Well-posedness and error estimates for coupled systems of nonlocal conservation laws Open Access

Abstract

1. Introduction

We here study the initial-value problem (IVP) for the coupled system of nonlocal hyperbolic conservation laws with discontinuous flux given by

Here the unknown is |$\boldsymbol{U}=(U^{1},\ldots ,U^{N}): [0,\infty )\times{\mathbb{R}} \rightarrow{\mathbb{R}}^{N}$|⁠, with |$\circ $| denoting the Hadamard (component-wise) product. Furthermore, |$\boldsymbol{\sigma }=( \sigma ^{1},\ldots , \sigma ^{N})\colon{\mathbb{R}} \rightarrow{\mathbb{R}}^{N}$|⁠, where each component |$\sigma ^{k}$| is allowed to have infinitely many discontinuities, including accumulation points. The system is coupled due to the nonlocal terms |$\boldsymbol{\mu } \circledast \boldsymbol{U}:{\mathbb{R}} \rightarrow{\mathbb{R}}^{N^{2}}$|⁠, with |$\boldsymbol{\mu }$| a smooth |$N\times N$| matrix, and |$(\boldsymbol{\mu }\circledast \boldsymbol{U})^{k}:= \left (\mu ^{k,1}*U^{1},\ldots ,\mu ^{k,N}*U^{N}\right )$| with convolution * over space only. We have |${\boldsymbol{\nu }}=(\nu ^{1},\ldots ,\nu ^{N})$|⁠, with each |$\nu ^{k}:{\mathbb{R}}^{N} \rightarrow{\mathbb{R}}$|⁠. The function |$\boldsymbol{F}=(f^{1},\ldots ,f^{N})\colon{\mathbb{R}} \rightarrow{\mathbb{R}}^{N}$| is Lipschitz in each component, with |$\boldsymbol{F}(\boldsymbol{0})=\boldsymbol{0}$| and |$\boldsymbol{\nu }(\boldsymbol{0})=0$|⁠, and |$|{\boldsymbol{\sigma }}|$| is strictly bounded away from zero. Precise assumptions are given below.

Hyperbolic conservation laws are partial differential equations (PDEs) that describe the conservation of |$\boldsymbol{U}$| in a physical system, where the temporal rate of change of |$\boldsymbol{U}$| is exactly balanced by the spatial rate of change of the flux function. The scalar case, i.e., with |$N=1$|⁠, is well understood, also in multidimensions, while the case of systems, i.e., |$N>1$|⁠, is considerably harder even in one dimension. The use of Hadamard products restricts the coupling in (1.1) to the convolution term |$\boldsymbol{\nu }(\boldsymbol{\mu } \circledast \boldsymbol{U})$|⁠, allowing us to use the scalar theory. Hyperbolic conservation laws were originally introduced in the context of fluid dynamics; however, recently these equations have been applied as models covering a wide range of phenomena, and this has necessitated a considerably wider class of flux functions. Here we include two effects—we let the flux function depend on space in a discontinuous manner through the function |$\boldsymbol{\sigma }(x)$|⁠, and, furthermore, we let the flux function depend on a local average of (the full vector) |$\boldsymbol{U}$| in the form of the convolution term |$\boldsymbol{\nu }(\boldsymbol{\mu } \circledast \boldsymbol{U})$|⁠.

Some of the recent applications include crowds (Colombo & Lécureux-Mercier, 2011; Colombo et al., 2011, 2012), traffic (Colombo et al., 2011; Blandin & Goatin, 2016), opinion formation (Aletti et al., 2007), sedimentation models (Betancourt et al., 2011), structured population dynamics (Perthame, 2007), supply chain models (Colombo et al., 2011), granular material dynamics (Shen, 2012) and conveyor belt dynamics (Göttlich et al., 2014). The case of smooth |$\boldsymbol{F}$| has been well studied in literature, a nonexhaustive list being Colombo & Lécureux-Mercier (2011); Colombo et al. (2011, 2012); Aggarwal et al. (2015); Amorim et al. (2015); Aggarwal & Goatin (2016); Blandin & Goatin (2016); Friedrich et al. (2022), and various references therein.

Let us explain why it is necessary to study the extensions given in this paper. As we know, roads may experience rapidly changing local conditions, and when we model traffic flow by hyperbolic conservation laws, we allow for this by having a nonsmooth, spatially depending function |$\boldsymbol{\sigma }(x)$| in the flux function. Furthermore, more practical problems like the multi-agent systems, as well as laser technology, dynamics of multiple crowds/cars, lane formation, sedimentation and conveyor belts, see, for example, Colombo & Marcellini (2015); Colombo et al. (2016); Gugat et al. (2016), one needs to analyse coupled systems of nonlocal conservation laws with discontinuous flux.

The literature of the following local counterpart of (1.1) with |$N=1$|⁠, viz.

is quite rich, with the results spanning to |$F$| being a nonlinear function of |$U$| and having general spatial discontinuities; see, for example, Adimurthi & Veerappa Gowda (2004, 2005, 2007); Perthame (2007); Andreianov et al. (2010); Aggarwal et al. (2020, 2021); Ghoshal et al. (2020, 2022a,b) and references therein.

However, the nonlocal counterpart (1.1) remains far from being settled, with the only exceptions being Keimer & Pflug (2021), Aggarwal & Vaidya (2023) and Chiarello et al. (2023). References Keimer & Pflug (2021); Chiarello et al. (2023) settle the problem for the case |$F(x,U)= s(x)U$|⁠, with Keimer & Pflug (2021) establishing this via a fixed-point argument with a bounded discontinuous |$s$| and monotone nonsymmetric kernels, and Chiarello et al. (2023) settling the same via a convergent numerical scheme for monotonically decreasing |$s$|⁠, with a single spatial discontinuity and |$\mu $| being a downstream kernel. The question of well-posedness for the cases of |$F(x,U)= s(x)U$| with |$ s$| having nondiscrete discontinuities, and for the case of |$F(x,U)= s(x)f(U)$| with a nonlinear |$f$| and |$ s$| having single or finitely many spatial discontinuities, with or without any constraints on monotonicity of |$\nu $| or |$\mu $|⁠, remained unexplored and unsettled for a long time. This question was recently settled in Aggarwal & Vaidya (2023), where the well-posedness was explored in a very general setup with |$F(x,U)=f( s(x)U),$| where |$f$| is nonlinear and there are no additional constraints on |$\nu $|⁠, |$\mu $| and the discreteness or monotonicity of spatial discontinuities.

The existence of the entropy solutions for (1.1) for |$N\ge 1$| with sufficiently smooth |$\boldsymbol{\sigma }$| has been studied in Betancourt et al. (2011); Göttlich et al. (2014); Aggarwal et al. (2015); Colombo & Marcellini (2015); Chiarello & Goatin (2023), while the uniqueness has been settled in Colombo & Marcellini (2015); Colombo et al. (2016); Colombo & Rossi (2018), for example. The question of existence as well as uniqueness of the entropy solutions for the system (1.1) with |$\boldsymbol{\sigma }$| having any kind of spatial discontinuities remains unexplored and unsettled as of now. This article settles these questions for IVPs (1.1), which we can write on component form as

for each |$k=1,2,\ldots ,N$|⁠. Here |$T$| is a given positive time. The assumptions that |$\sigma ^{k}\in \operatorname{BV}({\mathbb{R}},{\mathbb{R}}^{+})$| and |$\inf _{x\in{\mathbb{R}}}\sigma ^{k}(x)>0$| are justified, being the essential technical assumptions required for well-posedness of scalar local counterpart of (1.2)–(1.3), see Panov (2009); Towers (2020); Ghoshal et al. (2022a,b) and references therein. As stated earlier, the system (1.2)–(1.3) has been studied in Keimer & Pflug (2021) only for the linear case |$ f^{k}(u)=u,$| using a fixed point argument. However, in the setup of this paper, |$f^{k}$| is nonlinear and consequently, there can be multiple weak solutions, like in the local case. Hence, an entropy condition is required to single out the unique solution.

In this article, this question will be settled in Section 4, where it will be shown that any two entropy solutions (cf. (2)) of (1.2) corresponding to the same initial data (1.3) are in fact equal, and the approximate solutions obtained from the finite volume schemes proposed in Section 3 satisfy the discrete form of the entropy inequality (2). As a result, one will obtain that the approximate solutions converge to the unique entropy solution of (1.2)–(1.3). This uniqueness result is novel and is not dealt with in existing uniqueness results of Bürger et al. (2004); Blandin & Goatin (2016); Colombo et al. (2016); Colombo & Rossi (2018); Keimer & Pflug (2021), which deal with either local part of the flux being linear in state variable and possibly discontinuous in space variable, or nonlinear smooth local fluxes only. At the time of writing of this article, this is the first result establishing the existence and uniqueness of solutions of such coupled systems with discontinuous flux. It is to be noted that no technical restrictions on monotonicity of kernel or |$\boldsymbol{\sigma }$| have been made in the analysis, and hence the article presents the first result in a very general setting. However, to exercise this method, we need a uniform |$\operatorname{BV}$| bound on the entropy solution |$\boldsymbol{U}$| and its numerical approximation |$\boldsymbol{U}_{\varDelta }$|⁠, which is one of the objectives of this article.

The final objective of this article is the error analysis of our finite volume approximations, proposed in Section 3. To the best of our knowledge, this is the first result towards error analysis of numerical approximations for coupled systems of nonlocal conservation laws, including the case of smooth |$\sigma $|⁠. However, the literature on error estimates for local conservation laws is quite rich, see Kuznetsov (1976); Karlsen (1994); Sabac (1997); Karlsen & Risebro (2001); Cancès & Seguin (2012); Badwaik & Ruf (2020); Ghoshal et al. (2022b) for |$N=1$|⁠. This is achieved by following the techniques of Kuznetsov (1976), where the so-called doubling of variables argument is exploited to express the error between the entropy solution |$U=u$| (i.e., |$N = 1$|⁠) and its numerical approximation |$u_{\varDelta }$| in terms of their relative entropy. A first result towards scalar nonlocal conservation laws, i.e., with |$N=1$|⁠, with a homogeneous smooth local part of the flux, was recently done in Aggarwal et al. (2023). In general, for any |$N>1$|⁠, the local counterpart of (1.2) is fairly less explored with a limited list being Bressan et al. (2021); Bressan & Shen (2021). This paper establishes error estimates for |$N>1$| for the nonlocal counterpart (1.2) with smooth local fluxes in any dimension, and with a fairly large class of rough local fluxes (precise assumptions described later) in one dimension.

Thus, the novelties of this article for the system of nonlocal conservation laws (1.1), where |$k^{th}$| equation is given by (1.2), are as follows:

• Existence of the adapted entropy solutions via finite volume approximations.

• A Kuznetsov-type lemma and thereby the uniqueness of the entropy solution.

• Convergence rate estimates for the proposed numerical schemes.

The paper is organized as follows. In Section 2, we introduce some definitions and notation to be used in the article. In Section 3, we present the convergence of the Godunov type and the Lax–Friedrichs type scheme of the approximation of the IVP (1.2)–(1.3). In Section 4, we prove the uniqueness result, and, in Section 5, we discuss the convergence rates. Finally, in Section 6, we present some numerical experiments that illustrate the theory.

2. Definitions and notation

Throughout this paper, we use the following notation:

• (1) For |$\boldsymbol{z}:=(z^{1},\ldots ,z^{N})\in{\mathbb{R}}^{N},$| let |${\left \|\boldsymbol{z}\right \|}:=\sum _{k=1}^{N}|z^{k}|$| denote the usual |$1$|-norm.

• (2) For |$\boldsymbol{f}: = (f^{1},\ldots , f^{N})\in (L^{1}({\mathbb{R}}))^{N},$||${\left \|f\right \|}_{(L^{1}({\mathbb{R}}))^{N}}:=\sum _{k=1}^{N} {\left \|f^{k}\right \|}_{L^{1}({\mathbb{R}})}$| be the norm on the product space.

• (3) For the matrix valued function |$\boldsymbol{\mu } \in L^{\infty }({\mathbb{R}};{\mathbb{R}}^{N^{2}}),$|

  $$ \begin{align*} & {\left\|\boldsymbol{\mu}\right\|}_{(L^{\infty}({\mathbb{R}}))^{N^2}}:= \max\limits_{ 1 \leq i,j \leq N} {\left\|\mu^{i,j}\right\|}_{L^{\infty}({\mathbb{R}})}. \end{align*}$$

  In addition, if |$\boldsymbol{\mu }\in C^{2}({\mathbb{R}};{\mathbb{R}}^{N^{2}})$|⁠, then |$\boldsymbol{\mu }^{\prime}\in C^{1}({\mathbb{R}};{\mathbb{R}}^{N^{2}})$| and |$\boldsymbol{\mu }^{\prime\prime} \in C({\mathbb{R}};{\mathbb{R}}^{N^{2}})$| denote the component-wise derivative and second derivative, respectively.
• (4) For |$u: \overline{Q}_{T}\rightarrow{\mathbb{R}},$||$\boldsymbol{U}:\overline{Q}_{T} \rightarrow{\mathbb{R}}^{N}$| and |$\tau>0$|⁠,

  $$ \begin{align} \begin{aligned}|u|_{\operatorname{Lip}_{t}L^{1}_{x}}&:=\sup_{0\leq t_{1}<t_{2}\leq T}\frac{{\left\|u(t_{1})-u(t_{2})\right\|}_{L^{1}({\mathbb{R}})}}{|t_{1}-t_{2}|}, \\ |\boldsymbol{U}|_{(\operatorname{Lip}_{t}L^{1}_{x})^{N}}&:=\max_{1 \leq k\leq N}|U^{k}|_{\operatorname{Lip}_{t}L^{1}_{x}}, \\ |\boldsymbol{U}|_{(L^{\infty}_{t}\operatorname{BV}_{x})^{N}}&:=\max_{1 \leq k\leq N}\sup_{t\in[0,T]} TV(U^{k}(t,\, \cdot\,)),\\{\left\|\boldsymbol{U}\right\|}_{(L^{\infty}(\overline{Q}_{T}))^{N}}&:=\max_{1 \leq k\leq N} {\left\|U^{k}\right\|}_{L^{\infty}(\overline{Q}_{T})}, \\{\left\|\boldsymbol{U}\right\|}_{(L^{1}(\overline{Q}_{T}))^{N}}&:=\sum_{k=1}^{N} {\left\|U^{k}\right\|}_{L^{1}(\overline{Q}_{T})}, \\ \gamma(\boldsymbol{U},\tau)&:= \max\limits_{1 \leq k \leq N}\sup_{\substack{ |{t_{1}-t_{2}}| \leq \tau\\ 0\leq t_{1}\leq t_{2}\leq T }} {\left\|U^{k}(t_{1})-U^{k}(t_{2})\right\|}_{L^{1}({\mathbb{R}})}.\end{aligned}\end{align}$$

  (2.1)

We will make the following assumptions throughout this work:

• (H1)

  |$f^{k} \in \operatorname{Lip}({\mathbb{R}})$| with |$ f^{k}(0)=0$|⁠;

• (H2)

  |$\nu ^{k} \in (C^{2} \cap \operatorname{BV} \cap \, W^{2,\infty }) ({\mathbb{R}}^{N},{\mathbb{R}}^{+}), \boldsymbol{\mu }\in (C^{2} \cap \operatorname{BV} \cap \, W^{2,\infty }) ({\mathbb{R}},{\mathbb{R}}^{N^{2}})$|⁠, with |$\nu ^{k}(\boldsymbol{0})=0$|⁠;

• (H3)

  |$\sigma ^{k}\in (\operatorname{BV} \cap L^{1})({\mathbb{R}})$| with |$\inf \limits _{x\in{\mathbb{R}}}\sigma ^{k}(x)>0$|⁠.

3. Existence of the entropy solution

For |$\varDelta x,\varDelta t>0$| and |$\lambda =\varDelta t/\varDelta x,$| consider equidistant spatial grid points |$x_{i}:=i\varDelta x$| for |$i\in{\mathbb{Z}}$| and temporal grid points |$t^{n}:=n\varDelta t$| for integers |$n\in \{0,\ldots ,N_{T}\}$|⁠, such that |$T=N_{T} \varDelta t.$| Let |${1\kern-3.4pt1}_{i}(x)$| denote the indicator function of |$C_{i}:=[x_{i-1/2}, x_{i+1/2})$|⁠, where |$x_{i+1/2}=\frac 12(x_{i}+x_{i+1})$| and let |$\chi ^{n}(t)$| denote the indicator function of |$C^{n}:=[t^{n},t^{n+1})$|⁠, and let |$C_{i}^{n}:=C^{n}\times C_{i}$|⁠. For each |$k \in \{1, \ldots , N\},$| we approximate the initial data and |$\sigma ^{k}$| according to the following:

Further, we define a piecewise constant approximate solution |$\boldsymbol{U}_{\varDelta } = \left (U_{\varDelta }^{1}, \ldots , U_{\varDelta }^{N}\right )$|⁠, i.e., |$\boldsymbol{U}_{\varDelta }\approx \boldsymbol{U}$|⁠, to the IVP (1.2)–(1.3) by

The approximate solution |$\boldsymbol{U}_{\varDelta }$| is updated through the following marching formula:

where

• (1) for each |$n \in \{0, \ldots , N_{T}\}, i \in{\mathbb{Z}}$| and |$ k \in \{1, \ldots , N\}$|⁠, |$\overline{U}_{i}^{k,n}:= \sigma ^{k}_{i}U_{i}^{k,n}\approx \sigma ^{k}(x_{i})U^{k}(x_{i},t^{n})$| (pointwise value of the right continuous representative) and which is extended to the function defined in |${Q}_{T}$|⁠, via

  $$ \begin{align*}& \overline{U}^{k}_{\varDelta}(t,x) =\sum_{n=0}^{N_{T}-1} \sum_{i\in{\mathbb{Z}}} {1\kern-3.4pt1}_{i}(x) \chi^{n}(t) \overline{U}_{i}^{k,n}; \end{align*}$$
• (2) the convolution term is updated through the following formula:

  $$ \begin{align} {\boldsymbol{c}}^{k,n}_{ i+1/2}&=\left(c_{ i+1/2}^{1,k,n},\ldots, c_{ i+1/2}^{N,k,n}\right)\!,\end{align}$$

  (3.3)

  with, for each |$j\in \{1,\ldots ,N\}$|⁠, we have |$c_{ i+1/2}^{j,k,n}$| being computed through the following quadrature formula:

  $$ \begin{align*} c_{ i+1/2}^{j,k,n}&=\varDelta x\sum\limits_{p\in{\mathbb{Z}}} \mu^{j,k}_{i+1/2-p}U^{j,n}_{p+1/2} \approx c^{j,k}\left(t^{n},x_{i+1/2}\right)= \int_{{\mathbb{R}}} \mu^{j,k}(x_{i+1/2}-y)U^{j}_{\varDelta}(t^{n},y){\, \textrm{d}{y}}, \end{align*}$$

  with |$U^{k,n}_{p+1/2}$| being any convex combination of |$U^{k,n}_{p}$| and |$U^{k,n}_{p+1}$|⁠, and |$\mu ^{j,k}_{i+1/2} = \mu ^{j,k} (x_{i+1/2}).$|

• (3) |$F^{k}$| is chosen so as to make |$H^{k}$| increasing in its last three arguments.

The convolution terms satisfy the following estimates:

• (1) Lax–Friedrichs type flux:

  $$ \begin{align}\begin{aligned} F^{k}_{LF}(a,b,c) &= \frac{a}{2}\left( f^{k}(b) + f^{k}(c)\right) - \frac{\theta_{{k}}}{2\, \lambda}(c-b),\\ \theta_{{k}} &\in \left(0,\frac{2}{3{\left\|\sigma^{k}\right\|}_{L^{\infty}({\mathbb{R}})}} \right)\!.\end{aligned}\end{align}$$

  (3.4)

  Further, |$\varDelta t$| is chosen in order to satisfy the following Courant–Friedrichs–Lewy (CFL) like condition:

  $$ \begin{align}& \lambda \le \min_{k} \left\{\frac{\min\left(1, 4-6\theta_{{k}}{\left\|\sigma^{k}\right\|}_{L^{\infty}({\mathbb{R}})},6\theta_{{k}}{\left\|\sigma^{k}\right\|}_{L^{\infty}({\mathbb{R}})}\right)}{1+6{\left\|\sigma^{k}\right\|}_{L^{\infty}({\mathbb{R}})}|f^{k}|_{\operatorname{Lip({\mathbb{R}})}}{\left\|\nu^{k}\right\|}_{L^{\infty}({\mathbb{R}})}}\right\}.\end{align}$$

  (3.5)
• (2) Godunov type flux:

  $$ \begin{align*} F^{k}_{\textrm{Godunov}}(a,b,c)= a F^{k}_{\textrm{Godunov}}(b,c), \end{align*}$$

  where |$ F^{k}_{\textrm{Godunov}}$| is the Godunov flux for the corresponding local conservation law with discontinuous flux |$f^{k}(x,u)=f^{k}(\sigma ^{k}(x)u).$| Further, |$\varDelta t$| is chosen in order to satisfy the following CFL condition:

  $$ \begin{align}& \lambda \max_{k}\left\{{{\left\|\sigma^{k}\right\|}_{L^{\infty}({\mathbb{R}})}}|f^{k}|_{\operatorname{Lip({\mathbb{R}})}}{\left\|\nu^{k}\right\|}_{L^{\infty}({\mathbb{R}})}\right\} \leq \frac{1}{6}.\end{align}$$

  (3.6)

  The scheme (3.2) can now be rewritten in an incremental form as (see Ghoshal et al., 2022a; Aggarwal & Vaidya, 2023)

  $$ \begin{align} \overline{U}_{i}^{k,n+1} &= \overline{U}_{i}^{k,n} + \overline{U}_{i}^{k,n+1}-\overline{U}_{i}^{k,n} \nonumber\\ &= \overline{U}_{i}^{k,n} -\lambda \sigma_{i}^{k} \left[F^{k,n}_{i+1/2}\left(\overline{U}_{i}^{k,n},\overline{U}_{i+1}^{k,n}\right)-F^{k,n}_{i-1/2}\left(\overline{U}_{i-1}^{k,n}, \overline{U}_{i}^{k,n}\right)\right]\nonumber\\ &= \overline{U}_{{i}}^{k,n} - a_{i-1/2}^{k,n} \, \varDelta_{-}\overline{U}_{{i}}^{k,n} + b_{i+1/2}^{k,n} \, \varDelta_{+}\overline{U}_{{i}}^{k,n}\nonumber\\ & \quad - \lambda \sigma_{i}^{k}\left( F^{k,n}_{i+1/2} \left(\overline{U}_{{i}}^{k,n},\overline{U}_{{i}}^{k,n}\right) - F^{k,n}_{i-1/2} \left(\overline{U}_{{i}}^{k,n},\overline{U}_{{i}}^{k,n}\right) \right), \end{align}$$

  (3.7)

  where |$\varDelta _{+}z_{i}^{k,n}=z_{i+1}^{k,n}-z_{i}^{k,n}=\varDelta _{-}z^{k,n}_{i+1},$| for any vector |$z$|⁠, and

  $$ \begin{align*} a_{i+1/2}^{k,n} & = \lambda \sigma_{i}^{k} \, \frac{ {F^{k,n}_{i+1/2} \left(\overline{U}^{k,n}_{i+1},\overline{U}^{k,n}_{i+1}\right)} - F^{k,n}_{i+1/2}\left(\overline{U}^{k,n}_{i},\overline{U}^{k,n}_{i+1}\right)}{\varDelta_{+}\overline{U}_{{i}}^{k,n}} ,\\ b_{i+1/2}^{k,n} &= \lambda \sigma_{i}^{k} \, \frac{ {F^{k,n}_{i+1/2} \left(\overline{U}_{{i}}^{k,n},\overline{U}_{{i}}^{k,n}\right)} - F^{k,n}_{i+1/2}\left(\overline{U}_{{i}}^{k,n},\overline{U}^{k,n}_{i+1}\right)}{\varDelta_{+}\overline{U}_{{i}}^{k,n}} . \end{align*}$$

  The following theorem establishes the existence and the regularity of the entropy solutions of IVP (1.1):

4. Kuznetsov-type estimate and the uniqueness

For |$\epsilon ,\epsilon _{0}>0,$| define |$\varPhi : \overline{Q}_{T}^{2} \rightarrow{\mathbb{R}}$| by

where |$\omega _{a}(x)=\frac{1}{a}\omega \left (\frac{x}{a}\right ),$||$a>0$| and |$\omega $| is a standard symmetric mollifier with |$\operatorname{supp} (\omega ) \in [-1,1].$| Furthermore, we assume that |$\int _{{\mathbb{R}}} \omega _{a}(x)\, {\textrm{d}{x}} =1$| and |$\int _{{\mathbb{R}}} |{\omega ^{\prime}_{a}(x)}|\, {\textrm{d}{x}} =\frac{1}{a}.$| It is straight forward to see that |$\varPhi $| is symmetric, |$\varPhi _{x}=-\varPhi _{y}$| and |$\varPhi _{t}=-\varPhi _{s}$|⁠. Further,

Details of these estimates can be found in Holden & Risebro (2015, Sec. 3.3). We now list some of the notation to be used in the sequel. For |$\boldsymbol{U},\boldsymbol{V} \in ((L^{1} \cap L^{\infty })({\mathbb{R}}))^{N}$|⁠, all non-negative |$\phi \in C_{c}^{\infty }({\overline{Q}}_{T}),$| for each |$(t,x),(s,y)\in Q_{T}$| and for each |$k\in \{1,\dots ,N\},$| we define the following:

where

We now state and prove the Kuznetsov-type lemma for the coupled system of nonlocal conservation laws (1.1).

5. Error estimate

For every |$n \in \{0, \ldots , N_{T}\},~ i\in{\mathbb{Z}},~ k \in \{1, \ldots , N\}$|⁠, |$ (t,x)\in \overline{Q}_{T}$| and |$\alpha \in{\mathbb{R}},$| we define the following:

• |$\eta _{i}^{k,n}(\alpha ):=\left |U_{i}^{k,n}- \frac{\alpha }{\sigma _{i}^{k}}\right |$|⁠,

• |$p_{i}^{k,n}(\alpha ):=\mathop{\rm sgn} \left (\overline{U}_{i}^{k,n}-\alpha \right ) \left (f^{k}\left (\overline{U}_{i}^{k,n}\right )-f^{k}(\alpha )\right ) $|⁠,

• |$\mathcal{U}^{k}_{\varDelta }(t,x):= \nu ^{k}\left ((\boldsymbol{\mu }*\boldsymbol{U}_{\varDelta })^{k}(t,x)\right )$|⁠.

It is to be noted that owing to the discontinuous flux, |$\varLambda ^{k}_{\epsilon ,\epsilon _{0}}$| has nonpiecewise constant terms arising from |$\boldsymbol{\sigma }(x)$|⁠, unlike the classical cases considered in Holden & Risebro (2015). In comparison to a similar case considered in Ghoshal et al. (2022a), the PDE (1.2)–(1.3) consists of nonlocal coupled terms in addition, making it nontrivial. To address this issue, we first prove an estimate on |$\varLambda ^{k}_{\varDelta ,\epsilon ,\epsilon _{0}}$|⁠, an appropriate approximation of |$\varLambda ^{k}_{\epsilon ,\epsilon _{0}},$| (with |$\boldsymbol{\sigma }_{\varDelta }$| in place of |$\boldsymbol{\sigma }$|⁠) in Lemma 5.1, and then estimate the difference between |$\varLambda ^{k}_{\epsilon ,\epsilon _{0}}$| and |$\varLambda ^{k}_{\varDelta ,\epsilon ,\epsilon _{0}}.$| If the local part of the flux is homogeneous (i.e., |$\operatorname{TV}( s)=0$|⁠), then this step is not required and in fact, |$\varLambda ^{k}_{\epsilon ,\epsilon _{0}}=\varLambda ^{k}_{\varDelta ,\epsilon ,\epsilon _{0}}.$|

6.Numerical experiments

We now present some numerical experiments to illustrate the theory presented in the previous sections. We show the results for the Lax–Friedrichs scheme. The results obtained by the Godunov scheme are similar and are not shown here. Throughout the section, |$\theta =\theta _{x}=\theta _{y}$| is chosen to be |$1/3$| (satisfying the condition (3.4) for the choices of |$\sigma $| in the upcoming examples), and |$\lambda $| and |$\lambda _{x}=\lambda _{y}$| are chosen so as to satisfy the CFL condition (3.5).

6.1 One dimension

We consider the discontinuous generalization of the nonlocal Keyfitz–Kranzer system introduced in Aggarwal et al. (2015) in one space dimension, which reads as

We choose the kernel

where |$L$| is such that |$\int _{{\mathbb{R}}}\mu (x)\,{\textrm{d}{x}}=1,$| the velocity |$\nu (a,b)=(1-a^{2}-b^{2})^{3}$| and a |$\operatorname{BV}$| function

which admits infinitely many discontinuities at the points |$\{a_{n}\}_{n\in{\mathbb{N}}}.$| The system (6.1) fits into the setup of this article (cf. (1.1)) with

The local counterpart of (6.1), viz.

has not been studied yet in the literature and is of independent interest. However, the case |$\sigma \equiv 1$| is well understood, see, for example, Koley & Risebro (2013).

We present the numerical integration of (6.1) on the domain |$[0, \, 4]$| and the time interval |$[0, \, 0.3]$| with

Figure 1 displays the numerical approximations of (6.1) and (6.3) generated by the numerical scheme (3.2), with decreasing grid size |$\varDelta x$|⁠, starting with |$\varDelta x =0.00625$|⁠. It can be seen that the numerical scheme is able to capture both shocks and rarefactions well. Let |$e_{\varDelta x}(T)={\left \|\boldsymbol{U}_{\varDelta x}(T,\, \cdot \,)-\boldsymbol{U}_{\varDelta x/2}(T,\, \cdot \,)\right \|}_{(L^{1}({\mathbb{R}}))^{N}}.$| The convergence rate reads |$\alpha =\log _{2}(e_{\varDelta x}(T)/{e_{\varDelta x/2}(T)}).$| Errors and the convergence rates at time |$T=0.3$| obtained in the above experiment are recorded in Fig. 2. The convergence rates thus obtained obey the claims of Theorem 5.2. The results recorded in Fig. 2 show that the observed convergence rates exceed |$1.$|

Solution to (6.1) and (6.3) on the domain |$[0, \, 4]$| at times |$t = 0.00,\; 0.15,\;0.3$| with decreasing mesh size |$\varDelta x=0.00625\ ({\color{blue}{-\!-}}) $|⁠, |$\varDelta x=0.00625/2\ ({\color{red}{\text{-} \text{-} \text{-}\text{-}\text{-}\text{-}}}) $| and |$\varDelta x=0.00625/4 $|(- - -).

Fig. 1.

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Convergence rate |$\alpha $| for the numerical scheme (3.2) on the domain |$[0,\,4]$| at time |$T{=0.3}$| for the approximate solutions to the problem (6.1), (6.3)

Fig. 2.

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Convergence rate |$\alpha $| for the numerical scheme (3.2) on the domain |$[-1.1, \, 1.1]^{2}$| at time |$T=0.1$| for the approximate solutions to the problem (6.4)–(6.5).

Fig. 3.

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6.2 Two dimensions

As pointed out earlier, the analysis of the paper fits in multiple space dimensions with |$\boldsymbol{\sigma }\equiv 1$|⁠. To illustrate our results in two dimensions, we employ the two-dimensional nonlocal generalization of Keyfitz–Kranzer system, introduced in Aggarwal et al. (2015), given by

which is a system of two nonlocal conservation laws in two space dimensions. Here, we choose |$ \phi _{1} (a,b) = \sin (a^{2} + b^{2}), \phi _{2} (a,b) = \cos (a^{2}+b^{2})$| and

where |$L$| is chosen so that |$\iint _{{\mathbb{R}}^{2}} \mu (x,y)\, {\textrm{d}{x}}\, {\textrm{d}{y}}=1$|⁠. The system fits into the setup of this article (cf. (1.1)) with

and present the convergence rate analysis in this paper, with |$\eta =0.4.$| Errors and the convergence rates obtained in the experiment are recorded in Fig. 3. The convergence rates thus obtained obey the claims of Theorem 5.2. The results recorded in Figs 2–3 show that the observed convergence rates exceed |$0.5$|⁠.

Acknowledgements

The work was carried out during G.V.’s tenure of the ERCIM ‘Alain Bensoussan’ Fellowship Programme, and A.A.’s research visit at NTNU. It was supported in part by the project IMod—Partial differential equations, statistics and data: an interdisciplinary approach to data-based modelling, project number 325114, from the Research Council of Norway, and by A.A.’s Faculty Development Allowance, funded by IIM Indore. H.H. was also supported by the Swedish Research Council under grant no. 2021-06594 while in residence at Institut Mittag-Leffler in Djursholm, Sweden during the fall semester of 2023.

Conflict of interest

The authors have no conflicts of interest to declare.

References