Highly accurate compact difference scheme for fourth order parabolic equation with Dirichlet and Neumann boundary conditions: Application to good Boussinesq equation

Abstract

In this work, a three-level implicit compact difference scheme for the generalised form of fourth order parabolic partial differential equation is developed. The discretization is derived by approximating the lower order derivative terms using the governing differential equation with the imbedding technique and is fourth order accurate in space and second order accurate in time. The current approach is advantageous since the boundary conditions are completely satisfied and no further approximations are required to be carried out at the boundaries. The ability of the proposed scheme in handling linear singular problems is examined. The value of first order space derivative is computed alongwith the solution so it does not have to be estimated using the calculated value of the solution. The method successfully works for the highly nonlinear good Boussinesq equation for which more accurate solutions are obtained for the single and the double-soliton solutions in comparison with the existing numerical methods.