The truncation and stabilization error in multiphase moving particle semi-implicit method based on corrective matrix: Which is dominant?

The Lagrangian nature of the moving particle semi-implicit (MPS) method brings two challenges: disordered particle distribution and particle clumping. The former can cause large random discretization error for the original MPS models while corrective matrix can effectively reduce such large error to the high-order truncation error. The latter can trigger instability easily and thus some adjustment strategies for stability are indispensable, thereby causing non-negligible stabilization error. The purpose of this paper is to compare the relative magnitude of the truncation and stabilization error, which is of great significance for future improvements. An indirect approach is developed because of the difficulty of separating different error from total error in dynamic simulations. The basic idea is to check whether the total error decreases significantly after the truncation error is further reduced. First, a second order corrective matrix (SCM) is proposed for MPS to reduce the truncation error further, as demonstrated by theoretical error analysis. Second, error analysis reveals that the first order gradient model produces less numerical diffusion than the second order gradient model in interpolation after particle shifting. Then, several numerical examples, including Taylor-Green vortex, elliptical drop deformation, excited pressure oscillation flow and continuous oil spill flow, are simulated to test the variance of total error after SCM is applied. It is found that the SCM schemes basically did not remarkably decrease the total error for incompressible free surface flow, implying that truncation error is not dominant compared to the stabilization error. Therefore, reducing the stabilization error is of more significance in future.