A fast stable accurate artificial boundary condition for the linearized Green-Naghdi system

In this paper, we use a fully discrete Crank-Nicolson scheme to solve the Cauchy problem for one-dimensional linearized Green-Naghdi system with fast convolution boundary condition which is derived through the Padé approximation for the square root function. We also introduce a constant damping term to guarantee the stability. While the damping term meets certain conditions, the stability is proved for the numerical solutions with the fast convolution boundary condition. Therefore, the difficulty of numerical instability which rises in Kazakova and Noble (SIAM J. Num. Anal. 1, 657–683, 2020) is overcome. The computational cost of the convolution integral is reduced from O(N²) to O(Nln(N)) for the total number of time steps N. Numerical examples verify the theoretical results and demonstrate the performance for the fast numerical method.