## Numerical Methods for Singular Differential Equations Arising from Steady Flows in Channels and Ducts

### BibTeX

@MISC{Lemos_numericalmethods,

author = {A. C. Lemos},

title = {Numerical Methods for Singular Differential Equations Arising from Steady Flows in Channels and Ducts},

year = {}

}

### OpenURL

### Abstract

We study ordinary nonlinear differential equations which arise from steady nonlinear conservation laws with source terms. Two examples of conservation laws which lead to these equations are the Saint-Venant and the Euler equations. In each case there is a reduction to a scalar equation and we use the ideas of upwinding and discretisation of source terms to devise methods for the solution. Numerical results are presented with both the Engquist-Osher and the Roe scheme with different strategies for discretising the source terms based on balance ideas. Acknowledgements Firstly, I would like to express my gratitude to Professor Mike Baines. His supervision, support and patience were constant throughout this work and encouraged me to go on. My thanks go also to Professor Nancy Nichols. Her supervision and advice were very helpful. As a team, their supervision complemented each other and I benefited from their knowledge and teaching. I am grateful to my sponsors in Portugal, Funda ção para a Ciência e a Tecnologia (grant PRAXIS XXI/BD/15905/98 from the Subprograma Ciência e Tecnologia do 2o Quadro Comunitário de Apoio, andtheEscola Superior de Tecnologia e Gestão from the Instituto Politécnico de Leiria, who made this project viable. I wish to thank the help of staff and colleagues in the Mathematics Department in Reading who always made me feel welcome. Studying in Department of Mathematics of the University of Reading afforded the opportunity to learn with very good teachers and to meet colleagues and fellow researchers. In Reading, I met new friends and their friendship and support were very important in making me feel less lonely. I would like to thank especially Jessica, Ana Teresa, Hussain and Giovanni. We shared very happy moments that I will cherish forever. I extend my thanks to Helena, who made my stay in the University of Reading possible, and to Fernando, Cacilda, Teresa Mota and Cristine and other members of the Brazilian and Portuguese Speakers Society. Among the friends I met in in the Mathematics Department, I will remember with

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Citation Context ...n used to obtain numerical solutions since these numerical schemes have the ability to deal with all the features of the flow in the entire domain. For example, upwind schemes based in the Roe scheme =-=[75]-=- and the EngquistOsher [18] scheme are used. These are first-order methods which ensure correct application of the boundary conditions, in particular in the presence of shocks and expansions. An upwin... |

435 |
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Citation Context ...the steady problem and studying the convergence of this iteration. Proofs of convergence of this pseudo time iteration (which is a Picard iteration) rely on the contraction mapping theorem (see, e.g. =-=[70]-=-). Viscous steady problems of the form (3.13) with F of the form F (w) have been studied by several authors (e.g., [59, 54, 55, 56]). Less work has been done in the case F (x, w). The theory applied i... |

425 |
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Citation Context ...ithout the limit problem necessarily having to satisfy either of those boundary conditions (see [59]). Other methods of finding the physical correct solution make use of entropy conditions (see, e.g. =-=[49]-=-). In [59] MacDonald (see also [60]) presents theoretical results for the steady flow problem derived from the Saint-Venant equations. The solution of the steady flow problem is constructed as the van... |

418 |
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Citation Context ...shock capturing schemes. Furthermore, if the numerical scheme satisfies an additional condition known as entropy condition convergence is guaranteed to the unique physically relevant solution (Harten =-=[34]-=-). Examples of conservative schemes generating numerical solutions violating the entropy condition are given e.g. in Leveque [49] or Wesseling [101]. Nevertheless, as Toro [95] has pointed out, since ... |

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183 |
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Citation Context ...itten in the form (2.2) (see [15, 22, 2]). Other applications are possible (see [94] for more details). Some useful references on the numerical solution of hyperbolic systems of conservation laws are =-=[49, 44, 29, 9, 95, 93, 101]-=-. Other relevant references in the theory of conservation laws are [11, 86, 80]. In the next sections the Saint-Venant and Euler equations are introduced and we show how these equations can be reduced... |

155 |
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Citation Context ...les) has the same eigenvalues. The theory of characteristics leads us to study the signs of the eigenvalues (2.77)-(2.79) yielding the slope of characteristics where the Riemann invariants (see, e.g. =-=[7]-=-) are constant. (Note that the Riemann invariants are constant for the homogeneous case.) Furthermore, it is also know that boundary and initial conditions that lead to a wellposed problem for the Eul... |

141 |
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Citation Context ...relevant solution can be defined as the vanishing viscosity solution, i.e. the limit solution as the viscosity coefficient ɛ ↓ 0 of the parabolic equation ∂w ∂t ∂ + ∂x f(w) =ɛ∂2 w . (3.6) ∂x2 Oleinik =-=[68]-=- demonstrates the existence and uniqueness of a vanishing viscosity solution for given initial data w0(x) which satisfies the so-called Oleinik (entropy) condition f(w) − f(wL) w − wL ≥ s ≥ f(w) − f(w... |

137 |
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Citation Context ... of state is of the form e = e(p, ρ). (2.53) This equation of state depends on the gas under consideration. Even if viscous effects are taken in account it can be assumed that the gas is perfect (see =-=[40]-=-). For a perfect gas, the internal energy (per unit mass) e is a function of the temperature alone, e = e(T ) (see, e.g. [94, 49]), and we have a perfect gas law relating pressure, density and tempera... |

133 |
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Citation Context ...satisfied if ˜ J is taken to be the Jacobian evaluated at an averaged state Ũ, i.e. ˜ J(UL, UR) = ˜ J( Ũ). In general, an arithmetic average does not satisfy the last condition (see [49, 91] and also =-=[39]-=-) and a particular kind of geometric average is often used instead. This geometric average can be written in the form of an arithmetic mean of a parameter vector (see [75, 79, 27]). In [75] Roe showed... |

109 |
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Citation Context ...calar Cauchy problem (or initial value problem (IVP)) with smooth initial data where the flux function depends only on the conserved variable w, i.e. wt + F (w)x = 0 (4.16) 1Courant, Friedrichs, Lewy =-=[12]-=- whom firstly recognized the importance of a condition like (4.15) 56with w(x, 0) = w0(x). (4.17) A solution of the IVP can be constructed (for small t) by following characteristic curves x = x(t). T... |

102 |
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Citation Context ...etails). Some useful references on the numerical solution of hyperbolic systems of conservation laws are [49, 44, 29, 9, 95, 93, 101]. Other relevant references in the theory of conservation laws are =-=[11, 86, 80]-=-. In the next sections the Saint-Venant and Euler equations are introduced and we show how these equations can be reduced to a singular scalar ODE of the form (1.1) in the steady-state case. 2.2 The S... |

81 |
solvers, the entropy condition, and difference approximations
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Citation Context ...hock or by a fan, but not by both. The case of F being neither convex or concave is more complicated but a solution can still be found and it may involve both a shock and a rarefaction wave (e.g. see =-=[72, 101]-=-). An expression for the Godunov numerical flux which works even with nonconvex fluxes and that leads to entropy satisfying solutions of the Riemann problem is given by F ∗ j+ 1 2 (see LeVeque [49]). ... |

73 |
Definition and weak stability of nonconservative products
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(Show Context)
Citation Context ...oduct J(U)(U)x and to give it a meaning, in some way, when U is, say, a Heaviside function (a single jump). This can be done, as described in [29] following the theory of Dal Maso, Le Floch and Murat =-=[14]-=- who extended the work of Volpert (see [41, 92] for references). The definition of the jump [J(U)(U)x] φ depends on apathφ connecting the left state UL and the right state UR. A different approach is ... |

66 |
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(Show Context)
Citation Context ...r such nonhomogeneous systems is not so well-developed and the numerical treatment of source terms is also a subject of current research. Some recent work in this area and relevant for this thesis is =-=[3, 99, 42, 20, 43, 50, 21, 33]-=-. Some earlier work with useful discussions is presented in [53, 15, 90]. Numerical difficulties arise if the source terms are stiff and particularly in the steady3case if their magnitude is signific... |

66 |
High order essentially non-oscillatory schemes for hamilton-jacobi equaitons
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Citation Context ...1 j+ ] with midpoint 2 2 value w n j and slopes determined according to certain rules (c.f. van Leer [98]). Other approaches can be taken, for example, the essentially non-oscillatory or ENO approach =-=[36]-=- (see also [85]) and artificial viscosity schemes (see [101] for references). The Godunov, Engquist-Osher and Roe schemes under the hypothesis that an appropriate CFL condition holds, are examples of ... |

61 |
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(Show Context)
Citation Context ...itten in the form (2.2) (see [15, 22, 2]). Other applications are possible (see [94] for more details). Some useful references on the numerical solution of hyperbolic systems of conservation laws are =-=[49, 44, 29, 9, 95, 93, 101]-=-. Other relevant references in the theory of conservation laws are [11, 86, 80]. In the next sections the Saint-Venant and Euler equations are introduced and we show how these equations can be reduced... |

59 |
Self-adjusting grid methods for one-dimensional hyperbolic conservation laws
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(Show Context)
Citation Context ...ution contains only discontinuities and not expansion fans, leading to non entropy satisfying solutions. Different entropy fixes for Roe’s scheme have been proposed, though. See for example, [79] and =-=[37]-=- (the latter is also outlined in [49]). ForanytwoadjacentstatesULand UR the matrices ˜ J = ˜ J(UL, UR) should satisfy: (i). ˜ J(UL, UR) is diagonalisable (Hyperbolicity) (ii). ˜ J(UL, UR) → J(U) asUL,... |

56 | Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm
- LeVeque
- 1998
(Show Context)
Citation Context ... of the source terms corresponding to the way the derivative of the flux function is discretised (upwind). We also mention the work of Roe [78], Glaister [22, 24, 23, 25, 28, 26], Sweby [90], LeVeque =-=[50, 53]-=-, Emmerson [15], Gosse [31, 32], Greeenberg and LeRoux [33] and that of Jenny and Muller [43]. The use of the Engquist-Osher scheme [18] for problems of the form (2.1), even in the steady-state case, ... |

50 |
The Dynamics and Thermodynamics of Compressible Fluid Flow", Volume 1 and 2
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Citation Context ...unchanging in time, is important in Hydraulics and in Gas Dynamics. In fact, the steady form of the Saint-Venant equations and Euler equations has been studied in many classical texts such as [8] and =-=[84]-=-, respectively. 1The one-dimensional homogeneous unsteady Saint-Venant equations and the Euler equations are hyperbolic systems of conservation laws which admit discontinuous solutions (so-called wea... |

44 | Computational Fluid Dynamics: The basics with applications - Anderson - 1995 |

42 |
Characteristic-based schemes for the Euler equations
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(Show Context)
Citation Context ...nuity of the solution. The implementation of nonconservative schemes which fit the shock waves by explicitly computed discontinuities, may incorporate upwinding (of source terms as well) very cheaply =-=[77]-=-. Some references on using adaptive primitiveconservative schemes are given in Toro [95]. Moreover, the differential equations under study might include source terms. Examples of conservation laws wit... |

38 |
Upwind methods for hyperbolic conservation laws with source terms
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(Show Context)
Citation Context ...r such nonhomogeneous systems is not so well-developed and the numerical treatment of source terms is also a subject of current research. Some recent work in this area and relevant for this thesis is =-=[3, 99, 42, 20, 43, 50, 21, 33]-=-. Some earlier work with useful discussions is presented in [53, 15, 90]. Numerical difficulties arise if the source terms are stiff and particularly in the steady3case if their magnitude is signific... |

36 |
One-sided difference approximations for nonlinear conservation laws
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Citation Context ...solutions since these numerical schemes have the ability to deal with all the features of the flow in the entire domain. For example, upwind schemes based in the Roe scheme [75] and the EngquistOsher =-=[18]-=- scheme are used. These are first-order methods which ensure correct application of the boundary conditions, in particular in the presence of shocks and expansions. An upwind method does not need any ... |

34 |
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(Show Context)
Citation Context ...It can be shown that monotone conservative schemes are TVD and more importantly, that any conservative monotone scheme converges to the unique entropy satisfying solution of the conservation law (see =-=[49, 29, 13, 38]-=-). The drawback is that monotone schemes are first order accurate [38]. As described in [92], the limitation of first-order accuracy for monotone approximations can be avoided if L 1 -contractive solu... |

34 |
Why nonconservative schemes converge to wrong solutions: error analysis
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(Show Context)
Citation Context ...onservative schemes generating numerical solutions violating the entropy condition are given e.g. in Leveque [49] or Wesseling [101]. Nevertheless, as Toro [95] has pointed out, since Hou and LeFloch =-=[41]-=- proved that, in the case of a shock, if a scheme not written in conservative form converges it will converge to the solution of a new conservation law with a source term, this is another argument in ... |

33 |
Introduction to Singular Perturbations
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(Show Context)
Citation Context ...ation of the viscous problem (3.13) is second order, two boundary conditions are needed, the simplest being Dirichlet boundary conditions. Problem (3.13) is a singular perturbation problem (see, e.g. =-=[69]-=-) since the order of the differential equation reduces from second order to first order as ɛ vanishes. Furthermore, we cannot expect the solution of the limit problem to satisfy both boundary conditio... |

33 |
D.Systems of conservation laws. 1. Hyperbolicity, entropies, shock waves. Translated from the 1996 French original by
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(Show Context)
Citation Context ...etails). Some useful references on the numerical solution of hyperbolic systems of conservation laws are [49, 44, 29, 9, 95, 93, 101]. Other relevant references in the theory of conservation laws are =-=[11, 86, 80]-=-. In the next sections the Saint-Venant and Euler equations are introduced and we show how these equations can be reduced to a singular scalar ODE of the form (1.1) in the steady-state case. 2.2 The S... |

31 |
A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms
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(Show Context)
Citation Context ..., 15, 90]. Numerical difficulties arise if the source terms are stiff and particularly in the steady3case if their magnitude is significant. Much work has been done related to balancing source terms =-=[50, 3, 42, 5, 21, 31, 32, 33, 43, 73, 99]-=-. Problems arise when applying operator splitting methods to steady problems or near steady problems ([95]). Our choice of upwind methods to deal with steady conservation laws with source terms is sup... |

31 |
On finite-difference approximations and entropy conditions for shocks
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- 1976
(Show Context)
Citation Context ...It can be shown that monotone conservative schemes are TVD and more importantly, that any conservative monotone scheme converges to the unique entropy satisfying solution of the conservation law (see =-=[49, 29, 13, 38]-=-). The drawback is that monotone schemes are first order accurate [38]. As described in [92], the limitation of first-order accuracy for monotone approximations can be avoided if L 1 -contractive solu... |

30 |
On a class of high resolution total-variation-stable finite-difference schemes
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- 1984
(Show Context)
Citation Context ...s) and to avoid smearing discontinuities is to use the TVD criteria. Some references on constructing high order TVD schemes using limiter functions can be found in [59]. We mention the work of Harten =-=[35]-=- and Sweby [89]. Second order schemes can be obtained by solving a Generalized Riemann problem defined by assuming piecewise linear data in each interval [x 1 j− ,x 1 j+ ] with midpoint 2 2 value w n ... |

28 |
Stable and entropy satisfying approximations for transonicc flow calculations
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- 1980
(Show Context)
Citation Context ...er used the vanishing viscosity theory and artificial time stepping to prove convergence to a unique steady state solution of a nonlinear singular perturbation problem using the Engquist-Osher scheme =-=[17, 16, 18]-=-. Osher [71] recognised that the conservative approximation to the spatial derivative can be used in approximating singular perturbation problems of the form (3.13). The problem discussed in [71] is o... |

27 |
Calculation of plane steady transonic flows
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(Show Context)
Citation Context ...j+1 − wj) (4.31) 2 wj+1 ̸= wj wj+1 = wj ≥ 0 < 0 . (4.32) . (4.33) Theschemeisalsoknownasthefirst-order upwind scheme (FOU). We call it the Roe scheme although, as referred to in [59], Murman and Cole =-=[66, 67]-=- came up with a similar scheme earlier. The term upwind or upstream refers to the direction from which characteristic information propagates, with the grid points used in the spatial finite difference... |

23 | A class of approximate Riemann solvers and their relation to relaxation schemes
- LeVeque, Pelanti
(Show Context)
Citation Context ...xample, which is the best discretisation of the source terms, particularly if the Roe method is used and if the flux function F depends on x. Other interesting work is by LeVeque and co-authors (e.g. =-=[51, 52]-=-), Chinnayya and Le Roux [6] and also by Toro and coauthors (e.g. [96]). In Chapter 2 we showed that it is possible to reduce, in the steady state case, some systems of conservation laws to a single, ... |

21 |
Upwind differencing schemes for hyperbolic conservation laws with source terms, in ``Nonlinear hyperbolic problems
- ROE
- 1987
(Show Context)
Citation Context ...ource terms is supported by work based on using upwind methods combined with upwinding source terms (e.g., [23, 3, 99, 20, 42]). The idea of upwinding the source terms was first put forward by Roe in =-=[78]-=-. Further work was carried out in this direction in e.g. [3, 99]. The idea is to try to build discretisations of the source terms in a way similar to those used to construct the numerical flux functio... |

18 | A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics - Godunov - 1959 |

17 |
Fluid Mechanics, volume 6 of Course of Theoretical Physics
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(Show Context)
Citation Context ...Discontinuous Solutions Here we study the effect of irreversible thermodynamic processes that can occur in a nozzle having the particular form of a stationary shock front. It is known (see, e.g. [11],=-=[45]-=-,[84]) that under certain conditions (mass flow and pressure) a normal shock (shock which is locally perpendicular to a streamline) can occur in the divergent part of the nozzle. This discontinuous fl... |

16 |
Fluctuations and signals - a framework for numerical evolution problems
- Roe
- 1982
(Show Context)
Citation Context ...ms of conservation laws in the one-dimensional case with x-dependent flux function and possible extensions of two-dimensional upwinding via finite volumes methods. In a fluctuation-signal formulation =-=[76]-=-, the extra term V is thought of as a source term that is subtracted from the existing ones whereas in a numerical flux formulation this term V is included in the definition of the numerical flux func... |

13 |
Advanced Numerical Approximation of Nonlinear Hyperbolic Equations
- Cockburn, Johnson, et al.
- 1998
(Show Context)
Citation Context ...itten in the form (2.2) (see [15, 22, 2]). Other applications are possible (see [94] for more details). Some useful references on the numerical solution of hyperbolic systems of conservation laws are =-=[49, 44, 29, 9, 95, 93, 101]-=-. Other relevant references in the theory of conservation laws are [11, 86, 80]. In the next sections the Saint-Venant and Euler equations are introduced and we show how these equations can be reduced... |

12 |
B.: Computational hydraulics: elements of the theory of free surface flows
- Abbott
- 1992
(Show Context)
Citation Context ...2.12) ∫ h 0 √ 4+σ 2 η dη, (2.13) K = Ak1 nP k2 (2.14) and n is a constant representing the bed roughness of the channel. The friction slope Sf can be expressed by using Chezy’s or Manning’s laws (see =-=[8, 1]-=-). Here the Manning 9formulation for the friction slope Sf is adopted, i.e. k1 =5/3, k2 =2/3withtheManning coefficient, n, taking the value 0.03. A channel is said to be prismatic if its x-cross-sect... |

12 | Wave Propagation Methods for Conservation Laws with Source Terms
- LeVeque, Bale
- 1998
(Show Context)
Citation Context ...xample, which is the best discretisation of the source terms, particularly if the Roe method is used and if the flux function F depends on x. Other interesting work is by LeVeque and co-authors (e.g. =-=[51, 52]-=-), Chinnayya and Le Roux [6] and also by Toro and coauthors (e.g. [96]). In Chapter 2 we showed that it is possible to reduce, in the steady state case, some systems of conservation laws to a single, ... |

12 |
Mechanics of Fluids
- Massey
- 1995
(Show Context)
Citation Context ...mass (equal to the universal gas constant divided by the molecular mass of the gas) and we see that e is a function of p/ρ. For air, using the S.I. units, we have R = 287m.N/Kg.K (2.55) 19(see, e.g. =-=[65]-=-). Henceforth the quantities are defined in S.I. units unless stated otherwise. In the particular case of a polytropic (or calorically perfect) gas, the specific heat capacity cν is constant and we ha... |

9 |
A new General Riemann Solver for the Shallow Water equations, with friction and topography
- Chinnayya, LeRoux
- 1999
(Show Context)
Citation Context ...isation of the source terms, particularly if the Roe method is used and if the flux function F depends on x. Other interesting work is by LeVeque and co-authors (e.g. [51, 52]), Chinnayya and Le Roux =-=[6]-=- and also by Toro and coauthors (e.g. [96]). In Chapter 2 we showed that it is possible to reduce, in the steady state case, some systems of conservation laws to a single, singular ODE. That is the ca... |

9 |
On numercial treatment of the source terms in shallow water equations
- Garcia–Navarro, Vazquez-Cendom
(Show Context)
Citation Context ...tion of the source term determines the amount that is added to each grid point coming from the left-half cell and from the right-half cell. Both pointwise and upwind discretisation choices adopted in =-=[20, 19]-=- are analyzed in the context of satisfying a C-property (see [3, 99]) for the stationary solution of water 87at rest. Results are presented for the Saint-Venant equations in a nonprismatic channel wi... |

7 |
Difference schemes for the shallow water equations. Numerical Analysis Report 9/87
- Glaister
- 1987
(Show Context)
Citation Context ...dy problems ([95]). Our choice of upwind methods to deal with steady conservation laws with source terms is supported by work based on using upwind methods combined with upwinding source terms (e.g., =-=[23, 3, 99, 20, 42]-=-). The idea of upwinding the source terms was first put forward by Roe in [78]. Further work was carried out in this direction in e.g. [3, 99]. The idea is to try to build discretisations of the sourc... |

7 |
Rankine-Hugoniot-Riemann solver considering source terms and multidimensional effects
- Jenny, Müller
- 1998
(Show Context)
Citation Context ...r such nonhomogeneous systems is not so well-developed and the numerical treatment of source terms is also a subject of current research. Some recent work in this area and relevant for this thesis is =-=[3, 99, 42, 20, 43, 50, 21, 33]-=-. Some earlier work with useful discussions is presented in [53, 15, 90]. Numerical difficulties arise if the source terms are stiff and particularly in the steady3case if their magnitude is signific... |

5 |
Approximate Riemann Solutions of the Shallow Water Equations
- Glaister
- 1988
(Show Context)
Citation Context ...ns. Later Roe and Pike [79] presented an approach 59where the explicit construction of the matrix ˜ J can be avoided. The application of the Roe scheme to the Shallow Water Equations can be found in =-=[25]-=-. Roe’s scheme for the scalar ODE For the case of a scalar conservation law of the form (4.16) the third condition determines uniquely ˜ λ = ˜ J(wL,wR) as ˜λ = F (wR) − F (wL) . (4.29) wR − wL Hence, ... |

5 |
Test Problems with Analytic Solutions for Steady Open Channel flow. Numerical analysis report 6/94
- MACDONALD
- 1995
(Show Context)
Citation Context ... the Saint-Venant equations (rectangular prismatic channel case). The test problems used in this thesis for the Saint-Venant problem have been taken from [59] and were developed in previous work (see =-=[60, 61]-=-). These test problems have known analytical solutions and include features like varying channel geometries and discontinuous solutions. The test problems chosen for the Euler equations, namely a dive... |

4 |
Efficient construction of high-resolution TVD conservative schemes for equations with source terms: application to shallow water flows
- Burguete, Garcia-Navarro
- 2001
(Show Context)
Citation Context ..., 15, 90]. Numerical difficulties arise if the source terms are stiff and particularly in the steady3case if their magnitude is significant. Much work has been done related to balancing source terms =-=[50, 3, 42, 5, 21, 31, 32, 33, 43, 73, 99]-=-. Problems arise when applying operator splitting methods to steady problems or near steady problems ([95]). Our choice of upwind methods to deal with steady conservation laws with source terms is sup... |