Journal
SIAM JOURNAL ON APPLIED MATHEMATICS
Volume 72, Issue 1, Pages 464-487Publisher
SIAM PUBLICATIONS
DOI: 10.1137/110833233
Keywords
conservation laws; advection; nonlocal operator; integral operator; Burgers equation; peridynamics
Categories
Funding
- U.S. Department of Energy [DE-SC0005346, DE-AC04-94-AL85000]
- NSF [DMS-1016073]
- Sandia National Laboratories
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1016073] Funding Source: National Science Foundation
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We describe an approach to nonlocal, nonlinear advection in one dimension that extends the usual pointwise concepts to account for nonlocal contributions to the flux. The spatially nonlocal operators we consider do not involve derivatives. Instead, the spatial operator involves an integral that, in a distributional sense, reduces to a conventional nonlinear advective operator. In particular, we examine a nonlocal inviscid Burgers equation, which gives a basic form with which to characterize properties associated with well-posedness, and to examine numerical results for specific cases. We describe the connection to a nonlocal viscous regularization, which mimics the viscous Burgers equation in an appropriate limit. We present numerical results that compare the behavior of the nonlocal Burgers formulation to the standard local case. The developments presented in this paper form the preliminary building blocks upon which to build a theory of nonlocal advection phenomena consistent within the peridynamic theory of continuum mechanics.
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