Difference schemes of high accuracy for a Sobolev-type pseudoparabolic equation

In this work, numerical algorithms of higher-order accuracy are constructed and studied for a pseudoparabolic equation that describes the filtration process in fractured-porous media. The increase in the order of accuracy is achieved in various ways. First, only the spatial variables are approximated, as in the method of lines. Then, to solve the resulting system of linear ordinary differential equations, the finite difference method and the finite element method are applied. The application of these methods makes it possible to achieve a higher order of approximation for the difference schemes. Schemes of fourth-order accuracy in the spatial variables and second-order in time are presented, as well as schemes of fourth-order accuracy in all variables. Based on the stability theory of three-level difference schemes, stability conditions for the proposed algorithms are obtained. Using a special technique for solving the difference schemes, a priori estimates are derived, and based on them, theorems on convergence and accuracy are proven in the class of smooth solutions to the differential problem. An implementation algorithm is proposed for the difference scheme constructed using the finite element method. Test examples for one-dimensional and two-dimensional equations are also provided, demonstrating the higher-order accuracy of the proposed schemes.