High-order half-step compact numerical approximation for fourth-order parabolic PDEs

The aim of this study is to develop compact difference method to approximate parabolic PDEs of fourth order equipped with Dirichlet and Neumann boundary conditions involving half-step points. The proposed method converges quaternary and quadratically in space and time, respectively. The imbedding technique has been applied to approximate derivative terms of lower order by means of the governing differential equation to deduce the high-order method. The primary utility of this new discretization is that it can be straightaway applied to problems with singularities without necessitating fictitious nodes or special approach which has consequently lowered computational complicacy. We have examined linear stability of the proposed three-level implicit difference scheme using matrix stability analysis. In addition, we also obtained the solution of the first-order spatial derivative which is of significance in several physical problems. The efficacy of the proposed approximation is confirmed through numerical tests performed on a collection of physically relevant problems comprising the Euler Bernoulli beam equation and the highly nonlinear good Boussinesq equation. Numerical experiments evidently exhibit that the method provides more accurate results in contrast with the existing numerical techniques. The present method is able to simulate well the complex and intriguing long time dynamics of the good Boussinesq equation.