arXiv subjects

Nikhil Manoj

Publications and source records attributed to Nikhil Manoj.

3 recordsLinked to original sources

arXiv2025

A MUSCL-Hancock scheme for non-local conservation laws

In this article, we propose a MUSCL-Hancock-type second-order scheme for the discretization of a general class of non-local conservation laws and present its convergence analysis. The main difficulty in designing a MUSCL-Hancock-type scheme for non-local equations lies in the discretization of the convolution term, which we carefully formulate to ensure second-order accuracy and facilitate rigorous convergence analysis. We derive several essential estimates including $\mathrm{L}^\infty,$ bounded variation ($\mathrm{BV}$) and $\mathrm{L}^1$- Lipschitz continuity in time, which together with the Kolmogorov's compactness theorem yield the convergence of the approximate solutions to a weak solution. Further, by incorporating a mesh-dependent modification in the slope limiter, we establish convergence to the entropy solution. Numerical experiments are provided to validate the theoretical results and to demonstrate the improved accuracy of the proposed scheme over its first-order counterpart.

math.NA↗

arXiv2025

Analysis of a central MUSCL-type scheme for conservation laws with discontinuous flux

In this article, we propose a second-order central scheme of the Nessyahu-Tadmor-type for a class of scalar conservation laws with discontinuous flux and present its convergence analysis. Since solutions to problems with discontinuous flux generally do not belong to the space of bounded variation (BV), we employ the theory of compensated compactness to establish the convergence of approximate solutions. A major component of our analysis involves deriving the maximum principle and showing the $\mathrm{W}^{-1,2}_{\mathrm{loc}}$ compactness of a sequence constructed from approximate solutions. The latter is achieved through the derivation of several essential estimates on the approximate solutions. Furthermore, by incorporating a mesh-dependent correction term in the slope limiter, we show that the numerical solutions generated by the proposed second-order scheme converge to the entropy solution. Finally, we validate our theoretical results by presenting numerical examples.

math.NA↗

arXiv2024

A positivity preserving second-order scheme for multi-dimensional system of non-local conservation laws

Non-local systems of conservation laws play a crucial role in modeling flow mechanisms across various scenarios. The well-posedness of such problems is typically established by demonstrating the convergence of robust first-order schemes. However, achieving more accurate solutions necessitates the development of higher-order schemes. In this article, we present a fully discrete, second-order scheme for a general class of non-local conservation law systems in multiple spatial dimensions. The method employs a MUSCL-type spatial reconstruction coupled with Runge-Kutta time integration. The proposed scheme is proven to preserve positivity in all the unknowns and exhibits L-infinity stability. Numerical experiments conducted on both the non-local scalar and system cases illustrate the8 importance of second-order scheme when compared to its first-order counterpart.

math.NA↗