Computing near-equilibrium solutions for hyperbolic balance laws on networks

Mantri, Yogiraj; Noelle, Sebastian (Thesis advisor); Herty, Michael (Thesis advisor)

Aachen : RWTH Aachen University (2021)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2021


In this thesis, we study numerical schemes for flows modeled by hyperbolic balance laws near equilibrium. For flows near equilibrium, the imbalance between numerical approximations of flux and source terms, may lead to spurious oscillations in the solution. In order to avoid this it is essential to design so called well-balanced schemes which can preserve equilibrium flow exactly and are stable for flows close to equilibrium. Our approach for obtaining well-balancing is based on the paper by Chertock, Herty, Ozcan [2018]. The key idea is to rewrite the balance law in conservative form to obtain equilibrium variables which remain constant. The high order reconstruction of these equilibrium variables can be done accurately at equilibrium which helps designing well-balanced schemes. We study the extension of the scheme to two dimensional flows of hyperbolic balance laws. For 2D flows, in addition to imbalance between flux and source terms, we also need to ensure a balance between the components of flux in both space dimensions. In 2D, we particularly develop a well-balanced scheme for balance laws having a fluid dynamics structure. For these equations, we are able to obtain equilibrium variables which are constant along each of the spatial direction and reconstruct these variables along the respective direction. We also study the extension of the scheme by Chertock to high order discontinuous Galerkin scheme for any general hyperbolic balance law in 1D with sufficient regularity in source term. The key step is to represent the equilibrium variables in the space of piece-wise polynomials of the DG scheme and formulate the scheme in terms of these variables. In the final part of the thesis, we study the extension of the well-balanced schemes for flows on networks of hyperbolic balance laws. For flows on network, the numerical treatment of coupling conditions at nodes of the network also plays a crucial role. We model these coupling conditions in terms of the equilibrium variables to obtain high order well-balanced approximations for flows near equilibrium at the node.