values ensure that the dispersion (difference between numerical and exact wave propagation) is minimized.