Numerical Simulation And Analysis On Linear Nonlocal Conservation Laws With Variable Horizon

This paper mainly focuses on a class of linear nonlocal conservation laws with variable horizon,in which,we use upwinding nonlocal derivative.So that the nonlocal model can maintain good properties such as the maximum principle.The appearance of variable horizon makes our model helpful to the research on coupling problem of local model and nonlocal model.This paper is divided into the following parts:Firstly,we consider a quadrature-based finite difference discretization of onedimensional scalar linear nonlocal conservation laws.The range of nonlocal interactions is allowed to vary in the spatial domain.We are particularly concerned with the convergence of the discrete approximation both in the nonlocal setting and in the local limit.We present the first complete proof of the convergence of numerical discretization to both the nonlocal regime and the local limit of all feasible kernels,which in particular,establishes the asymptotically compatibility of the numerical scheme,guaranteed the reliability and robustness of the numerical scheme.We also present numerical results to demonstrate the effect of the variable horizon on the wave propagation described by the nonlocal model.Secondly,we study the propagation of singularities in solutions of linear convection equations with spatially heterogeneous nonlocal interactions.A spatially varying nonlocal horizon parameter is adopted in the model,which measures the range of nonlocal interactions.Via heterogeneous localization,this can lead to the seamless coupling of the local and nonlocal models.We are interested in understanding the impact on singularity propagation due to the heterogeneities of nonlocal horizon and the local and nonlocal transition.We first analytically derive equations to characterize the propagation of different types of singularities for various forms of nonlocal horizon parameters in the nonlocal regime.We then use the derived equations to calculate the development of singularities of the solution and its derivative.Meanwhile,asymptotically compatible schemes are employed to discretize the equations and carry out numerical simulations to verify the correctness of the theoretical analysis results.Through the analysis and comparison of several possible situations,a more complete conclusion is given on the propagation patterns of singularity in different scenarios.Finally,we consider the discontinuous Galerkin scheme of one-dimensional scalar linear nonlocal conservation laws.Since nonlocal models are often used to deal with problems related to singularity or discontinuity,it is natural to use discontinuous Galerkin methods for numerical simulation of nonlocal models.We present the discontinuous Galerkin scheme of nonlocal model with constant horizon,and prove the L2-stability in both local limit and nonlocal setting.We also provide the optimal error estimates in nonlocal setting and the suboptimal error estimates in the local limit setting.That is,the asymptotically compatible property of the discontinuous Galerkin scheme is obtained,which shows that the numerical solution obtained by the discontinuous Galerkin scheme is reliable even when the horizon approaches zero in proportion to the grid size.The numerical results are consistent with the theoretical analysis results.For the nonlocal model with variable horizon,we also provide the discontinuous Galerkin scheme,and numerically verify its convergence order in both the nonlocal regime and the local limit.