Advanced search
Start date
Betweenand
Related content
(Reference retrieved automatically from Web of Science through information on FAPESP grant and its corresponding number as mentioned in the publication by the authors.)

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

Full text
Author(s):
Abreu, E. [1] ; Matos, V. [2, 3] ; Perez, J. [4] ; Rodriguez-Bermudez, P. [5]
Total Authors: 4
Affiliation:
[1] Univ Estadual Campinas, IMECC Dept Appl Math, Campinas, SP - Brazil
[2] Univ Porto, Ctr Matemat, Fac Econ, Porto - Portugal
[3] Univ Nacl Timor Lorosae, Dili - Timor-Leste
[4] ITM Inst Univ, Medellin - Colombia
[5] Fluminense Fed Univ, Volta Redonda, RJ - Brazil
Total Affiliations: 5
Document type: Journal article
Source: JOURNAL OF SCIENTIFIC COMPUTING; v. 86, n. 1 JAN 2 2021.
Web of Science Citations: 1
Abstract

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 delta -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 Krukov'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. (AU)

FAPESP's process: 19/20991-8 - Study of partial differential algebraic equations of hyperbolic-parabolic dominance with relaxation: theory, numerics and applications
Grantee:Eduardo Cardoso de Abreu
Support Opportunities: Research Grants - Visiting Researcher Grant - Brazil