Nonlocal Conservation Laws. A New Class of Monotonicity-Preserving Models

Abstract

We introduce a new class of nonlocal nonlinear conservation laws in one space dimension that allow for nonlocal interactions over a finite horizon. The proposed model, which we refer to as the nonlocal pair interaction model, inherits at the continuum level the unwinding feature of finite difference schemes for local hyperbolic conservation laws, so that the maximum principle and certain monotonicity properties hold and, consequently, the entropy inequalities are naturally satisfied. We establish a global-in-time well-posedness theory for these models which covers a broad class of initial data. Moreover, in the limit when the horizon parameter approaches zero, we are able to prove that our nonlocal model reduces to the conventional class of local hyperbolic conservation laws. Furthermore, we propose a numerical discretization method adapted to our nonlocal model, which relies on a monotone numerical flux and a uniform mesh, and we establish that these numerical solutions converge to a solution, providing as by-product both the existence theory for the nonlocal model and the convergence property relating the nonlocal regime and the asymptotic local regime.where the given numerical flux g is a Lipschitz continuous function consistent with the flux f = f (u), in the sense that g(u, u) = f (u). It is well known that if g = g(u, v) is a monotone flux, i.e, is non-decreasing in u and non-increasing in v, then the scheme (1.2) is Total Variation Diminishing (TVD) and enjoys the Maximum Principle [11,32]. By picking appropriate choice of g satisfying the

Numerical methods for nonlocal and fractional models

“…On the other hand, it is possible to introduce nonlocal conservation laws for which appropriate entropy conditions are automatically satisfied (Du, Huang and LeFloch 2017a), thus retaining important physical features in the modelling process that are missing from local models. The model of Du et al (2017a) also improves the model studied by Du, Kamm, Lehoucq and Parks (2012b). With AC discretization (Du and Huang 2017), the numerical convergence has been demonstrated with or without singular solutions.…”

Nonlocal Conservation Laws in Bounded Domains

“…Often, these models are set in the whole space R N , although the physics might require their stating in domains with boundaries. Two difficulties typically motivate this simplification: the rigorous treatment of boundaries and boundary data in conservation laws is technically quite demanding, see [7,16], and the very meaning of non local operators in the presence of a boundary is not straightforward, see [18,22] for recent different approaches.…”

Nonlinear semigroups for nonlocal conservation laws

“…Should the underlying model allow such considerations, this case corresponds to interactions that can be unfolded into subinteractions along the individual axes. A clear advantage of this example is the ease of numerical approximation of the integral as described in [16,Section 3.1]. If n = 1 and supp(ω 1 ) = (0, δ 1 ) instead, then again, we obtain the one-dimensional unidirectional nonlocal pair-interaction model of [16], as in the previous special case.…”

“…This case describes a natural multidirectional generalization of the one-dimensional unidirectional nonlocal pair-interaction model investigated in [16]. In fact, if n = 1 and supp(ω) ⊂ R + , the law (2) coincides with the latter.…”