Extended particle difference method for weak and strong discontinuity problems: part I. Derivation of the extended particle derivative approximation for the representation of weak and strong discontinuities

In this paper, the extended particle derivative approximation (EPDA) scheme is developed to solve weak and strong discontinuity problems. In this approximation scheme, the Taylor polynomial is extended with enrichment functions, i.e. the step function, the wedge function, and the scissors function, based on the moving least squares procedure in terms of nodal discretization. Throughout numerical examples, we demonstrate that the EPDA scheme reproduces weak and strong discontinuities in a singular solution quite well, and effectively copes with the difficulties in computing the derivatives of the singular solution. The governing partial differential equations, including the interface conditions, are directly discretized in terms of the EPDA scheme, and the total system of equations is derived from the formulation of difference equations which is constructed at the nodes and points representing the problem domain and the interfacial boundary, respectively.