arXiv subjects

Aekta Aggarwal

Publications and source records attributed to Aekta Aggarwal.

8 recordsLinked to original sources

arXiv2026

Stability estimates for systems of nonlocal balance laws with memory

In this work, we investigate entropy solutions for a class of systems of nonlocal {balance laws in which the convective flux and the source involves terms where the state variable convolved with kernels} in both spatial and temporal variables. This formulation captures the dependence of the flux on the solution within its spatial neighborhood (spatial nonlocality) as well as on its past states (temporal nonlocality), thereby incorporating memory effects. The resulting systems are coupled through these nonlocal interactions. We establish stability estimates for entropy solutions with respect to perturbations in the flux, the spatial and temporal kernels, and the initial data for the corresponding initial value problems. Finally, we present numerical experiments to illustrate the theoretical results and to highlight the influence of memory and source terms on the solution dynamics.

math.AP↗

arXiv2026

Systems of Nonlocal Conservation Laws with Memory and Their Zero Retention Limit

We study the entropy solution for a class of systems of nonlocal conservation laws in which the convective flux is convoluted with a kernel in both spatial and temporal variables. This formulation models the flux dependence on the solution within its spatial neighbourhood (nonlocal in space) as well as on prior states in time (nonlocal in time), thereby incorporating memory effects. In addition, employing a convergent finite volume approximation, the existence of the entropy solution is discussed. The uniqueness of such entropy solutions is also established. In addition, we analyze the asymptotic behaviour of the solutions as the support of the temporal convolution kernel shrinks, demonstrating the "memory-to-memoryless" effect and convergence to the entropy solution of the corresponding nonlocal conservation law without memory (i.e., nonlocal only in space). Convergence rate estimates are derived. In addition, the proposed numerical approximations are shown to be asymptotically compatible with this passage to the memoryless limit by deriving the corresponding asymptotic convergence rate estimates. The analysis is carried out in a very general setting, without imposing any geometric restrictions such as the convexity of the spatial and temporal convolution kernels, unlike the existing literature on the asymptotic analysis of nonlocal-in-space only conservation laws. To the best of our knowledge, this provides the first convergence and asymptotic analysis for finite volume schemes applied to nonlocal conservation laws with memory. Numerical experiments are included to illustrate the theory.

math.NA↗

arXiv2023

Error Estimates for Systems of Nonlocal Balance Laws Modeling Dense Multilane Vehicular Traffic

We discuss a class of coupled systems of nonlocal nonlinear balance laws modeling multilane traffic, with the nonlocality present in both convective and source terms. The uniqueness and existence of the entropy solution are proven via doubling of the variables arguments and convergent finite volume approximations, respectively. The primary goal is to establish that the finite volume numerical approximations of the system converge to the unique entropy solution at a rate of $\sqrt{\Delta t}$, even when using relatively less regular one-sided kernels, compared to the globally smooth kernels analyzed in [Num. Math., 156(1):237-271, 2024] and [IMA J. Numer. Anal., 44(6):3354-3392, 2024]. The applicability of the proven theory to a general class of systems of nonlocal balance laws coupled strongly through the convective part and weakly through the source part, is indicated. As the support of the kernel tends to zero, the convergence of the entropy solutions of the proposed model to its local counterparts [SIAM J. Math. Anal., 51: 3694--3713, 2019] is also discussed. Numerical simulations illustrating the behavior of the entropy solutions of the coupled nonlocal systems are also shown.

math.NA↗

arXiv2023

A well-balanced second-order finite volume approximation for a coupled system of granular flow

A well-balanced second-order finite volume scheme is proposed and analyzed for a 2 X 2 system of non-linear partial differential equations which describes the dynamics of growing sandpiles created by a vertical source on a flat, bounded rectangular table in multiple dimensions. To derive a second-order scheme, we combine a MUSCL type spatial reconstruction with strong stability preserving Runge-Kutta time stepping method. The resulting scheme is ensured to be well-balanced through a modified limiting approach that allows the scheme to reduce to well-balanced first-order scheme near the steady state while maintaining the second-order accuracy away from it. The well-balanced property of the scheme is proven analytically in one dimension and demonstrated numerically in two dimensions. Additionally, numerical experiments reveal that the second-order scheme reduces finite time oscillations, takes fewer time iterations for achieving the steady state and gives sharper resolutions of the physical structure of the sandpile, as compared to the existing first-order schemes of the literature.

math.NA↗

arXiv2023

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

This article deals with the error estimates for numerical approximations of the entropy solutions of coupled systems of nonlocal hyperbolic conservation laws. The systems can be strongly coupled through the nonlocal coefficient present in the convection term. A fairly general class of fluxes is being considered, where the local part of the flux can be discontinuous at infinitely many points, with possible accumulation points. The aims of the paper are threefold: 1. Establishing existence of entropy solutions with rough local flux for such systems, by deriving a uniform BV bound on the numerical approximations; 2. Deriving a general Kuznetsov-type lemma (and hence uniqueness) for such systems with both smooth and rough local fluxes; 3. Proving the convergence rate of the finite volume approximations to the entropy solutions of the system as $1/2$ and $1/3$, with homogeneous (in any dimension) and rough local parts (in one dimension), respectively. Numerical experiments are included to illustrate the convergence rates.

math.NA↗

arXiv2023

Convergence of the numerical approximations and well-posedness: Nonlocal conservation laws with rough flux

We study a class of nonlinear nonlocal conservation laws with discontinuous flux, modeling crowd dynamics and traffic flow, without any additional conditions on finiteness/discreteness of the set of discontinuities or on the monotonicity of the kernel/the discontinuous coefficient. Strong compactness of the Godunov and Lax-Friedrichs type approximations is proved, providing the existence of entropy solutions. A proof of the uniqueness of the adapted entropy solutions is provided, establishing the convergence of the entire sequence of finite volume approximations to the adapted entropy solution. As per the current literature, this is the first well-posedness result for the aforesaid class and connects the theory of nonlocal conservation laws (with discontinuous flux), with its local counterpart in a generic setup. Some numerical examples are presented to display the performance of the schemes and explore the limiting behavior of these nonlocal conservation laws to their local counterparts.

math.NA↗

arXiv2023

On the accuracy of the finite volume approximations to nonlocal conservation laws

In this article, we discuss the error analysis for a certain class of monotone finite volume schemes approximating nonlocal scalar conservation laws, modeling traffic flow and crowd dynamics, without any additional assumptions on monotonicity or linearity of the kernel $μ$ or the flux $f$. We first prove a novel Kuznetsov-type lemma for this class of PDEs and thereby show that the finite volume approximations converge to the entropy solution at the rate of $\sqrt{Δt}$ in $L^1(\mathbb{R})$. To the best of our knowledge, this is the first proof of any type of convergence rate for this class of conservation laws. We also present numerical experiments to illustrate this result.

math.NA↗

arXiv2021

Positivity--preserving numerical scheme for hyperbolic systems with $δ\,-$ shock solutions and its convergence analysis

Godunov type numerical schemes for the class of hyperbolic systems, admitting non-classical $δ-$ shocks are proposed. It is shown that the numerical approximations converge to the solution and preserve the physical properties of the system such as positive density and bounded velocity. The scheme has been extended to positivity preserving and velocity bound preserving second-order accurate scheme by using appropriate slope limiters. The numerical results are compared with the existing the literature and the scheme is shown to capture the solution efficiently. The paper presents a hyperbolic system, for which an entropy satisfying scheme is constructed through an appropriate decoupling of the system into two scalar conservation laws with discontinuous flux.

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