On the partition noise in chosen particle weighted methods and its consequences for weakly-compressible flow models

Abstract

The Meshless Finite Volume (MFV) method, which is a member of the particle weighted methods (PWM) family, is analysed in the context of the so-called partition noise problem. This issue is inherent to PWM-type discretisation which is based on compact support kernel functions and particle sampling. The partition noise exhibits itself as spurious oscillations observed in the solution; they are caused by unphysical source terms in the discretised equations and approximate local volumes computation. Unfortunately, decreasing the discrete length scale which controls the spatial resolution reduces the partition noise only weakly (and never completely). When choosing numerical schemes, a special care is required since in naive implementations the partition noise may even lead to loss of stability. However, even when stability is assured, PWM does not fulfil the assumptions of the Lax equivalence theorem (i.e. the discretised equations are not consistent), therefore convergence in the classical meaning cannot be guaranteed.As an example of a problematic use of MFV, we investigate its application to the solution of fluid flow equations at low Mach numbers, aiming to mimic truly incompressible flow conditions. Taken here as isothermal Euler equations, they are the basis of the so-called weakly-compressible (WC) flow models. Such modelling does not involve any elliptic-type equation and, due to the algorithmic and computational reasons, seems to be well-suited for particle methods. The WC flow model imposes, however, a serious challenge for PWM since: (i) the method is not gauge invariant and the disparity between the speed of sound and the convective velocity scale causes a significant spurious source of momentum; (ii) the inaccuracy in the local volumes approximation leads to errors in the computed density which are comparable to physically relevant slight variations, inherent to the WC approach. We propose specific numerical and modelling measures: controlled relaxation of initial particle distribution, particle volumes computation using an optimal kernel function, a choice of gradient limiting techniques and a carefully chosen background pressure. Although these improvements significantly increase the accuracy (however at remarkably increased computational cost), the theoretical 2nd-order convergence is achieved only at lower spatial resolutions when the truncation errors dominate. The accuracy of mesh-based methods is achieved in PWM only for the pressure field; the errors in the velocity field are much larger and convergence is at most 1st-order. We point some contradictory requirements of the model equations and the numerical method. One of the conclusions is that it is doubtful to construct a convergent truly Lagrangean method of PWM type at reasonable computational cost. The present findings also apply to conservative variants of the weakly-compressible Smoothed Particle Hydrodynamics which, being first order consistent in the best case, suffers from the partition noise even more severely than the considered MFV approach which is theoretically second order consistent. The conclusions pose a question about a rationale for the use of WC-PWM approach in chosen applications, in particular free surface and interfacial flows.