On approximation of local conservation laws by nonlocal conservation laws

Abstract:The potential systems and nonlocal conservation laws of Prandtl boundary layer equations on the surface of a sphere have been investigated. The multiplier approach yields two local conservation laws for the Prandtl boundary layer equations on the surface of a sphere. Two potential variables ψ and ϕ are introduced corresponding to first and second conservation law. Moreover, another potential variable p is introduced by considering the linear combination of both conservation laws. Two level one potential systems involving a single nonlocal variable ψ or ϕ are constructed. One level two potential system involving both nonlocal variables ψ and ϕ is established. The nonlocal variable p is utilised to derive a spectral potential system. The nonlocal conservation laws of Prandtl boundary layer equations on the surface of a sphere are derived by computing the local conservation laws of its potential systems. The nonlocal conservation laws are utilised to derive the further nonlocally related systems.

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Conservation Laws and Nonlocally Related Systems of Two-Dimensional Boundary Layer Models

Zeitschrift für Naturforschung A ◽

10.1515/zna-2017-0238 ◽

2017 ◽

Vol 72 (11) ◽

pp. 1031-1051

Author(s):

R. Naz ◽

A.F. Cheviakov

Keyword(s):

Boundary Layer ◽

Conservation Laws ◽

Direct Method ◽

Local Conservation ◽

Solution Sets ◽

Nonlocal Symmetries ◽

Pde Systems ◽

Nonlocal Conservation Laws ◽

Constitutive Parameters ◽

Nonlocally Related Systems

AbstractLocal conservation laws, potential systems, and nonlocal conservation laws are systematically computed for three-equilibrium two-component boundary layer models that describe different physical situations: a plate flow, a flow parallel to the axis of a circular cylinder, and a radial jet striking a planar wall. First, local conservation laws of each model are computed using the direct method. For each of the three boundary layer models, two local conservation laws are found. The corresponding potential variables are introduced, and nonlocally related potential systems and subsystems are formed. Then nonlocal conservation laws are sought, arising as local conservation laws of nonlocally related systems. For each of the three physical models, similar nonlocal conservation laws arise. Further nonlocal variables that lead to further potential systems are considered. Trees of nonlocally related systems are constructed; their structure coincides for all three models. The three boundary layer models considered in this work provide rich and interesting examples of the construction of trees of nonlocally related systems. In particular, the trees involve spectral potential systems depending on a parameter; these spectral potential systems lead to nonlocal conservation laws. Moreover, potential variables that are not locally related on solution sets of some potential systems become local functions of each other on solution sets of other systems. The point symmetry analysis shows that the plate and radial jet flow models possess infinite-dimensional Lie algebras of point symmetries, whereas the Lie algebra of point symmetries for the cylinder flow model is three-dimensional. The computation of nonlocal symmetries reveals none that arise for the original model equations, which is common for partial differential equations (PDE) systems without constitutive parameters or functions, but does reveal nonlocal symmetries for some nonlocally related PDE systems.

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A convergent finite volume method for the Kuramoto equation and related nonlocal conservation laws

IMA Journal of Numerical Analysis ◽

10.1093/imanum/dry074 ◽

2018 ◽

Vol 40 (1) ◽

pp. 405-421 ◽

Cited By ~ 1

Author(s):

N Chatterjee ◽

U S Fjordholm

Keyword(s):

Initial Data ◽

Conservation Laws ◽

Finite Volume Method ◽

Finite Volume ◽

Numerical Examples ◽

Continuity Equations ◽

Multiple Dimensions ◽

Nonlocal Conservation Laws ◽

Volume Method ◽

Singular Data

Abstract We derive and study a Lax–Friedrichs-type finite volume method for a large class of nonlocal continuity equations in multiple dimensions. We prove that the method converges weakly to the measure-valued solution and converges strongly if the initial data is of bounded variation. Several numerical examples for the kinetic Kuramoto equation are provided, demonstrating that the method works well for both regular and singular data.

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