Numerical Analysis of Conservation Laws Involving Non-local Terms

Abstract

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.