A new reconstruction of numerical fluxes for conservation laws using fuzzy operators

Abstract

This article develops a new hybrid flux‐limited scheme for a numerical solution of the hyperbolic conservation laws by applying fuzzy logic‐based operator functions. The construction of the proposed flux‐limiter is explored using a fuzzy modifier function, having a suitable intensity. The purpose of this article is to present an efficient finite volume flux‐limited technique, derived from an entirely different subject of fuzzy mathematics, for tackling hyperbolic partial differential equations. Several standard test cases in one and two dimensions are solved numerically for demonstrating the robustness of the proposed new hybrid flux‐limited scheme.

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

“…For this work, some important test cases are considered for evaluation [17] which are based on the advection equation and the Euler system of equations. This nonlinear Euler system is often used in the previous studies to evaluate the accuracy of shock-capturing frameworks [18,19,39].…”

“…Some prominent research work on these problems can be seen in the following literatures [26,29,43]. Also, several more applications of HRMs such as solving advection problems, Burger's problem, dam-break problems can be observed in these papers [15][16][17]39]. Toro recently has recently presented his work on the HLLC method showcasing various applications on real life models like blood flow in arteries, shallow water problems over varying topography, Euler system mainly [32].…”

“…">1.Test cases in one dimension: (a)The shock tube problem by G.A Sod, refer [5] (TP2). (b)The shock tube problem by Peter Lax, refer [9] (TP3). (c)Shock tube problem with stationary shock, refer [5] (TP4). 2.Test cases in two dimensions: (a)Advection problem with smooth profile [17] (TP1). (b)Riemann problem 1: It is the configuration 4 of the paper [40] (TP5). (c)Riemann problem 2: the configuration 5 of the paper, refer [40] (TP6). …”

“…In all other techniques for tackling hyperbolic PDEs, limiters are imposed at the start of each time unit; this is the conventional style of using limiters. During the last several years, a lot of important schemes for simulating compressible gas dynamics have been developed [17][18][19][20]. Concepts like monotonicity, total variation bound (TVB), total variation diminishing (TVD), essentially nonoscillatory (ENO), monotonicity preserving are often used to construct effective numerical techniques for such difficulties.…”

“…TVD methods are generally built by using flux/slope limiters, which are bound to total variation nonincreasing (TVNI) or simply known as the TVD limitations to restrict solution variance and prevent the occurrence of new local extrema. Currently, limiters [14][15][16][17] have become necessary for any form of simulation with complicated flow regimes consisting of discontinuous profiles, especially the high-order computational approximations on uniform/nonuniform grids. Apart from these, the traditional schemes of high order, such as second-order upwind, central differencing, quadratic and cubic upwind interpolation, etc.…”