A Newton linearized compact finite difference scheme for one class of Sobolev equations

In this article, a Newton linearized compact finite difference scheme is proposed to numerically solve a class of Sobolev equations. The unique solvability, convergence, and stability of the proposed scheme are proved. It is shown that the proposed method is of order 2 in temporal direction and order 4 in spatial direction. Moreover, compare to the classical extrapolated Crank-Nicolson method or the second-order multistep implicit-explicit methods, the proposed scheme is easier to be implemented as it only requires one starting value. Finally, numerical experiments on one and two-dimensional problems are presented to illustrate our theoretical results.