We are very glad to announce the Pre-Synopsis Seminar Talk by Mr. Nikhil Manoj.
The details are as follows:
Name of the Student Speaker : Mr. Nikhil Manoj
Date: 30/05/2025
Time: 4 pm
Venue: PSB1207
Please find below the title and abstract of the talk.
Title: On the discretization of conservation laws with non-local and discontinuous fluxes.
Abstract: In this talk, we focus on the analysis of numerical discretizations for two classes of conservation laws with spatially dependent flux: (i) conservation laws with non-local flux, where the flux at a point depends on the convolution of the unknown with a given kernel function, and (ii) conservation laws with discontinuous flux, where the flux function changes abruptly across interfaces in space. While robust first-order finite volume methods exist for these problems, they tend to be overly diffusive, necessitating the development of second-and higher-order schemes supported by rigorous theoretical analysis. In the first part, we develop second-order schemes for certain classes of one-dimensional non-local conservation laws. The convergence of these schemes is shown by deriving a series of a priori estimates including $\mathrm{L}^\infty$ and bounded variation bounds, followed by a mesh-dependent modification to ensure entropy convergence. Next, for multi-dimensional non-local systems, we design a second-order method that is provably positivity-preserving and $\mathrm{L}^\infty$ stable. In the final part, for a general class of conservation laws with discontinuous flux, we propose a single-stage second-order central scheme and establish its convergence using the theory of compensated compactness. The effectiveness and accuracy of the proposed schemes are demonstrated through a series of numerical experiments in comparison with existing first-order methods.