scholarly journals Convergence of the Finite Volume method for scalar conservation laws with multiplicative noise: an approach by kinetic formulation

2019 ◽  
Vol 8 (2) ◽  
pp. 265-310 ◽  
Author(s):  
Sylvain Dotti ◽  
Julien Vovelle
Keyword(s):  
Conservation Laws ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Multiplicative Noise ◽  
Scalar Conservation Laws ◽  
Kinetic Formulation ◽  
Scalar Conservation ◽  
Volume Method

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.

A note on strong solutions to the variational kinetic equation for scalar conservation laws

2014 ◽  
Vol 11 (03) ◽  
pp. 621-632
Author(s):  
Misha Perepelitsa
Keyword(s):  
Conservation Laws ◽  
Differential Inclusion ◽  
Maximal Monotone ◽  
Strong Solutions ◽  
Indicator Function ◽  
Closed Convex Cone ◽  
Scalar Conservation Laws ◽  
Kinetic Formulation ◽  
Degeneracy Condition ◽  
Scalar Conservation

We consider a variational kinetic formulation for weak, entropy solutions of scalar conservation laws due to Brenier. The solutions in this formulation are represented by a kinetic density function Y that solves a differential inclusion ∂tY ∈ -A(Y) = -∂vf ⋅ ∇xY -∂ IK(Y), where IK is the indicator function of a closed, convex cone K. Under a certain "non-degeneracy" condition we determine a maximal monotone extension of A and use it to prove the existence of strong and weak solutions of the differential inclusion for a general, possibly degenerate, flux ∂vf(v). Furthermore, we discuss several properties of strong solutions.

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