# Convergence of a linearly transformed particle method for aggregation equations

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## Abstract

We study a linearly transformed particle method for the aggregation equation with smooth or singular interaction forces. For the smooth interaction forces, we provide convergence estimates in \(L^1\) and \(L^\infty \) norms depending on the regularity of the initial data. Moreover, we give convergence estimates in bounded Lipschitz distance for measure valued solutions. For singular interaction forces, we establish the convergence of the error between the approximated and exact flows up to the existence time of the solutions in \(L^1 \cap L^p\) norm.

## Mathematics Subject Classification

65M12 65M50 82C22 35Q70## 1 Introduction

*W*the interaction potential. Since the total mass is preserved, without loss of generality, we assume

*W*are given by

Equation (1.1) has attracted lots of attention in the recent years for three reasons: its gradient flow structure [2, 32, 33, 61, 73], the blow-up dynamics for fully attractive potentials [12, 14, 26, 31], and the rich variety of steady states and their bifurcations both at the discrete (1.2) and the continuous (1.1) level of descriptions [3, 4, 5, 11, 14, 22, 25, 27, 28, 49, 50, 67, 74, 75]. Furthermore, these systems are ubiquitous in mathematical modelling appearing in granular media models [10, 61], swarming models for animal collective behavior [30, 46, 59], equilibrium states for self-assembly and molecules [47, 54, 70, 76], and mean-field games in socioeconomics [17, 43] among others.

On the other hand, global in time unique weak measure solutions can be constructed for any probability measure as initial data under suitable smoothness assumptions on the interaction potential. In this work, whenever we refer to *smooth potentials*, we mean that the interaction potential satisfies \(\nabla W \in \mathcal {W}^{1,\infty }(\mathbb R^d)\). For smooth potentials, the approach introduced by Dobrushin for the Vlasov equation [45] using the bounded Lipschitz distance between probability measures, see [21, 24, 53] for further details, gives a well-posedness theory of weak measure solutions.

*singular potentials*we mean that the interaction potential is not smooth but satisfies

*q*, and in [12, 14] a local-in-time well-posedness theory for initial data in \((L^1\cap L^\infty ) (\mathbb R^d)\) was developed for singularities up to and including a Newtonian singularity at the origin, corresponding to \(\alpha = d-1\). In this work, we will use the setting introduced in [24]. The Newtonian case is very specific because of the relation between the divergence of the velocity field and the density becomes local.

Smooth particle methods are an extremely popular tool for the numerical simulation of a large variety of problems, mostly due to their conceptual simplicity and their automatic, mesh-free adaptation properties. They are usually referred to as Particle-In-Cell (PIC) methods in the plasma physics community where they are coupled with grid-based (Finite Difference or Finite Element) solvers for the electromagnetic field [16, 42, 48], Vortex-Blob methods for incompressible Navier–Stokes and Euler equations, see e.g. [37, 40, 62] and the references therein, and Smoothed Particle Hydrodynamics in astrophysics, see e.g. [52, 68]. More recently they have been adapted to the aggregation equation in [13] where the approximate densities are shown to converge with arbitrary orders but only in negative Sobolev norms.

This weak convergence relates to a general feature of particle methods, namely that the particle approximations to the transported density are less accurate than the approximated trajectories. On a theoretical level this is supported by the classical analysis of vortex-blob methods [7, 66] and simplified PIC schemes [39], and it is also consistent with the common observation that particle codes can provide a satisfactory description of the problem dynamics despite “noisy”, i.e. oscillating, density reconstructions.

To mitigate these oscillations, several approaches have been developed over the years. Extending the interaction radius of the smooth particles [68] (or the number of particles per cell in PIC methods [16]) is a legitimate choice that is also consistent with the standard error estimates [7, 39, 66], but leads to numerically intensive simulations. Another option is to resort to remapped or resampled particle methods where new weighted particles are periodically used to approximate the transported density [44, 60]. The resulting schemes are sometimes referred to as forward semi-Lagrangian [36, 41, 65] and their improved convergence properties can be explained by the fact that the frequent reinitializations prevent the particles to become too irregularly distributed over time. This also has a cost: reinitializing the particles can be computationally expensive, it may hamper the natural adaptivity of the particle distribution and it usually introduces numerical diffusion which may conflict with the low dissipative essence of the method.

An interesting tool is then offered by using deformed particle shapes. In these methods the particles are pushed on to discrete times according to an approximation of the exact flow as in the standard case, but each particle has its own shape which is transformed in the discrete evolution in order to better approach the local flow. Again several methods have been developed in the respective communities, and examples include transformed Gaussian shapes for plasma physics problems [6], Navier–Stokes equations [71] and astrophysics [1]. Some methods have been studied for more general transport problems, such as the spatially varying blob sizes based on appropriate mappings [8, 38, 56], the Finite Mass method [51] and the Linearly Transformed Particle (LTP) method [18]. By carefully choosing the transformation parameters as time evolves, these works obtain significant improvements in the accuracy, and mathematical proofs show that the strong convergence of the transported densities indeed holds without requiring periodic remappings or extended overlapping for the particles, see [18, 35]. In practice it has been observed that periodic remappings were still necessary to obtain satisfactory results for physically relevant problems, mostly because of the elongation of the deformed shapes, however these remappings can be done at a much lower rate than with the fixed-shape methods [18, 20].

In this article we propose and study an extension of the LTP method [18, 20] to aggregation equations. In this method each particle shape is transported by the linearized flow around its trajectory. To our knowledge the convergence of the LTP method has only been proved for a linear transport equation [18] and for a Vlasov-Poisson system [19] involving measure-preserving characteristic flows. The technical difficulties posed by the deformation of the flows in our present case have been overcome by detailed estimates of the Jacobian matrices and determinants. These estimates have allowed us to control the error on the densities via the errors of the flows to finally obtain the convergence results. Certain Sobolev regularity is needed on the initial data to obtain convergence of the LTP method in Lebesgue spaces for both smooth and singular potentials. However, a general result of convergence for weak measure solutions is obtained in an appropriate distance for measures.

We also remark that particles methods have been combined with remeshing and adaptive mesh refinement for transport and convection-diffusion equations, see [8, 9, 69] and the references therein, which also require global transforms or mapping functions related to the distortion of the flow.

Let us finally mention that other numerical methods have been proposed in the literature for the aggregation equation. In [23], the authors proposed a finite volume scheme which is shown to be energy preserving, i.e., it keeps the property that the energy functional is dissipated along the semidiscrete flow. Finite volume and finite difference schemes have been shown to be convergent to weak measure solutions of the aggregation equations for mildly singular potentials in [31, 58].

In this work, we extend the LTP method to the aggregation equation seen as one of the most important representatives of a class of nonlinear continuity equations with non divergence free velocity fields in any dimensions. We start by summarizing the basic ideas of the numerical LTP method in Sect. 2 together with the preliminaries and notations used in this work. Section 3 is devoted to give convergence results for smooth potentials in Lebesgue spaces. Depending on the regularity of the initial data, we will be able for smooth potentials to control errors in \(L^1\) and \(L^\infty \). For initial data just being a probability measure, we will show in Sect. 4 the convergence in bounded Lipschitz distance. In the case of singular potentials, we will control in Sect. 5 the error up to the existence time of the solution of (1.1) in \(L^1\) and \(L^p\) with *p* suitably chosen. We finally show in Sect. 6 the performance of this method in one dimension validating the numerical implementation with explicit solutions and making use of it to study certain not well-known qualitative features of the evolution of (1.1) with several smooth and singular potentials.

## 2 Preliminaries

### 2.1 Basic properties of the exact flow

*t*and Lipschitz continuous in

*x*. The solution of the characteristic system

*T*] for locally Lipschitz velocity fields \(u \in L^\infty (0,T;\mathcal {W}^{1,\infty }(\mathbb R^d))\) for some \(T>0\), with constant \(L_T:=\sup _{t\in [0,T]} \Vert u(t,\cdot )\Vert _{\mathcal {W}^{1,\infty }}\). These estimates will be used in Sect. 5, where the dependence on T of the Lipschitz constant will be omitted for the sake of simplicity.

### 2.2 Linearly transformed particles

## Proposition 1

*C*independent of \(\rho ^0\).

*q*is the conjugate exponent of

*p*and \(\langle w,v\rangle \) is the duality pair that coincides with the integral of the product

*wv*as soon as the latter is integrable.

## Proposition 2

*C*independent of

*h*.

## Proof

*k*. In particular, given \(v \in \mathcal {W}^{1,\infty }(\mathbb R^d)\) we have

### 2.3 Approximated Jacobian matrices and particle positions

*x*and

*y*, then the exact solution to the ODE (2.2) takes an exponential form. However, in the general case the matrix \(J^{t_n,t_{n+1}}(x)\) is

*not*equal to

## Proposition 3

*C*independent of \(n \le N-1\) and \(\Delta t\).

## Proof

*B*we readily find \((a) \le \sum _{m=2}^\infty \frac{1}{m !}(C\Delta t)^m \le C(\Delta t)^2 \). Turning to (

*b*), we use again (2.3) to write

## Remark 1

### 2.4 General strategy of the convergence proofs

*k*-th particle is pushed forward by the approximated flow \(F_{h,k}^n\) during the time interval \([t_n,t_{n+1}]\), we need to control the local error between this approximation and the exact flow \(F^{t_n,t_{n+1}}\). To this end we define a first error term on the support of the smooth particles,

## 3 \(L^1\) and \(L^\infty \) convergence for smooth potentials

*u*is bounded by \(\Vert \nabla W\Vert _{\mathcal {W}^{1,\infty }}\): indeed letting \(|\cdot |\) denote the Euclidean norm in \(\mathbb R^d\) as well as its associated matrix norm, we have for all \(x \in \mathbb R^d\), \(t \in [0,T]\),

*u*, so that estimates (2.5)–(2.9) hold with \(L = C \Vert \nabla W\Vert _{\mathcal {W}^{1,\infty }}\). However, to obtain convergence rates in \(L^p\)-spaces we need more regularity on the solutions. In turn we assume that \(\rho ^0 \in \mathcal {W}^{1,1}_+(\mathbb R^d)\) in this section and we compute the weights with the formula (2.16) involving the dual kernels. According to the propagation of regularity of solutions to (1.1) in Proposition 9 in the “Appendix”, this ensures that the unique solution to (1.1) satisfies \(\rho \in L^\infty (0,T;\mathcal {W}^{1,1}(\mathbb R^d))\) for all \(T>0\).

Given the solution \(\rho \) to (1.1), we will use the shortcut notation, \(\rho ^n(x):= \rho (t_n,x)\) for \(x \in \mathbb R^d\). From now on, *C* denotes a generic constant independent of *h* and \(\Delta t\), depending only on \(L= \sup _{t\le T} |u(t)|_{\mathcal {W}^{1,\infty }}\), *d* and the exact solution.

*h*and \(\Delta t\) are bounded by an absolute constant. We denote by

Concept | Continuum | Discrete | Error |
---|---|---|---|

Density | \(\rho (t,x)\) | \(\rho ^n_h(x) = \sum _{k \in \mathbb Z^d}\omega _k \varphi ^n_{h,k}(x)\) | \(\theta _n\): \(L^1\)-error |

\(\varepsilon _n\): \(L^\infty \)-error | |||

\(\Gamma ^n_h\): \(L^1 \cap L^p\)-error | |||

Local flow map | \(F^{s,t}(x)\) | \(F^{n}_{h,k}(x) = x^{n+1}_k + J^n_k(x - x^n_k)\) | \(e^n_F\), \(\tilde{e}_F^n\): \(L^\infty \)-errors |

Iterated flow map | \(F^{0,t}(x)\) | \( \overline{F}^n_{h,k}(x) = x^n_k + (D^n_k)^{-1}(x - x^0_k)\) | \(\overline{e_F}^n\): \(L^\infty \)-error |

Jacobian matrix | \(J^{s,t}(x)\) | \(J^n_k\) | – |

Deformation matrix | – | \(D^{n+1}_k = D^n_k(J^n_k)^{-1}\) | – |

Jacobian determinant | \(j^{s,t}(x)\) | \(j^n_k\) | \(e^n_j\): \(L^\infty \)-error |

Particle volume | – | \(h^{n+1}_k = j^n_k h^n_k\) | – |

Particle shape | – | \(\varphi ^n_{h,k}(x) = \frac{1}{h^n_k}\varphi \left( \frac{D^n_k(x - x^n_k)}{h}\right) \) | – |

### 3.1 Estimates on the flows and related terms

We first control the particle overlapping from the approximation error on the flow.

## Lemma 1

*C*independent on

*h*and \(\Delta t\) such that

## Proof

Using the formulas (2.31), (2.32) and the a priori \(L^1\) bound (2.28) on the approximated densities \(\rho ^n_h\) we easily derive uniform estimates for the approximated Jacobian matrices and the particle supports.

## Lemma 2

*C*is uniform in

*k*and \(n \le N\), depending only on the \(L^1\)-norm of the initial data \(\rho _0\).

We next show that the support of the particle approximation is of order *h*.

## Lemma 3

*C*independent of \(\Delta t\) and

*h*.

## Proof

*x*is such that \(F^{t_n,t_{n+1}}(x) \in {{\mathrm{supp}}}(\varphi ^{n+1}_{h,k})\), we have

## Proposition 4

*C*independent of \(\Delta t\) and

*h*.

## Proof

*b*), using the Lipschitz regularity of the flow (2.6) and the error bound (2.19) on the initial data we find

## Proposition 5

*C*depending only on

*d*,

*T*,

*L*, and \(\Vert \rho ^0\Vert _{\mathcal {W}^{1,1}}\) for \(\Delta t\) small enough. Moreover, at \(x=x_k^n\), we have

## Proof

*c*) we use the one-to-one mapping \(\Phi : y \mapsto F^{\tau ,t_n}(F^{t_n,\tau }(x)-y)\) with Jacobian determinant \(|\det \Phi (y)| = j^{\tau ,t_n}(F^{t_n,\tau }(x)-y)\). The change of variable formula yields

*d*) term, we introduce

*h*, \(\Delta t\) and

*n*, such that

We can now compute an estimate for the error of the Jacobian determinants.

## Corollary 1

*C*is a positive constant depending only on

*T*,

*L*, and \(\Vert \rho \Vert _{L^\infty (0,T:\mathcal {W}^{1,1})}\).

## Proof

From Proposition 5 we also derive an estimate for the error between Jacobian matrices.

## Corollary 2

*C*independent of \(\Delta t\) and

*h*. At \(x = x_k^n\), we have

## Proof

## Remark 2

If \(\rho ^0\) is only assumed to be an \(L^1(\mathbb R^d)\) function (or a Radon measure), then \(\xi _n(D^2 W)\) can be bounded by a constant using the \(L^1\) bound on \(\rho ^n_h\), see (2.28), and the \(\mathcal {W}^{1,\infty }(\mathbb R^d)\) smoothness of \(\nabla W\). Arguing as in the proof above we then find an error estimate for the Jacobian matrices on the order of \(\Delta t\).

We next turn to the approximation errors involving the forward characteristic flows and we establish a series of estimates.

## Lemma 4

*C*independent of \(\Delta t\) and

*h*.

## Proof

## Proposition 6

*C*independent of \(\Delta t\) and

*h*.

## Proof

*C*depends on \(||\rho ^0||_{\mathcal {W}^{1,1}}\)). For (

*b*), we easily get using estimate (3.11) in Corollary 2 and Lemma 3 that

*c*) we next differentiate (2.3) and obtain for \(1 \le i,j,m \le d\),

*C*, see (2.5). Invoking the Gronwall Lemma, we then obtain

*C*only depends on

*d*,

*T*,

*L*and \(||\rho ^0||_{\mathcal {W}^{1,1}}\). With a Taylor expansion this gives

*x*and \(x_{k}^n\) and a constant

*C*that only depends on

*d*,

*T*,

*L*and \(||\rho ^0||_{\mathcal {W}^{1,1}}\). Combining the above estimates yields the desired result. \(\square \)

We finally provide estimates for \(\overline{e_F}^n\) and \(\tilde{e}_F^n\).

## Corollary 3

*C*independent of \(\Delta t\) and

*h*.

## Proof

### 3.2 Proof of \(L^1\) and \(L^{\infty }\) convergence results

## Theorem 1

*C*depending only on

*d*,

*T*,

*L*, and \(||\rho ^0||_{\mathcal {W}^{1,1}}\).

## Remark 3

From this result, it is clear that we need a restrictive constraint on \(\Delta t=o(h)\) for convergence. This is a consequence of the low order time discretization considered in Sect. 2.3, implying the need of small time stepping. As for the factor \(\frac{1}{h}\), it comes from the Lipschitz constant of the particle shape functions, and it is classical in the analysis of particle methods. It is not clear how to improve these error estimates even if high order time integrators are used to improve the ODE solver for the particle positions.

## Proof

*x*can be taken in \(S^{n-1}_{h,k}\) in the

*k*-th term.

*k*-th term, we must consider the cases where \(y \in S^n_{h,k}\) and those where \(x \in S^{n-1}_{h,k}\). Thus,

*x*must be taken in the extended particle support \(\tilde{S}^{n-1}_{h,k}\), see (2.34). Using the incremental relation (2.24) we then estimate

We next derive \(L^\infty \)-estimates. Here the required regularity propagates in time. As proved in the “Appendix”, Proposition 9, the unique solution to (1.1) belongs to \(\rho \in L^\infty (0,T;(\mathcal {W}^{1,1}_+\cap L^\infty )(\mathbb R^d))\) provided that \(\rho ^0 \in (\mathcal {W}^{1,1}_+ \cap L^\infty )(\mathbb R^d)\).

## Theorem 2

*h*and \(\Delta t\).

## Proof

Given \(y\in \mathbb R^d\), we decompose \( \rho (t_n,y) - \rho _h^n(y)\) into three terms as in (3.14).

*k*-th term vanishes if \(x \not \in S^{n-1}_{h,k}\). In particular, the sum can be restricted to the indices

*k*in the set \(\mathcal {K}_{n-1}(x)\). Gathering the bounds (3.4) on \(h^n_k\), (2.20) on \(\omega _k\) and (3.3) on \(\kappa _n := \sup _{x\in \mathbb R^d} \#(\mathcal {K}_{n-1}(x))\), we compute

*k*-th summand in \(C_n(y)\) must be considered when \(y \in S^n_{h,k}\) or when \(x \in S^{n-1}_{h,k}\) (or both). Clearly the cardinality of the corresponding index set satisfies

## 4 Convergence for measure solutions with smooth potentials

*W*satisfies \(\nabla W \in \mathcal {W}^{1,\infty }(\mathbb R^d)\), a well-posedness theory for measure valued solutions to (1.1) can be developed by using the classical results of Dobrushin [45], see [24, 53] for related results.

To estimate the error between the exact flow and its local linearizations we now revisit some results from the previous section, namely Proposition 6, given the low regularity of the solutions. As in the previous section, we denote \(\rho ^n = \rho (t_n)\).

## Proposition 7

*W*satisfies \(\nabla W \in \mathcal {W}^{1,\infty }(\mathbb R^d)\), then the flow error defined on the particles support (2.33)

*C*independent of

*h*and \(\Delta t\).

## Proof

*b*), we easily get from Remark 2 that \(|(b)| \le C h \Delta t\). Finally, we observe that (

*c*) cannot be estimated as in the proof of Proposition 6, due to the lesser regularity of the densities. We then proceed as follows,

We now show that our LTP method is unconditionally stable in the weak norm between measures \(d_{BL}\).

## Theorem 3

*W*satisfies \(\nabla W \in \mathcal {W}^{1,\infty }(\mathbb R^d)\), then the estimate

*C*depends only on

*d*,

*L*and

*T*.

## Remark 4

Observe that a convergence condition on the approximation of the initial data in Theorem 3 such as \(d_{BL}(\rho ^0,\rho ^0_h) \lesssim h\) is easily achieved by using a uniform quadrangular mesh of size \(h^d\) and approximating the initial data \(\rho ^0\) by a sum of Dirac deltas via transporting the mass of \(\rho ^0\) inside each *d*-dimensional cube to its center. A cut-off procedure to leave small mass outside a large ball allows us to reduce to a finite number of Dirac deltas in this approximation. Finally, the error produced between smoothed particles and Dirac deltas is obviously of order *h* in the \(d_{BL}\) distance.

## Proof of Theorem 3

*b*) with

*h*. The proof is then completed using Gronwall’s inequality as in Theorem 1. \(\square \)

## 5 \(L^1\) and \(L^p\) convergence for singular potentials

In this part, we are interested in \(L^p\)-convergence between the solution and its approximation allowing for more singular potentials. With this aim, we consider the solutions of the Eq. (1.1) in \(L^\infty (0,T; L^\infty (\mathbb R^d) \cap \mathcal {W}^{1,1}(\mathbb R^d) \cap \mathcal {W}^{1,p}(\mathbb R^d))\) with \(1 \le p \le \infty \) to be determined depending on the singularity of the potential. Since we are dealing with both attractive and repulsive potentials, we can only expect local in time existence and uniqueness of solutions as in [15, 24]. In those references, a local in time well-posedness theory in \(L^\infty (0,T; L^1(\mathbb R^d) \cap L^{p}(\mathbb R^d))\) was developed under suitable assumptions on the potentials. The solutions are constructed by characteristics since the velocity fields are still Lipschitz continuous in *x*. However, to prove convergence rates we need more regularity on the solutions. For the existence of solutions to (1.1) in \(L^\infty (0,T; L^\infty (\mathbb R^d) \cap \mathcal {W}^{1,1}(\mathbb R^d) \cap \mathcal {W}^{1,p}(\mathbb R^d))\), we provide a priori estimates in “Appendix A”, Proposition 10. These estimates combined with the existing literature [15, 24] show the well-posedness of solutions in the desired class. In our presentation we will follow the setting of local existence introduced in [24].

*x*locally in time for densities in \((L^1\cap L^p)(\mathbb R^d)\) where

*p*is the conjugate exponent of

*q*. Note that \( q = p' < \frac{d}{\alpha + 1}\) is equivalent to \(\alpha < -1 + \frac{d}{p'}\), giving us the condition on the initial data for the well-posedness theory. Indeed, it follows from (5.1) that

*C*depending on \(\tilde{L}\),

*q*and

*d*, and a similar estimate holds for

*u*using (5.2) and the fact that \(\nabla W\) is bounded away from the origin.

*N*. We introduce the following notations:

*h*and \(\Delta t\), which we need to estimate \((\nabla W * \rho _h^n)\) and \((D ^2 W * \rho _h^n)\). In order to do that, we will prove by induction that there is some \(h_* > 0\) for which

Under the (induction) assumption that \(\widetilde{\Gamma ^n_h}\) is bounded uniformly in *h* and \(\Delta t\), we can derive the following estimates.

## Lemma 5

*M*but not on

*h*and \(\Delta t\).

## Proof

*M*. \(\square \)

We next give the estimates of \( u(\tau ,F^{t_m,\tau })-u_k^m\) for \(\tau \in [t_{m},t_{m+1}]\) and \({\tilde{\xi }}_m(D^2 W)\) for \(0 \le m \le n-1\). The proof can be obtained by using similar arguments as in Proposition 5 with the help of Lemma 5 and a second-order estimate provided either by Proposition 2 or by a standard \(L^p\) error estimate as described in Proposition 1. We omit its proof, but point out that the crucial point is the smoothness assumptions (5.3) on the singular potential and the Lipschitz bound (5.4) on the velocity field.

## Lemma 6

*M*but not on

*h*and \(\Delta t\).

We can also adapt the proof of Corollary 1, Lemma 6, and Proposition 6 to obtain the following result.

## Lemma 7

*M*but not on

*h*and \(\Delta t\).

We finally connect the errors to the \(L^1\cap L^p\) bounds on the densities.

## Lemma 8

*M*but not on

*h*and \(\Delta t\).

## Proof

*m*. Then from (5.6) we derive

We are now in a position to show the uniform \(L^1 \cap L^p\) bounds on the density.

## Proposition 8

*W*is singular in the sense of (5.1) and (5.2), and let \(\rho \) be a solution to the Eq. (1.1) up to time \(T>0\), such that \(\rho \in L^\infty (0,T;(\mathcal {W}^{1,1} \cap \mathcal {W}^{1,p} \cap L^\infty )(\mathbb R^d))\) with initial data \(\rho ^0 \in \mathcal {W}^{2,p}(\mathbb R^d)\), \(-1 \le \alpha < -1 + d/p'\), and \(1 < p \le \infty \). Assume in addition that \(\Delta t \lesssim h^2 \le 1\). Then for all \(M > 0\), there exists \(h_*(M) > 0\) such that

## Proof

*n*. Since \(\widetilde{\Gamma ^0_h} = \Gamma ^0_h \lesssim h^2\), clearly there exists \(h_0(M)\) such that \(\Gamma ^0_h \le M\) for all \(h < h_0(M)\). We then assume that \(n < N\) and \(h_n(M) > 0\) are such that

*M*. Decomposing the error as in Theorem 1, we write

*M*but independent of

*h*and \(\Delta t\). In particular, setting \(h_{n+1}(M) := \min (h_n(M), M/C_M)\) allows to write

Putting together all the results in this section, we obtain the main convergence result in \((L^1 \cap L^p)(\mathbb R^d)\). We note that, as above, the condition on the time step is a result of the low order time discretization (see Remark 3).

## Theorem 4

*W*is singular in the sense of (5.1) and (5.2), and let \(\rho \) be a solution to the Eq. (1.1) up to time \(T>0\), such that \(\rho \in L^\infty (0,T;(\mathcal {W}^{1,1} \cap \mathcal {W}^{1,p} \cap L^\infty )(\mathbb R^d))\) with initial data \(\rho ^0 \in \mathcal {W}^{2,p}(\mathbb R^d)\), \(-1 \le \alpha < -1 + d/p'\), and \(1 < p \le \infty \). Assume in addition that \(\Delta t \lesssim h^2 \le 1\). Then

*C*independent of

*h*and \(\Delta t\).

## 6 Numerical results

We will present in this Section some numerical examples in one dimension, with different interaction potentials and initial densities to showcase some of the features already observed in numerical and theoretical analysis of the aggregation equation (1.1) in [4, 11, 14, 49, 50, 57]. In this way, we first validate our numerical implementation in order to explore some less-known properties about the behavior of its solutions in one dimension. A further more complete numerical study in 2D of this method will be reported elsewhere. These examples already show the wide range of different behaviors of solutions to the aggregation equation.

### 6.1 Numerical method: computation of the velocity field

Obviously, the cost of the above strategies depends on the numbers of particles and grid points. The first one scales like \(N_\mathrm{parts}^2\), the second one like \(N_\mathrm{parts} N_\mathrm{grid}\) and the third one like \(N_\mathrm{grid}\log (N_\mathrm{grid})\). For our simulations we have tried these three strategies (with \(N_\mathrm{grid} \sim N_\mathrm{parts}\)) and have observed no significant differences in the resulting densities.

It is worth mentioning that the vortex method for the 2D Euler equations is studied in [72], where the vorticity is approximated by a piecewise interpolation polynomial on a triangulation of the vortices and the vertices of the triangulation move with the fluid velocity. In our case, the particles could have a piecewise affine shape, but their supports can intersect as mentioned before. Thus our method of computing the velocity fields in general cannot be reduced to a *P*1-finite element discretization of the velocity fields as in [72], see also [37] for other remeshed particle methods and general convergence proofs.

### 6.2 Numerical method: validation and comparison to classical particle methods

*h*between two particles, the reconstructed density will oscillate or even vanish between nearby particles and thus become inaccurate; if \(\varepsilon \) is too large the reconstructed density will be too spread out and the results will again lack accuracy, as demonstrated in Fig. 2(right).

In Figs. 3 and 4 we further compare the Smooth Particle (SP) and the LTP approximations by showing \(L^1\), \(L^{\infty }\) and \(d_{BL}\) error curves, using several values of *h* and \(\varepsilon \). Again the potential is \(W(x)=x^2\) and the exact solution is given by (6.6).

Together with Fig. 2, these error curves show not only the higher accuracy reached by the LTP method but also the sensitivity of the final error with respect to the particle size \(\varepsilon \). An interesting feature of the LTP approach is the automatic adaptation of the particle size, and for the cases considered here, Figs. 3 and 4 show that such an approach outperforms any uniform choice of \(\varepsilon \). In Fig. 5 some comparisons are shown with B1 and B3 spline shape functions, and again the gain of accuracy reached by the LTP method is clear.

*x*. In the case of potential \(W(x)=x^2\), the explicit expression of the density (6.6) shows that \(j^{0,t}(x)=e^{-2t}\), and thus the average distance between two particles decreases exponentially in time. Consequently, the optimal size \(\varepsilon \) for reconstruction in classical particle method is not the same during the whole simulation: An evolution in time of \(\varepsilon \) may be better adapted. A similar issue may also appear regarding the space dependency. In fact for the potential \(W(x)=x^2\) considered in Figs. 2, 3 and 4, the Jacobian determinant \(j^{0,t}\) was constant with respect to

*x*and all the particles in the LTP method had the same size at a given time

*t*. However in general this is not the case: with the potential \(W(x)= \frac{x^4}{4}- \frac{|x|^{2.5}}{2.5}\) considered in Figs. 6 and 10 with various initial densities, we see that the Smooth Particle method with \(\varepsilon =h\) leads to solutions that seem accurate in some regions but strongly oscillate in some others. And in this case, Fig. 10 (bottom) shows that the size of the LTP particles evolves in space, thus giving a hint that an optimal particle size is indeed space-dependent.

### 6.3 Numerical simulations: singular potentials and qualitative properties of steady states

*W*is cut-off at infinity or if the initial data is compactly supported since the effective values of the potential lie on a bounded set and

*W*can be cut-off at infinity without changing the solution. Figure 7 presents the numerical results obtained by the LTP method in the case of \(a=1.5\) and \(a=2.5\). We represent the approximate density \(\rho _h^n\), the reconstructed velocity \(u_h^n\) and the reconstructed particles sizes \(h^n\) using piecewise linear interpolation such that

Next we further analyze the blow-up behavior by looking at the case of attractive-repulsive potentials \(W(x)=\tfrac{|x|^a}{a} -\tfrac{|x|^b}{b}\), \(1<b<a\). Notice again that for \(b\ge 2\) the potential is smooth while for \(1<b<2\) it is singular once *W* is cut-off at infinity or if the initial data is compactly supported as discussed above. Figure 8 presents the approximate density \(\rho _h^n\), the reconstructed velocity \(u_h^n\) and the particles sizes \(h^n\) obtained by the LTP method in the case of the attractive-repulsive potentials with \((a,b)=(3,1.5)\) and \((a,b)=(3,2.5)\). In this case \(\rho ^0\) is given by (6.3).

We observe that the long time asymptotics for \(b=2.5\) are characterized by the concentration of mass equally onto Dirac deltas at two points in infinite time, while for \(b=1.5\) we obtain a convergence in time towards a steady \(L^1\) density profile seemingly diverging at the boundary of the support. This last behavior has been reported in several simulations and related problems [11]. However, it has not been rigorously proven yet. Let us point out that the set of stationary states where the interaction potential is analytic in 1D consists of a finite number of Dirac deltas as proven in [49, 50]. This result also holds for \(W(x)=\tfrac{|x|^a}{a} -\tfrac{|x|^b}{b}\), \(2<b<a\), as it will be reported in [29].

## Notes

### Acknowledgements

JAC was partially supported by the EPSRC Grant Number EP/P031587/1 and from the Royal Society by a Wolfson Research Merit Award. YPC was supported by National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (Nos. 2017R1C1B2012918 and 2017R1A4A1014735). JAC and YPC were supported by EPSRC grant with Reference EP/K008404/1. This work was partially done when FC was visiting Imperial College funded by the EPSRC EP/I019111/1 (platform grant).

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