A finite volume method for nonlocal competition-mutation equations with a gradient flow structure

2017 ◽  
Vol 51 (4) ◽  
pp. 1223-1243 ◽  
Author(s):  
Wenli Cai ◽  
Hailiang Liu
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Flow Structure ◽  
Gradient Flow ◽  
Volume Method

scholarly journals A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure

2014 ◽  
Vol 17 (1) ◽  
pp. 233-258 ◽  
Author(s):  
José A. Carrillo ◽  
Alina Chertock ◽  
Yanghong Huang
Keyword(s):  
Finite Volume ◽  
Flow Structure ◽  
Gradient Flow ◽  
Finite Volume Scheme ◽  
Stationary States ◽  
Nonlocal Equations ◽  
Long Time ◽  
Long Time Asymptotics ◽  
Volume Method ◽  
Accurate Computations

AbstractWe propose a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with a gradient flow structure. These properties allow for accurate computations of stationary states and long-time asymptotics demonstrated by suitably chosen test cases in which these features of the scheme are essential. The proposed scheme is able to cope with non-smooth stationary states, different time scales including metastability, as well as concentrations and self-similar behavior induced by singular nonlocal kernels. We use the scheme to explore properties of these equations beyond their present theoretical knowledge.

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

2018 ◽  
Vol 40 (1) ◽  
pp. 405-421 ◽  
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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