Weak asymptotic method for scalar conservation laws with nonlocal flux: numerical analysis and applications

Abstract

The main purpose of the research project is to investigate conditions to guarantee existence and uniqueness of -- analitically and numerically of unique Kruzhkov-type solution in the Riemann and Cauchy problems associated with general nonlocal hyperbolic-transport conservation laws with potential applications in Sustainable Transportation and Smart Cities with impact in Global Climate Changes in close connection to applied mathematics and numerical analysis. The interplay between local and nonlocal conservation laws have attracted much interest in the past few years motivated by either the need of novel analysis and techniques or by demand to applied sciences. Nonlocal models appear in fluid mechanics and geophysical problems and relatedflows given by nonlocal balance laws and fractional diffusion models, for instance, porous medium equation, diffusion-reaction, surface quasi-geostrophic equation, just to name a few important problems. For instance, very recently in, the authors considered a class of convection-diffusion equations with memory effects. Moreover, nonlocality and nonlocal models with finite time wave break-down occur very often in modeling problems in several applied sciences, Cucker-Smale models and related problems, optimal control problems, kinematic sedimentation process, traffic flow problems, Wave breaking, dispersive water waves, collective motion of biological cells, Models of relaxing medium and pressureless gas dynamics. Therefore, it is clear that nonlocal1D and Multi-D models are fundamental in the study of several flow problems in pure and applied sciences and, thus, numerical analysis with applications is a relevant foundation in applied mathematics for such study, covering: 1) To investigate the convergence of the Weak asymptotic method for 1D nonlocal scalar conservation laws with source term. 2) To investigate numerically conditions to guarantee existence and uniqueness of Weak asymptotic solutions to obtain a family of approximate solution to 1D nonlocal scalar conservation laws where this family satisfies the Kruzhkov entropy inequalities and has someregularity properties in the sense of bounded variations functions or the numerical scheme has total variation nonincreasing estimates. 3) To investigate numerically conditions to guarantee existence and uniqueness Weak asymptotic solutions for class of multidimensional nonlocal conservation laws. 4) To construct and implement computationally a new semidiscrete Lagrangian-Eulerian numerical method for the treatment of 1D and multidimensional nonlocal conservation laws with nonlocal flux with potential applications. In this context, the study of weak asymptotic method for scalar conservation laws with nonlocal flux via numerical analysis and applications is of relevance for improve comprehension on the applied mathematics involving potential of applications to sustainable transportation, smart cities and also related nonlocal hyperbolic-transport problems. (AU)