Convergence-improved source term of pressure Poisson equation for moving particle semi-implicit

Abstract

The moving particle semi-implicit (MPS), one of the projection-based particle methods, suffers from a poor performance in the convergence with a time step size, although this is essential to ensure the reliability and robustness of numerical simulations. Under the formulation of a kernel-based representation, the finite-difference of the pressure Poisson equation (PPE) yields resolution ratio (particle size/time step size ℓ0/Δt) dependency in the source term. As a result, the solution diverges as Δt→0. This study proposes two new concepts that remove the dependency in the source term and result in a good performance in the convergence with numerical resolutions. The first proposal is the new source term derived based on the second-order derivative of fluid density (particle number density). This is named the second-order differential source term (SDS). The SDS separates the source term into predictor step and present time components, and the resolution ratio dependency is found only in the present time component. The second proposal is the resolution-free representation (RF) of the source term that replaces the resolution ratio with the Courant–Friedrichs–Lewy condition. Applying the RF to the SDS, the new source term RF-SDS is obtained. The proposed source term is verified and validated by four typical free surface flow problems (wedge slamming test, hydrostatic test, dam breaking test, and sloshing test). Performances of three different source terms are compared, and the proposed RF-SDS effectively improves the convergence performance with the time step size.