Nonlocal Models with Convection Effects

Open Access
Author:
Huang, Zhan
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
May 18, 2015
Committee Members:
  • Qiang Du, Dissertation Advisor
  • Qiang Du, Committee Chair
  • Xiantao Li, Committee Member
  • Chun Liu, Committee Member
  • John Fricks, Committee Member
Keywords:
  • nonlocal
  • volume constrain
  • convection diffusion
  • conservation law
  • entropy solution
  • shock formation
  • numerical
  • convergence
Abstract:
This dissertation focuses on proposing and studying nonlocal models in the form of partial-integral-differential equations, as a generalization of local models, in the form of classical differential equations. It consists of two parts: the first part works on the volume-constraint problems associated with nonlocal convection-diffusion equations, the local counterpart part of which is the boundary-value problems associated with classical convection-diffusion differential equations. The second part deals with the initial-value problems of scalar nonlocal hyperbolic conservation laws. In both parts, we show that the nonlocal models we propose are reasonable extensions of their local counterparts, by demonstrating the consistency of their most important properties. In particular, both nonlocal problems enjoy maximum principle, just as their local counterparts do. Additionally, in the first part, we identify the underlying stochastic jump processes for the nonlocal convection-diffusion problems with Dirichlet volume-constraint, Robin volume-constraint, or on the whole space. The local counterpart of these jump processes are the Brownian motions with reflective or censored behavior near the domain boundary, or Brownian motions in the free space. Both Monte Carlo simulations and Finite Difference Methods are performed to observe the nonlocal solutions. In the second part, for nonlocal hyperbolic conservation laws, we prove the uniqueness and existence of the nonlocal entropy solution, and establish a condition on the kernel function, under which the nonlocal solution develop no shocks from smooth initial condition. Numerically, we propose a monotone scheme, the solution of which, as the horizon parameter is fixed and grid-size goes to zero, converges to the entropy solution of the nonlocal conservation law, while as horizon parameter and grid-size both vanish, converges to the entropy solution of the local conservation law. Numerical experiments are performed to further study the solution behaviors.