A Class of Lagrangian¿Eulerian Shock-Capturing Schemes for First-Order Hyperbolic Problems with Forcing Terms

• Type: Article in International Scientific Journal
• Date: 2021

Authors

Abreu, E (Author) Other; Vítor Matos (Author) FEP; Pérez, J (Author) Other; Rodríguez Bermúdez, P (Author) Other

Journal

Title: Journal of Scientific Computing

Vol. 86

ISSN: 0885-7474

Publisher: Springer Nature

Indexing

ISI Web of Knowledge - 13 Citations

Scopus - 15 Citations

Other information

Authenticus ID: P-00T-9FY

DOI: 10.1007/s10915-020-01392-w

Abstract (EN): In this work, we develop an improved shock-capturing and high-resolution Lagrangian¿Eulerian method for hyperbolic systems and balance laws. This is a new method to deal with discontinuous flux and complicated source terms having concentrations for a wide range of applications science and engineering, namely, 1D shallow-water equations, sedimentation processes, Geophysical flows in 2D, N-Wave models, and Riccati-type problems with forcing terms. We also include numerical simulations of a 1D two-phase flow model in porous media, with gravity and a nontrivial singular ¿-source term representing an injection point. Moreover, we present approximate solutions for 2D nonlinear systems (Compressible Euler Flows and Shallow-Water Equations) for distinct benchmark configurations available in the literature aiming to present convincing and robust numerical results. In addition, for the linear advection model in 1D and for a smooth solution of the nonlinear Burgers¿ problem, second order approximations were obtained. We also present a high-resolution approximation of the nonlinear non-convex Buckley¿Leverett problem. Based on the work of A. Harten, we derive a convergent Lagrangian¿Eulerian scheme that is total variation diminishing and second-order accurate, away from local extrema and discontinuous data. Additionally, using a suitable Kru¸kov¿s entropy definition, introduced by K. H. Karlsen and J. D. Towers, we can verify that our improved Lagrangian¿Eulerian scheme converges to the unique entropy solution for conservation laws with a discontinuous space-time dependent flux. A key hallmark of our method is the dynamic tracking forward of the no-flow curves, which are locally conservative and preserve the natural setting of weak entropic solutions related to hyperbolic problems that are not reversible systems in general. In the end, we have a general procedure to construct a class of Lagrangian¿Eulerian schemes to deal with hyperbolic problems with or without forcing terms. The proposed scheme is free of Riemann problem solutions and no adaptive space-time discretizations are needed. The numerical experiments verify the efficiency and accuracy of our new Lagrangian¿Eulerian method. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.

Language: English

Type (Professor's evaluation): Scientific