Compact difference scheme with high accuracy for one dimensional unsteady quasi-linear biharmonic problem of second kind: Application to physical problems

The present paper uses a new two level implicit difference formula for the numerical study of one dimensional unsteady biharmonic equation with appropriate initial and boundary conditions. The proposed difference scheme is second order accurate in time and third order accurate in space on a non-uniform grid and in case of a uniform mesh, it is of order two in time and four in space. The approximate solutions are computed without using any transformation and linearization. The simplicity of the proposed scheme lies in its three point spatial discretization which yields a block tri-diagonal matrix structure without the use of any fictitious nodes for handling the boundary conditions. The proposed scheme is directly applicable to singular problems, which is the main utility of our work. The method is shown to be unconditionally stable for a model linear problem for a uniform mesh. The efficacy of the proposed approach has been tested on several physical problems, including complex fourth-order nonlinear equations like the Kuramoto–Sivashinsky equation and the extended Fisher–Kolmogorov equation, where comparison is made with some earlier work. It is clear from the numerical experiments that the obtained results are not only in good agreement with the exact solutions but also competitive with the solutions derived in earlier research studies.