A review on approaches to solving Poisson’s equation in projectionbased meshless methods for modelling strongly nonlinear water waves
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Abstract
Three meshless methods, including incompressible smooth particle hydrodynamic (ISPH), moving particle semiimplicit (MPS) and meshless local Petrov–Galerkin method based on Rankine source solution (MLPG_R) methods, are often employed to model nonlinear or violent water waves and their interaction with marine structures. They are all based on the projection procedure, in which solving Poisson’s equation about pressure at each time step is a major task. There are three different approaches to solving Poisson’s equation, i.e. (1) discretizing Laplacian directly by approximating the secondorder derivatives, (2) transferring Poisson’s equation into a weak form containing only gradient of pressure and (3) transferring Poisson’s equation into a weak form that does not contain any derivatives of functions to be solved. The first approach is often adopted in ISPH and MPS, while the third one is implemented by the MLPG_R method. This paper attempts to review the most popular, though not all, approaches available in literature for solving the equation.
Keywords
Nonlinear water waves ISPH MPS MLPG_R Projection scheme Particle methods Meshless methods Poisson’s equation1 Introduction
Marine structures are widely used in ocean transportation, exploitation and exploration of offshore oil and gas, utilization of marine renewable energy and so on. All these are vulnerable to harsh weather and so to very violent waves. Under action of violent waves, they may suffer from serious damages. Therefore, it is crucial to be able to model the interaction between violent waves and structures for designing safe and costeffective marine structures. The available numerical models for strongly nonlinear interactions between water waves and marine structures are mainly based on solving either the fully nonlinear potential flow theory (FNPT) or the Navier–Stokes (NS) equations. For dealing with the problems associated with violent waves, the NS model should be employed.
The NS model may be solved by either meshbased methods or meshless methods. The former is usually based on the Eulerian formulation, but the latter on the Lagrangian formulation. In the meshless methods, the fluid particles are largely followed and so the methods are also referred to as particle methods. The meshbased methods have been developed for several decades and mainly based on finite volume and finite different methods (Greaves 2010; Causon et al. 2010; Chen et al. 2010; Zhu et al. 2013). The meshless (or particle) methods are of relative new development, but have been recognized as promising alternative methods in recent years, particularly for modelling violent waves and their interaction with structures owing to their advantages that meshes are not required and numerical diffusion associated with convection terms is eliminated in contrast to meshbased methods. Extensive review of all the methods would divert the focus of this paper. A brief overview for meshless methods is given below, as this paper is concerned only on topics related to them. For more information about meshbased methods, the readers are referred to other publications, such as Causon et al. (2010) and Zhu et al. (2013).
1.1 Overview of meshless methods
Many meshless methods have been developed and reported in literature, such as the moving particle semiimplicit method (MPS) (e.g. Koshizuka 1996; Gotoh and Sakai 2006; Khayyer and Gotoh 2010), the smooth particle hydrodynamic method (SPH) (e.g. Monaghan 1994; Shao et al. 2006; Khayyer et al. 2008; Lind et al. 2012), the finite point method (e.g. Onate et al. 1996), the element free Galerkin method (e.g. Belytschko et al. 1994), the diffusion element method (Nayroles et al. 1992), the method of fundamental solution (e.g., Wu et al. 2006), the meshless local Petrov–Galerkin method based on Rankine source solution (MLPG_R method) (e.g. Ma 2008) and so on. Among them, the MPS, SPH and MLPG_R methods have been used to simulate violent wave problems.
When the meshless methods are applied to model strongly nonlinear or violent waves, two formulations are employed. One is based on the assumption that the fluid can be weakly compressed, while the other just assumes that the fluid is incompressible. The first one is mainly adopted for SPH, e.g. Monaghan (1994), Dalrymple and Rogers (2006), Gomezgesteira et al. (2010) and so on. More references can be found in Violeau and Rogers (2016). The second formulation has been implemented in SPH, MPS and MLPG_R methods. The SPH based on incompressible assumption is called as incompressible smooth particle hydrodynamic (ISPH). Most of the publications that employ the three meshless methods for modelling incompressible flow are based on the projection scheme developed by Chorin (1968). One of the main tasks associated with the projectionbased meshless methods is to find the pressure through solving Poisson’s equation. Various SPH methods have been reviewed very recently by Violeau and Rogers (2016). All the aspects of ISPH and MPS have also been discussed by Gotoh and Khayyer (2016). This paper tries to only review the approaches of solving Poisson’s equation in the meshless methods for incompressible flow.
1.2 Mathematical formulation of projectionbased meshless methods
 (1)Calculate the intermediate velocity (\(\vec {u}^*)\) and position (\(\vec {r}^*)\) of particles using$$\begin{aligned} \vec {u}^*=\vec {u}^n+\vec {g}\Delta t+\upsilon \nabla ^2\vec {u}^n\Delta t, \end{aligned}$$(5)where \(\vec {r}\) is the position vector of particles and the superscript n represents the nth time step; \(\Delta t\) is the increment of the time step.$$\begin{aligned} \vec {r}^*=\vec {r}^n+\vec {u}^*\Delta t, \end{aligned}$$(6)
 (2)Evaluate the pressure \(p^{n+1}\) usingwhere \(\Lambda \) is a coefficient taking a value between 0 and 1. \(\rho ^{n+1}\) and \(\rho ^*\) are the fluid densities at \((n+1)\) time step and intermediate fluid density, respectively.$$\begin{aligned} {\nabla ^2}{p^{n + 1}} = \Lambda \frac{{{\rho ^{n + 1}}  \rho ^ *}}{{\Delta {t^2}}} + (1  \Lambda )\frac{\rho }{{\Delta t}}\nabla .\mathop u\limits ^{ \rightarrow *}, \end{aligned}$$(7)
 (3)Calculate the fluid velocity and update the position of the particles using$$\begin{aligned} \vec {u}^{**}=\frac{\Delta t}{\rho }\nabla p^{n+1}, \end{aligned}$$(8a)$$\begin{aligned} \vec {u}^{n+1}=u^*+\vec {u}^{**}=\vec {u}^*\frac{\Delta t}{\rho }\nabla p^{n+1}, \end{aligned}$$(8b)where \(\beta \) is often taken as 0 (such as in Ma and Zhou 2009) or 0.5 (such as Cummins and Rudman 1999).$$\begin{aligned} \vec {r}^{n+1}=\vec {r}^n+( {1\beta })\vec {u}^{n+1}\Delta t+\beta \vec {u}^n\Delta t, \end{aligned}$$(9)
 (4)
Go to (1) for the next time step.
It is noted that multiple substeps in each time step in the above procedure may be applied as in Hu and Adams (2007). No matter how many substeps are used, the solution of Poisson’s equation is always concerned in the projectionbased meshless methods. Again, as our focus here is on the approaches for solving Poisson’s equation, readers who are interested in multiple substeps procedure may refer to relevant publications such as Hu and Adams (2007).
The right hand side of Eq. (7) is the source term of the Poisson’s equation combining the terms of density invariant and velocity divergence. The appropriate choice of the \(\Lambda \) value has been discussed for achieving relatively more ordered particle distribution and more smoothing pressure field by, e.g. Ma and Zhou (2009), Gui et al. (2014, (2015). In addition, attempts are also made by improving the source term. Khayyer et al. (2009) replaced the source term by a higherorder source term, while Kondo and Koshizuka (2011), Khayyer and Gotoh (2011), Khayyer and Gotoh (2013), Gotoh and Khayyer (2016) and Gotoh et al. (2014) introduced an errorcompensating term (including a highorder main term and two errormitigating terms multiplied by dynamic coefficients). The higherorder source and the errorcompensating terms help to enhance the pressure field calculation, volume conservation and uniform particle distributions throughout the simulation that minimizes the perturbations in particle motions. As the work related to improving the formulation and evaluation of the source term has been well covered by the cited papers, further details will not be given in this paper. This review hereafter focuses on the ways to deal with the Laplacian on the left hand side of Eq. (7).
Another issue in solving Eq. (7) is related to the boundary conditions satisfied by pressure on the free surface, rigid (fixed or moving) wall and arbitrary boundaries (also called in/outlets) introduced for computation purpose. To numerically implement the boundary condition on the rigid wall, several approaches have been suggested, including, for example, addition of dummy particles (e.g. Lo and Shao 2002; Gotoh and Sakai 2006) and unified semianalytical wall boundary condition (e.g. Leroy et al. 2014). To numerically implement the boundary condition on the free surface, the key issue is how to identify the particles on it. There are several approaches for doing so, such as detecting if the density (particle number density) is smaller than a specified value (e.g. Lo and Shao 2002; Gotoh and Sakai 2006), mixed particle number density and auxiliary function method (Ma and Zhou 2009) and an auxiliary condition proposed by Khayyer et al. (2009). For implementing in/outlet conditions and other more details about the treatment of boundary conditions, the readers are referred to the recent review papers by Violeau and Rogers (2016) and Gotoh and Khayyer (2016).
2 Approaches of ISPH in solving Poisson’s equation
As far as we know, most publications based on the ISPH method adopt an approach that is to discretize Poisson’s equation directly. In such an approach, discretization of Laplacian is a key. Various different formulations of Laplacian discretization for the ISPH method are discussed in this section.
Use of the ISPH method appears to start in Cummins and Rudman (1999), which just gave the results for 2D problems irrelevant to water waves. In that paper, they employed the following approach to approximate the Laplacian in Eq. (7), i.e.
The above formulation was followed by Lo and Shao (2002), which considered the water waves propagating near shore. In their work, the Laplacian in Eq. (7) was approximated by
Hu and Adams (2007) suggested the following approximation by considering particleaveraged spatial derivative:
Hu and Adams (2009) suggested another approximation with double summations:
Hosseini and Feng (2011) used the following approximation, which was derived to ensure that the gradient of a linear function is accurately evaluated as proposed by Oger et al. (2007).
Gotoh et al. (2014) derived the following expression using the divergence of the pressure gradient,
3 Approaches of MPS in solving Poisson’s equation
Movingparticle semiimplicit (MPS) method was proposed by Koshizuka et al. (1995) and Koshizuka (1996). In this method, Poisson’s equation (Eq. 7) is solved also by directly approximating the Laplacian. In the cited papers, the Laplacian was approximated by
The expression of C in Eq. (23b) is written only for 2D problems, though it can be straightforwardly extended to 3D problems. Theoretically, LPMPS03 should be reduced to LPMPS02 if C is a unit matrix, but actually it is not. The reason is perhaps attributed to the approximation adopted when deriving the LPMPS03. Interested readers can find more details about this from the cited papers. Very recently, Tamai et al. (2016) proposed another formation given by
Apart from these, Tamai and Koshizuka (2014) proposed a scheme based on a least square method, but Tamai et al. (2016) pointed out that this scheme needed inversion of a larger size matrix (an order of 5 for 2D cases and 9 for 3D cases) and also a larger support domain (or smoothing length) to keep the matrix invertible. More discussions can be found in the cited papers.
4 Patch tests on different discrete Laplacians
To investigate the behaviours of different forms of Laplacian discretization, a few papers carried out patch tests. In some patch tests, a Laplacian discretization is applied to estimate the value of Laplacian for a specified function, which is defined on a specified domain, giving the exact evaluation of the error. This section will summarize these tests available in published papers.
For irregular or disorderly particle distributions, the behaviour of discrete Laplacians also depend on how to choose the smoothing length. Schwaiger (2008) studied two options, one is h = 1.2S and the other is h \(=\) \(0.268\sqrt{S} \). Based on the relative errors in the region 2.25 \(< x<\) 2.75 and 2.25 \(< y<\) 2.75 without accounting for the particles near the boundaries, they found that for \(h =\) 1.2S, both LPSPH04 and CSPM did not show a fully convergent behaviour, but remained at a fairly constant relative error,reducing the average particle distance. They also found that LPSPH03 became divergent, i.e. the relative error increasing with reducing the particle distance. CSPM should give converged results even for irregular particle distribution. The reason it did not do so is perhaps because there were no sufficient number of particles within the region of size \(h = 1.2S,\) due to irregular shifting of particle positions, yielding that the property of matrixes involved in the CSPM became worse, and so leading to nonconvergent results. For \(h = 0.268\sqrt{S} \), they showed in Fig. 5 of their paper that all approximations exhibited convergent behaviour and that LPSPH04 and CSPM results converged much faster, with the rate being near the second order, while LPSPH03 results converged much slower with its convergent rate being less than first order. The reason for LPSPH04 and CSPM results to be in secondorder convergent rate in this case is perhaps because \(0.268\sqrt{S} \) is much larger than 1.2S, and so there were always sufficient number of particles involved in the cases studied. Schwaiger (2008) mentioned that the smoothing length h was often set proportional to S, but the divergence behaviour corresponding to the case is perhaps troubling.
Lind et al. (2012) carried out similar investigations by comparing LPSPH03 with LPSPH04 for the functions of \(x^m+y^m\) with \(m =\) 1, 2 and 3 defined on the same domain as that by Schwaiger (2008). They just confirmed that the relative error of LPSPH04 could reach 70 %, while LPSPH03 yielded an error of 4000 % on the boundary. At the row next to the boundary, the relative error of LPSPH04 reduced to 4 %, while that of LPSPH03 remained to be 500 %.
Lind et al. (2012) also carried out investigations by solving the equation of \(( {\nabla ^2p})_i =1\) for 1D problem with the boundary conditions of dp/dy \(=\) 10 at y \(=\) 0 and \(p =\) 1 at \(y =\) 1 using LPSPH04. For this purpose, they employed both uniform and nonuniform particle configurations. The latter was produced by specifying different small random perturbations of \((\pm 0.1 \sim \pm 0.5)S\) to the particle distance for uniform distribution. They particularly indicated that the relative error of the solution became larger with increased random perturbation: 1.5 % corresponding to \((\pm 0.1)S,\) but 17 % to \((\pm 0.5)S\). They also demonstrated that the LPSPH04 may lead to results with a convergent rate of 1.2–1.3 (less than 2 as shown for \(h=0.268\sqrt{S} \) by Schwaiger 2008) with the particle shifting scheme to maintain the particle orderliness.
In their tests, \(S=0.1,0.05,0.02,0.0125,0.01,0.08\) and \(k=0,0.2,0.4,0.8,1.0,1.2\) were considered. Some of their results are reproduced in Figs. 1, 2, and 3. Figure 1a presents the average relative errors for different values of S with a value of k being fixed to be 0.8, i.e. with the random shift up to ±0.4S, the same as that in Schwaiger (2008). From the figure, one can see that the average error of LPSPH04 is consistently reduced with reduction of S. This trend is similar to the results of Schwaiger (2008) for a function of \(x^m+y^m\) obtained using \(h=0.268\sqrt{S} ,\) but different from those of Schwaiger (2008) obtained using \(h=1.2S\) which is shown to be constant with the reduction of S in their papers. The reason is perhaps because the smooth length used in Zheng et al. (2014) was larger, though it was still proportional to S. The average errors of LPSPH03 can increase with the reduction of S, which is a divergent behaviour. Figure 1b demonstrates that the maximum error of LPSPH04 consistently decreases until \(S=0.0125\) or Log\((S)\approx 1.9,\) but increase with increasing the resolution of the particles after that. In addition, the smallest value of the error is Log\((\mathrm{Er}_\mathrm{max})>0.6\), corresponding to \(\mathrm{Er}_\mathrm{max} =25~{\% }\), which is considerably larger than the average errors for the same case (Fig. 1a) and may be considered to be significant as the error occurs inside the domain. Again, the maximum error of LPSPH03 shows a divergent behaviour when \(S<0.05\) (Log\((S)<1.3)\). It is noted that overall, the accuracy of numerical methods are controlled by the maximum error, and not the average error.
To demonstrate if there is a significant number of particles with a large error, Zheng et al. (2014) plotted a figure similar to Fig. 3. In this figure, the horizontal axis shows the different ranges of relative error, e.g. [20, 30 %], while the vertical axis shows the number of particles whose error lies in a range. For example, in the range of [20, 30 %], there are about 230 particles for the LPSPH04. The relative error at each individual particle used in this figure is estimated by \(\mathrm{Er}_i =\left {\frac{\nabla ^2f_{i,c} \nabla ^2f_{i,a} }{\nabla ^2f_{i,a,m} }} \right \quad (i=1,2,3\ldots N)\). This figure demonstrates that a quite large relative error (>20 %) can happen at a considerable number of particles for the approximations even when they are applied to computing the Laplacian of the quite simple function, though the number for the LPSPH04 is much smaller than for the LPSPH03.
In most of the above tests (except for some cases in Schwaiger 2008), the value of S / h is fixed with the smoothing length varying and sometimes with different randomness. Quinlan et al. (2006) discussed the theoretical convergence of approximating the gradient of a function used in SPH. They showed that the error caused by numerical approximations to the gradient did not only depend on the smoothing length and randomness (nonuniformity), but also on the ratio S / h. Specifically speaking, the error increases with the larger randomness and can be proportional to 1 / h if S / h is not small enough, which is consistent with that observed in the above results. When S / h is small enough, the convergent behaviour of the approximation to the gradient can be improved. Graham and Hughes (2007) particularly investigated the behaviour of LPSPH03 with \(\eta =0\) by varying the value of S / h. They studied the pressuredriven flow between parallel plates with a constant pressure gradient with the diffusion term estimated by LPSPH03 (\(\eta \) = 0) for three values (1, 1/1.25 and 1/1.5) of S / h. They showed that the method was not convergent in several cases they studied and that random particle configurations could have a dramatic effect on the accuracy of the SPH approximations. More specially, the results are divergent if their random factor is larger than 0.25, and their best results are these obtained by using \(S/h = \) 1/1.5 with the particles fixed, among which the error reduces at a rate less than first order when their random factor is relatively small. Fatehi and Manzari (2011) also carried out tests by varying S / h from 1/1.5 to 1/3.5 on a scheme similar to LPSPH03 with \(\eta =0\) and their new scheme which is similar to CSPM (discussed above) by solving a thermal diffusivity problem defined on a unit square \(0\le x\le 1\) and \(0\le y\le 1\), which has a similar equation to the problem with the zero pressure gradient considered by Graham and Hughes (2007) . In their tests, regular and irregular particle distributions were considered, and the relative errors of numerical results to the analytical ones at a time near steady state were presented in their paper. The random perturbation they employed was \(\left \epsilon \right \le 0.05S\) or \(\left \epsilon \right \le 0.1S\), much less than \(\left \epsilon \right \le 0.4S\) used by Schwaiger (2008). Their results showed that the scheme LPSPH03 with \(\eta =0\) had a convergent rate of first order at the best, and that their new scheme similar to CSPM had a convergent rate of second order. However, they indicated that the scheme did not work when smoothing length was 1.5S, consistent with the analysis of Quinlan et al. (2006). This is perhaps because the number of neighbouring particle is not sufficient, which may make the matrixes involved in CSPM invertible. It is not sure if the convergent rate would maintain when the random perturbation is larger.
Gotoh et al. (2014) presented some convergent test results on LPSPH08. For this purpose, the approximation was used together with their errorcompensation term to simulate a pressure field caused by a modified gravitational acceleration. Their results showed that for irregular particle distributions (the initial distribution randomly altered and half of the fluid particles displaced by ±0.02S), the normalized root mean square error reduced with decrease of the initial particle distance, a convergent behaviour. According to the cited paper, the errors are 0.108, 0.068 and 0.065 corresponding to S \(=\) 0.004, 0.003 and 0.002, respectively, which gives an average convergent rate at about 0.7, though it is about 1.6 from 0.004 to 0.003.
Ikari et al. (2015) tested the discrete Laplacians (LPMPS02 and LPMPS03) for the MPS method. Their results are summarized here. The first case they presented was about the computation of a pressure field due to a sinusoidal disturbance to gravitational acceleration. The particles were randomly shifted by \(\pm 0.05S\) on the basis of uniform distribution. As they indicated, the results of LPMPS03 were better than those of LPMPS02. They also showed that there were some spurious fluctuations in the pressure time histories from LPMPS02 on reducing the particle distance. Their second case was similar to their first case except for a difference that the sinusoidal disturbance was multiplied by an exponential growing factor. The results for this case also showed the outperformance of LPMPS03 compared to LPMPS02. They pointed out that the clear convergence of results from LPMPS03 was not observed in terms of root mean square error of numerical results relative to the analytical solution for the case. The third case they investigated was about a 2D diffusion problem on a square domain. For this case, the performances of both LPMPS02 and LPMPS03 were satisfactory, though LPMPS03 was slightly better. The convergent trend was not, however, exhibited. For example, the root mean square error of LPMPS03 is 19.2850, 30.8089 and 24.2292 corresponding to the mean particle distance of 10, 5 and 2.5 mm. The convergent property of LPMPS02 with a higherorder source term on the right hand side of Eq. (7) was also examined by Khayyer and Gotoh (2012), showing an improved and more stabilized (without fast fluctuation) pressure for the similar case (but in 3D here) to that in Gotoh et al. (2014) discussed above. In this test, the initial distribution of particles is randomly altered and half of the fluid particles are displaced by \({\mp } 0.05 S\), similar to that in Ikari et al. (2015). The results demonstrated that the normalized root mean square error reduced with decrease of the initial particle distance. In other words, convergent behaviour was observed. The specific information is that the errors are 0.241, 0.228 and 0.192 corresponding to S \(=\) 0.012, 0.010 and 0.008, respectively. The average convergent rate is near 0.8.
Summary of the main features of different discrete Laplacians
Scheme  Type  Number of matrix inversion  Typical behaviours observed in tests described above 

LPSPH01  1  NO  For incompressible flow, similar to LPSPH03 
LPSPH02  1  NO  For incompressible flow, similar to LPSPH03 
LPSPH03  1  NO  Schwaiger (2008), Zheng et al. (2014) and Tamai et al. (2016) showed it is divergent for severe randomness of particle distribution if smoothing length is proportional to the particle distance; Schwaiger (2008) and Lind et al. (2012) showed a very large error near the boundaries; Schwaiger (2008) showed it converged at a rate less than first order when using a smooth length proportional to the square root of the particle distance; Fatehi and Manzari (2011) and Graham and Hughes (2007) also showed it converged at a rate less than first order when the randomness of the particle distribution is not very severe 
LPSPH04  3  2, their sizes are 2 \(\times \) 2 for 2D cases and 3 \(\times \) 3 for 3D cases  Error near the boundary can be large if the random level of particle distribution is high; secondorder convergent rate is observed by Schwaiger (2008) but can be much less than second order shown by Lind et al. (2012), and depending on the randomness of particle distribution (Zheng et al. 2014). The error inside the domain can also be large when the level of the randomness of particle distribution is very high (Zheng et al. 2014) 
LPSPH05  1  NO  For incompressible flow, similar to LPSPH03 
LPSPH06  1  NO but double summation  Not popular; patch tests not available 
LPSPH07  2  1, its size is 2 \(\times \) 2 for 2D cases and 3 \(\times \) 3 for 3D cases  Patch tests not available 
LPSPH08  1  NO  Convergent at a rate less than first order for a lower level of irregularity of particle distribution shown by Gotoh et al. (2014) 
CSPM  3  2, one with a larger size of 3 \(\times \) 3 for 2D cases and 6 \(\times \) 6 for 3D cases  Error near boundary can be large if random level of particle distribution is high; computational cost is high, though secondorder convergent rate is observed in Schwaiger (2008) 
LPMPS01  1  NO  Not convergent for a higher level of irregularity of particle distribution shown by Tamai et al. (2016) 
LPMPS02  1  NO  Convergent at a rate less than first order for a lower level of irregularity of particle distribution shown by Khayyer and Gotoh (2012); not convergent for a higher level of irregularity of particle distribution shown by Tamai et al. (2016) 
LPMPS03  2  1, its size is 2 \(\times \) 2 for 2D cases and 3 \(\times \) 3 for 3D cases  Clear convergent behaviour is not observed for random particle distribution, though its results are better than LPMPS02 in Ikari et al. (2015) 
LPMPS04  3  2, one with a larger size of 3 \(\times \) 3 for 2D cases and 6 \(\times \) 6 for 3D cases  Computational cost is high; though secondorder convergent rate is observed shown by Tamai et al. (2016). As it is just suggested recently, its behaviours need to be confirmed by more applications 

The discretization of Laplacians can be a notable issue at particles near a boundary, especially for disordered particle distributions, such as water surface, without additional appropriate treatment. This may not be a big issue in some applications where the solution near the boundary is not mainly concerned, but would be a critical issue for modelling water waves and their interaction with structures in marine or coastal engineering, in which the accuracy of pressure near the water and body surface is important. The corrected Laplacian operator in integral formulation (SoutoIglesias 2013) improves the Laplacian evaluation near the boundary and gives convergent solutions of Poisson’s equation with boundary conditions which is also applied to evaluate the curvature in Khayyer et al. (2014). This may suggest that the discretization schemes of Laplacians discussed above should be employed together with the correction to improve their behaviour near boundaries.

The error of discrete Laplacians can become larger when the degree of particle disorderliness is higher or results converge slower even inside computational domains. In the cases for violent water waves, the particle distribution always becomes highly disordered even though they are uniformly and regularly located initially. More effort may be required to make them less sensitive to particle disorderliness.

It is observed that Type 1 Laplacian approximations may not converge for a high degree of particle distribution randomness (or disorderliness), but they may converge at a rate less than first order for a low degree of particle distribution randomness (or disorderliness). That means that the results obtained from approximations may become worse with reduction of particle distance or increase of the number of particles used when particle distribution randomness level is high. Type 2 may have similar problem, though it may be more accurate for the same number of particles.

Type 3 has a convergent rate of 2nd order if the random level of particle distribution is not very high, but the rate may become lower with the increase of the random level. The computation costs of the type are high compared with others. In addition, the number of neighbouring particles must always be high enough to ensure the matrixes to be invertible. This is not necessarily guaranteed when modelling violent water waves as the configuration of particles can dramatically and dynamically vary during simulation, which is not a priori predictable.
These results demonstrate that although a lot of effort has been undertaken to develop better approximations to the Laplacian discretization, it still needs to be improved, in particular for modelling violent water waves where the water particles can become severely disordered. To improve the accuracy of computation, one may adjust the distribution of particles as done by Xu et al. (2009) to reduce the disorderliness of particles. The other way to circumvent the issues is by avoiding the direct discretization of Laplacian. This approach will be discussed in the next section.
5 Approaches of MLPG_R in solving Poisson’s equation
It is noted that the right hand side of MLPGR02 is a volumetric integration, i.e. the integration domain is a sphere in 3D and a circle in 2D cases, rather than a surface integration as in MLPGR01. Use of normal numerical integration techniques, like Gaussian quadrature, may need a significant amount of computational time for estimating the term. To overcome this problem, Ma (2005b) and Zhou et al. (2010) developed semianalytical techniques for 2D and 3D cases. With use of these techniques, the computational costs for evaluating the right hand term in MLPGR02 is similar to that for evaluating the right hand term in MLPGR01.
6 Patch tests of MLPGR02 and comparative studies
The errors estimated by \(\mathrm{Er}_p ={\sqrt{\sum \nolimits _{i=1}^{N_t } {\left {p_i p_{i,a} } \right ^2} } } / {\sqrt{\sum \nolimits _{i=1}^{N_t } {\left {p_{i,a} } \right ^2} } }\) for \(k=0.2\) and 0.3 are shown in Fig. 5, where \(p_i \) is the numerical results, while \(p_{i,a} \) is the analytical solution. From this figure, one can see that the convergent rate is close to the second order. Compared with the methods using Type 1 and Type 2 Laplacians which converge at a lower rate, the convergent properties of the MLPGR02 scheme is much better. Compared with these adopting the Type 3 Laplacian, the computational efficiency of the MLPGR02 scheme is higher as the inversion of a matrix with a size of 3x3 for 2D cases and 6x6 for 3D cases is required by the Type 3 Laplacian, but the inversion of such a higher order matrix is not needed in the MLPGR02 scheme.
The second case was considered by Zheng et al. (2014) who developed a hybrid method. In the hybrid method, pressure is solved using MLPGR02 and all others are the same as ISPH. They named the hybrid method as incompressible smoothed particle hydrodynamics based on Rankin source solution (ISPH_R) method. They applied the ISPH_R method together with the ISPH method based on LPSPH04 (named as CISPH) and with the traditional weakly compressible SPH (named as SPH) to simulate the sloshing waves in a tank (Fig. 6) that is subject to the motion \(X_s =a_0 \Omega /\sqrt{gd} (1\cos \Omega t)\). The error of the numerical solution against the analytical solution is evaluated by \(Er_\eta ={\sqrt{\sum \nolimits _{i=1}^{N_t } {\left {\eta _i \eta _{i,a} } \right ^2} } } / {\sqrt{\sum \nolimits _{i=1}^{N_t } {\left {\eta _{i,a} } \right ^2} } },\) where \(\eta _i \) is the numerical result at the time instant, \(N_t \) is the total time steps in the simulation duration of \(\tilde{t}=t\sqrt{L/g} =50.0\), and \(\eta _{i,a} \) is the analytical solution at \(i\mathrm{th}\) time step, given by Faltinsen (1976).
To explore the properties of the methods in another way, Fig. 8 depicts the CPU time spent by all the methods corresponding to different numerical errors on the same computer, which is also originally presented in Zheng et al. (2014). One can see from Fig. 8 that the ISPH_R method needs much less CPU time to achieve the same level of accuracy. For example, corresponding to \(\mathrm{Log}(\mathrm{Er}_\eta )\approx 3.545\), the CPU times spent by the ISPH_R and CISPH are \(\mathrm{Log}(\mathrm{CPU}\_t)\approx \) 2.97 and 4.02, corresponding to \(\mathrm{CPU}\_t\approx 925\) and 10, 531 (about 11 times of the former) seconds, respectively. It is noted that the CPU time for running a case may depend on the choice of solver and preconditioner for solving the system of linear algebraic equations resulting from the discretized equations. As far as we know, the results in Fig. 8 were obtained by a solver combining the GMRES with Gauss–Seidel method. At each time step, they firstly run the Gauss–Seidel procedure for a specified number of iterations and then run the GMRES if necessary. If different procedure would have been used, the CPU time would be different.
To further show the performance of different methods in cases involving violent waves, Fig. 9 presents the pressure distribution at two time instants for sloshing in a rectangular tank shown in Fig. 6, but with the parameters of \(L =\) 0.6 m, \(d = 0.12\) m \(= 0.2\) L, \(a_0 (\text{ moiton } \text{ amplitude })=0.05\) m and \(T_0 (\text{ moiton } \text{ peirod })=1.5\) s. Figure 10 depicts the pressure time histories recorded at a point on the left wall with a height of 0.1667L, resulting from three approaches (SPH, ISPH_R and CISPH defined above). The results are also compared with Kishev and Kashiwagi (2006).
Gotoh et al. (2014) presented some results for the same cases as in Figs. 9 and 10, which are produced using CISPHHS and CISPHHSHLECS. LPSPH02 together with a higher order source term was used in CISPHHS. LPSPH08 was employed in CISPHHSHLECS together with the errorcompensating source (ECS) term. Their original figures for pressure fields and pressure time histories at the same point as in Fig. 10 are duplicated in Figs. 11 and 12, respectively. As they indicated, the pressure trace by CISPHHS is characterized by frequent and, relatively, largeamplitude unphysical oscillations, while the results of CISPHHSHLECS are much smoother. If comparing the results in Figs. 10 and 12, one may find that the pressure time histories produced by ISPH_R and CISPHHSHLECS have a similar level of smoothness and agreement with the experimental data. This indicated that the ECS term is quite effective, because there are still visible unphysical oscillations if it is not applied as shown in Fig. 4 of Gotoh et al. (2014). It may be interesting to see more comparisons of different approaches to improve their performance further.
7 Conclusions
This paper has reviewed the approaches to solve Poisson’s equation for pressure involved in incompressible smoothed particle hydrodynamic (ISPH), moving particle semiimplicit (MPS) and meshless local Petrov–Galerkin method, based on Rankine source solution (MLPG_R) methods for simulating nonlinear or violent water waves assuming fluids are the incompressible. As summarized in Table 2, there are three different approaches, i.e. discretizing Laplacian directly (DLD) by approximating the secondorder derivatives, transferring Poisson’s equation into a weak form containing only gradient of pressure (WCG) and transferring Poisson’s equation into a weak form that does not contain any derivatives of functions to be solved (WCF). The first approach DLD has been employed by most publications related to ISPH and MPS, while the third approach WCF has been implemented by the MLPG_R method for modelling water waves.
Three approaches for solving Poisson’s equation
Approaches  Discretising Laplacian directly (DLD)  Weak form containing only gradient (WCG)  Weak form containing no derivatives of functions to be solved (WCF)  

Features  Requiring approximation to second derivatives of function to be solved  Requiring approximation to first derivatives of function to be solved  Requiring approximation to function to be solved, not derivatives  
Methods involved  ISPH  MPS  ISPH; MLPG_R  MLPG_R 
Discretization  Type 1:  Type 1:  
LPSPH01  LPMPS01  
LPSPH03  LPMPS02  
LPSPH0506  
LPSPH08  Bonet and Kulasegaram (2002)  MLPGR02  
Type 2:  Type 2:  MLPGR01  
LPSPH07  LPMPS03  
Type 3:  Type 3:  
LPSPH04,  LPMPS04  
CSPM  
Inversion of matrixes  Type 1: not need  
Type 2: inversion of 1 matrix with size of 2 \(\times \) 2 for 2D cases and 3 \(\times \) 3 for 3D cases  
Refer to texts in Sect. 5, not repeated here as this approach is not popular  inversion of 1 matrix with a size of \(2 \times 2\) for 2D cases and \(3 \times 3\) for 3D cases, respectively  
Type 3: LPSPH04: Inversion of two matrixes, both with a size of \(2 \times 2\) for 2D cases and \(3 \times 3\) for 3D cases;  
CSPM and LPMPS04: inversion of two matrixes with a size of the larger matrix being \(3 \times 3\) for 2D cases and \(6 \times 6\) for 3D cases;  
Patch test results  Type 1: divergent for a higher level of particle distribution randomness; and converges at a rate less than first order for a lower level of particle distribution randomness  
Type 3: LPSPH04: Secondorder convergent rate in some cases but the rate being significantly less than second order when applying it to solve simplified fluid problems  Not available  Secondorder convergent rate for disorderly distribution of particles in solving a simplified problem and also in solving sloshing waves with small amplitudes, but not so popular until now  
CSPM and LPMPS04: secondorder convergent rate for disorderly distribution of particles 
Type 1 discrete Laplacians in the DLD approach is relatively easier to implement and have been the most popular one so far, particularly in the community which employs the ISPH and MPS methods. Their performance can be improved by applying the errorcompensating term on the right hand side of Poisson’s equation, by reducing the randomness of particle distribution and by adding a correction term near the boundaries. More efforts may be made to improve their performance by enhancing the convergent rate.
The third approach (WCF) adopted by the MLPG_R method does not need to deal with any derivative and so has no issue related to discretizing derivatives when solving Poisson’s equation for pressure. Using relatively simple approximation (requiring only inversion of one matrix with a size of 2 \(\times \) 2 for 2D cases and 3 \(\times \) 3 for 3D cases) to the pressure, one can achieve the secondorder convergent rate in patch pests, as that achieved by using Type 3 discrete Laplacians in the first approach, and also in solving the sloshing waves with small amplitudes. In addition, limited tests in literature available demonstrate that the ISPH based on the third approach (WCF) requires less CPU time to achieve the results with the same accuracy compared to ISPH based on the first approach (DLD) for simulating sloshing waves. Nevertheless, the WCF approach is currently less popular than others, perhaps because it is relatively new as it has just started to be used since 2005.
More comparative studies are encouraged, in particular for applying all the schemes to the same cases. The studies should compare convergent rate, accuracy and computational efficiency of different approaches. Such studies help fully understand the behaviours of the different approaches and select the best for modelling water waves in general cases.
Notes
Acknowledgments
The authors acknowledge the support of UK EPSRC grants, EP/L01467X/1, EP/N008863/1 and EP/M022382/1. Figures 9 and 10 are reprinted from Journal of Computational Physics, 276, Zheng X., Ma Q. W., Duan, W. Y., Incompressible SPH method based on Rankine source solution for violent water wave simulation, 291–314, 2014, with permission from Elsevier. Figures 11 and 12 are reprinted from Applied Ocean Research, 46, Gotoh H., Khayyer A., Ikaria H., Arikawa T., Shimosak K., On enhancement of Incompressible SPH method for simulation of violent sloshing flows, 104–115, 2014, with permission from Elsevier.
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