Singularly perturbed reaction–diffusion problems on a k‐star graph

Abstract

Singularly perturbed reaction-diffusion equations on a star graph (having k + 1 nodes and k edges) resulting in a system with k individual partial differential equations along the edges with coupling conditions at the common junction are presented. In the singular limit, as the diffusion parameter tends to zero, possibly individually along each edge, boundary layers may occur at the multiple nodes as well as at the simple nodes. Numerically, the proposed equations are solved using central finite difference schemes on properly extended Shishkin meshes.Error estimates are discussed and validated by solving a test problem on a graph with three edges (tripod). A more general graph problem with eight edges and three connecting nodes has also been solved numerically.

A comparative study of high‐resolution methods for nonlinear hyperbolic problems

“…In the paper, [26] the concept of HR limiting approaches in a topological sense has been geometrically expressed for homogeneous grids. The findings in this latest study [26, 30] provide significant evidence for the advantages of employing high‐order computational techniques over low‐order computational methods when simulating natural circulation phenomena, which has attracted the curiosity of many existing and advanced nuclear reactor designs. A very nice extensive review of almost all existing TVD approaches found in the literature based on the one‐step time space coupled unsteady (OTU) TVD formulation, the multi step time space divisional unsteady (MTU) TVD formulation and the semidiscrete steady state (SS) TVD formulation has been discussed in this work [18].…”

On the approximation of dam-break problems using a fuzzified HR-TVD scheme

“…Three specific problems occur in the numerical simulation of the shallow water models for various wave propagation phenomena over real domains [21] . The first issue is the approximation of the fronts or abrupt waves of fluid that could be interpreted as a numerically propagating discontinuity [19] , [22] , [23] . The second issue stems from sudden alterations in bathymetry (the measurement of the depth of water in water bodies).…”

Distributed optimal control problems driven by space-time fractional parabolic equations

“…Motivated by their work and the genuine interest in fractional Sturm-Liouville problems, outlined above, in this paper, we consider optimal control problems governed by space-time fractional parabolic equations of Sturm-Liouville type in an interval. The analysis and discretization of such problems on metric graphs, in the spirit of Leugering (2019, 2021), and Kumar and Leugering (2021) will be considered in a forthcoming paper. To the authors' best knowledge, even the well-posedness and the numerical study of optimal control problems for STFPEs of Sturm-Liouville type on intervals has not been investigated yet.…”