RT Journal Article
T1 Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws
A1 Gómez Bueno, Irene
A1 Castro-Díaz, Manuel Jesús
A1 Parés-Madroñal, Carlos María
A1 Russo, Giovanni
K1 Matemáticas aplicadas
AB In some previous works, two of the authors introduced a technique to design high-order numerical methods for one-dimensional balance laws that preserve all their stationary solutions. The basis of these methods is a well-balanced reconstruction operator. Moreover, they introduced a procedure to modify any standard reconstruction operator, like MUSCL, ENO, CWENO, etc., in order to be well-balanced. This strategy involves a non-linear problem at every cell at every time step that consists in finding the stationary solution whose average is the given cell value. In a recent paper, a fully well-balanced method is presented where the non-linear problems to be solved in the reconstruction procedure are interpreted as control problems. The goal of this paper is to introduce a new technique to solve these local non-linear problems based on the application of the collocation RK methods. Special care is put to analyze the effects of computing the averages and the source terms using quadrature formulas. A general technique which allows us to deal with resonant problems is also introduced. To check the efficiency of the methods and their well-balance property, they have been applied to a number of tests, ranging from easy academic systems of balance laws consisting of Burgers equation with some non-linear source terms to the shallow water equations—without and with Manning friction—or Euler equations of gas dynamics with gravity effects.
PB MPDI
YR 2021
FD 2021
LK https://hdl.handle.net/10630/32674
UL https://hdl.handle.net/10630/32674
LA eng
NO Gómez-Bueno, I.; Díaz, M.J.C.; Parés, C.; Russo, G. Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws. Mathematics 2021, 9, 1799. https://doi.org/10.3390/math9151799
DS RIUMA. Repositorio Institucional de la Universidad de Málaga
RD 19 ene 2026