Pseudospectral meshless radial point interpolation for generalized biharmonic equation in the presence of Cahn–Hilliard conditions

Abstract

In this study, we develop an approximate formulation for a generalization form of biharmonic problem based on pseudospectral meshless radial point interpolation (PSMRPI). The boundary conditions are considered as Cahn-Hilliard type boundary conditions with application to spinodal decomposition. Since the rigorous steps to analyze such a problem is of high-order derivatives, implementing multiple boundary conditions and especially when the geometry of domain of the problem is complex. In PSMRPI method, the nodal points do not need to be regularly distributed and can even be quite arbitrary. It is easy to have high-order derivatives of unknowns in terms of the values at nodal points by constructing operational matrices. Furthermore, it is observed that the multiple boundary conditions can be imposed by an erudite application of PSMRPI on nodal points near the boundaries of the domain. The main results of generalized biharmonic problem are demonstrated by some examples to show validity and trustworthy of PSMRPI technique.

Hybrid radial basis function methods of lines for the numerical solution of viscous Burgers’ equation

“…RBF method in strong form was formulated by Kansa (1990) using collocation approach for numerical solution of elliptic, parabolic and hyperbolic PDEs. Further enrichment to RBF collocation method has been made by various researchers in different problems of engineering and applied sciences [see (Cavoretto 2015;Golbabai et al 2015;Shivanian and Abbasbandy 2020;Jiwari et al 2020;) and (Shu et al 2003;Cecil et al 2004;Shu et al 2005;Chan et al 2014)]. For a detailed overview and some other recent advances, the reader is referred to the texts (Fasshauer and McCourt 2015;Chen et al 2014;Biancolini 2017;Sarra and Kansa 2009).…”

Numerical simulation and analysis of the Swift–Hohenberg equation by the stabilized Lagrange multiplier approach

“…In this work, we focus on developing an efficient linear numerical scheme with secondorder temporal accuracy and energy stability for the SH equation. In the past few years, a Lagrange multiplier-type auxiliary variable approach has been proved to be an effective way to construct linear high-order schemes (Guillén-González and Tierra 2013) for the phasefield models with fourth-order polynomial potential, such as the Allen-Cahn (AC) equation (Church et al 2019), the Cahn-Hilliard (CH) equation (Dong et al 2020;Li et al 2019;Shivanian and Abbasbandy 2020), etc. Its basic idea is to define a new variable to replace some parts in nonlinear terms, then an extra evolution equation of the new auxiliary variable is solved.…”