Interface Immersed Particle Difference Method for weak discontinuity in elliptic boundary value problems

Abstract

In this paper, the Interface Immersed Particle Difference Method (IIPDM) for weak discontinuity in elliptic problems is presented. Heat conduction and potential flow problems with smooth and non-smooth interfaces are considered. The previously developed Particle Difference Method (PDM) accurately solves this class of interfacial singularity problems, however, it requires additional difference equations on the interfacial points. In contrast, the IIPDM no longer requires the additional interfacial equations since the interface condition is already immersed in the particle derivative approximation through the moving least squares procedure. The method successfully captures both singularities and discontinuities in the derivative field due to the interface geometry regardless of its smoothness. In fact, enforcement of the interface condition is conducted both in an implicit manner and in an explicit manner enriching the polynomial basis as well as the local approximation. Under the constrained derivative approximation, discretization of the elliptic PDE in its strong form leads to a difference scheme or a point collocation scheme involving only the particles. Consequently, the strong formulation can avoid the increase in total system size and improves the computational efficiency. Numerical experiments show that the IIPDM can sharply capture discontinuities and singularities within a solution field. Furthermore, convergence studies of various elliptic interface problems demonstrate efficiencies captured from the IIPDM when comparing convergence rates against the PDM.