Well-posedness of the initial value problem for the Ostrovsky-Hunter equation with spatially dependent flux

In this paper we study the Ostrovsky-Hunter equation for the case where the flux function $f(x, u)$ may depend on the spatial variable with certain smoothness. Our main results are that if the flux function is smooth enough (specified later), then there exists a unique entropy solution. To show the existence, after proving some a priori estimates we have used the method of compensated compactness and to prove the uniqueness we have employed the method of doubling of variables

A convergent finite volume method for the Kuramoto equation and related non-local conservation laws

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 both for regular and singular data

Numerical Analysis of Conservation Laws Involving Non-local Terms

A particular class of Partial differential Equations (PDEs) is the hyperbolic conservation laws which play an instrumental role in numerous real life applications such as synchronization of cardiac pacemakers, traffic flow models, shallow water waves in rotating fluid and so on. In this thesis, I designed and investigated numerical methods which approximate the solutions of these kind of models, which often involve a non-local term as a source term or within the flux term, making the problem more involving. In my doctoral dissertation, I have used finite volume method to approximate the "exact" PDEs numerically, so that computer simulations can be performed to check if the numerical methods developed, actually lead to a solution which can be "visualized". To be precise, the results obtained in my thesis involve finite volume methods, which approximate conservation laws, taking into account the effect of nonlocal term present as in the source/sink term or as in the flux term of the conservation laws. In the thesis theoretical convergence has been proved and the schemes are verified using suitable numerical examples. Also, the results include theoretical proof of convergence for a second order numerical method, namely TeCNO scheme, in multiple spatial dimension which satisfies an entropy stability relation

A convergent finite volume method for the Kuramoto equation and related non-local conservation laws

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

Convergence of second-order, entropy stable methods for multi-dimensional conservation laws

High-order accurate, entropy stable numerical methods for hyperbolic conservation laws have attracted much interest over the last decade, but only a few rigorous convergence results are available, particularly in multiple space dimensions. In this paper we show how the entropy stability of one such method, which is semi-discrete in time, yields a (weak) bound on oscillations. Under the assumption of L∞-boundedness of the approximations we use compensated compactness to prove convergence to a weak solution satisfying at least one entropy condition

Well-posedness of the Initial Value Problem for the Ostrovsky–Hunter Equation with Spatially Dependent Flux

In this paper we study the Ostrovsky–Hunter equation for the case where the flux function f(x, u) may depend on the spatial variable with certain smoothness. Our main results are that if the flux function is smooth enough (namely fx(x, u) is uniformly Lipschitz locally in u and fu(x, u) is uniformly bounded), then there exists a unique entropy solution. To show the existence, after proving some a priori estimates we have used the method of compensated compactness and to prove the uniqueness we have employed the method of doubling of variables