On some fast well-balanced first order solvers for nonconservative systems. Díaz, Manuel J., C., Pardo, A., Parés, C., & Toro, E. MATHEMATICS OF COMPUTATION, 79(271):1427–1472, 2010. abstract bibtex The goal of this article is to design robust and simple first order explicit solvers for one-dimensional nonconservative hyperbolic systems. These solvers are intended to be used as the basis for higher order methods for one or multidimensional problems. The starting point for the development of these solvers is the general definition of a Roe linearization introduced by Toumi in 1992 based on the use of a family of paths. Using this concept, Roe methods can be extended to nonconservative systems. These methods have good wellbalanced and robustness properties, but they have also some drawbacks: in particular, their implementation requires the explicit knowledge of the eigenstructure of the intermediate matrices. Our goal here is to design numerical methods based on a Roe linearization which overcome this drawback. The idea is to split the Roe matrices into two parts which are used to calculate the contributions at the cells to the right and to the left, respectively. This strategy is used to generate two different one-parameter families of schemes which contain, as particular cases, some generalizations to nonconservative systems of the well-known Lax-Friedrichs, Lax-Wendroff, FORCE, and GFORCE schemes. Some numerical experiments are presented to compare the behaviors of the schemes introduced here with Roe methods.
@Article{castro2010force,
author = {Castro D{\'i}az, Manuel J. and Pardo, Alberto and Par{\'e}s, Carlos and Toro, E.-F.},
title = {{O}n some fast well-balanced first order solvers for nonconservative systems.},
journal = {MATHEMATICS OF COMPUTATION},
year = {2010},
volume = {79},
number = {271},
pages = {1427–1472},
abstract = {The goal of this article is to design robust and simple first order
explicit solvers for one-dimensional nonconservative hyperbolic systems. These
solvers are intended to be used as the basis for higher order methods for one or
multidimensional problems. The starting point for the development of these
solvers is the general definition of a Roe linearization introduced by Toumi in
1992 based on the use of a family of paths. Using this concept, Roe methods
can be extended to nonconservative systems. These methods have good wellbalanced
and robustness properties, but they have also some drawbacks: in
particular, their implementation requires the explicit knowledge of the eigenstructure
of the intermediate matrices. Our goal here is to design numerical
methods based on a Roe linearization which overcome this drawback. The
idea is to split the Roe matrices into two parts which are used to calculate
the contributions at the cells to the right and to the left, respectively. This
strategy is used to generate two different one-parameter families of schemes
which contain, as particular cases, some generalizations to nonconservative systems
of the well-known Lax-Friedrichs, Lax-Wendroff, FORCE, and GFORCE
schemes. Some numerical experiments are presented to compare the behaviors
of the schemes introduced here with Roe methods.},
}
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