A new finite volume approach for transport models and related applications with balancing source terms

Author(s)

Abreu, E. ; Lambert, W. ; Perez, J. ; Santo, A.

• Document type: Journal article
• Source: MATHEMATICS AND COMPUTERS IN SIMULATION; v. 137, p. 27-pg., 2017-07-01.

Abstract

We develop a new finite volume scheme for numerically solving transport models associated with hyperbolic problems and balance laws. The numerical scheme is obtained via a Lagrangian Eulerian approach that retains the fundamental principle of conservation of the governing equations as it is linked to the classical finite volume framework. As features of the novel algorithm we highlight: the new scheme is locally conservative in balancing the flux and source term gradients and preserves a component wise structure at a discrete level for systems of equations. The novel approach is applied to several nontrivial examples to evidence that we are calculating the correct qualitatively good solutions with accurate resolution of small perturbations around the stationary solution. We discuss applications of the new method to classical and nonclassical nonlinear hyperbolic conservation and balance laws such as the classical inviscid Burgers equation, two-phase and three-phase flow problems in porous media as well as numerical experiments for nonlinear shallow water equations with friction terms. In addition, we consider the case of the source term which is discontinuous as a function of space x. We also extend the Lagrangian Eulerian framework to the two-dimensional scalar conservation law, along with pertinent numerical experiments to show the performance of the new method. (C) 2017 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved. (AU)

• FAPESP's process: 11/23628-0 - Systems of Balance Laws Problems in Fluid Dynamics in Porous Media: Mathematical Modeling and Numerical Approximation
• Grantee: Abel Alvarez Bustos
• Support Opportunities: Scholarships in Brazil - Doctorate
• FAPESP's process: 14/03204-9 - Computing qualitatively correct approximations of partial differential equations in porous media transport phenomena
• Grantee: Eduardo Cardoso de Abreu
• Support Opportunities: Regular Research Grants