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On entropy stable finite volume methods for scalar conservation laws

Kristoffersson, Loke LU (2023) In Bachelor's Theses in Mathematical Sciences NUMK11 20231
Centre for Mathematical Sciences
Mathematics (Faculty of Sciences)
Abstract
In this thesis, entropy conservative and entropy stable finite volume methods for one-dimensional scalar conservation laws are studied. The need for weak solutions is shown and entropy is explored to obtain uniqueness. Using discrete analogues of properties of entropy solutions, entropy stable finite volume methods are constructed and implemented to solve a number of scalar conservation laws, in particular the transport equation, Burgers' equation, a traffic flow problem and a model for enhanced oil recovery. The solutions obtained are accurate leading up to shocks, at which point the solutions break down. The viscous approximation to the scalar conservation law is solved to further understand the behaviour of solutions after shocks. It is... (More)
In this thesis, entropy conservative and entropy stable finite volume methods for one-dimensional scalar conservation laws are studied. The need for weak solutions is shown and entropy is explored to obtain uniqueness. Using discrete analogues of properties of entropy solutions, entropy stable finite volume methods are constructed and implemented to solve a number of scalar conservation laws, in particular the transport equation, Burgers' equation, a traffic flow problem and a model for enhanced oil recovery. The solutions obtained are accurate leading up to shocks, at which point the solutions break down. The viscous approximation to the scalar conservation law is solved to further understand the behaviour of solutions after shocks. It is theoretically shown and experimentally validated that the finite volume schemes are stable. Second order convergence is observed for all test problems with several different choices of numerical fluxes. The experiments indicate that numerical fluxes conserving or dissipating the squared entropy have the smallest errors. (Less)
Please use this url to cite or link to this publication:
author
Kristoffersson, Loke LU
supervisor
organization
course
NUMK11 20231
year
type
M2 - Bachelor Degree
subject
keywords
Conservation laws, Finite volume methods
publication/series
Bachelor's Theses in Mathematical Sciences
report number
LUNFNA-4049-2023
ISSN
1654-6229
other publication id
2023:K21
language
English
id
9131154
date added to LUP
2024-04-15 16:44:40
date last changed
2024-04-15 16:44:40
@misc{9131154,
  abstract     = {{In this thesis, entropy conservative and entropy stable finite volume methods for one-dimensional scalar conservation laws are studied. The need for weak solutions is shown and entropy is explored to obtain uniqueness. Using discrete analogues of properties of entropy solutions, entropy stable finite volume methods are constructed and implemented to solve a number of scalar conservation laws, in particular the transport equation, Burgers' equation, a traffic flow problem and a model for enhanced oil recovery. The solutions obtained are accurate leading up to shocks, at which point the solutions break down. The viscous approximation to the scalar conservation law is solved to further understand the behaviour of solutions after shocks. It is theoretically shown and experimentally validated that the finite volume schemes are stable. Second order convergence is observed for all test problems with several different choices of numerical fluxes. The experiments indicate that numerical fluxes conserving or dissipating the squared entropy have the smallest errors.}},
  author       = {{Kristoffersson, Loke}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor's Theses in Mathematical Sciences}},
  title        = {{On entropy stable finite volume methods for scalar conservation laws}},
  year         = {{2023}},
}