## Front tracking for scalar balance equations

Venue: | J. Hyperbolic Differ. Equ |

Citations: | 4 - 2 self |

### BibTeX

@ARTICLE{Karlsen_fronttracking,

author = {K. H. Karlsen and N. H. Risebro and J. D. Towers},

title = {Front tracking for scalar balance equations},

journal = {J. Hyperbolic Differ. Equ},

year = {},

pages = {2005--35156}

}

### OpenURL

### Abstract

Abstract. We propose and prove convergence of a front tracking method for scalar conservation laws with source term. The method is based on writing the single conservation law as a 2 × 2 quasilinear system without a source term, and employ the solution of the Riemann problem for this system in the front tracking procedure. In this way the source term is processed in the Riemann solver, and one avoids using operator splitting. Since we want to treat the resonant regime, classical arguments for bounding the total variation of numerical solutions do not apply here. Instead compactness of a sequence of front tracking solutions is achieved using a variant of the singular mapping technique invented by Temple [69]. The front tracking method has no CFL–condition associated with it, and it does not discriminate between stiff and non-stiff source terms. This makes it an attractive approach for stiff problems, as is demonstrated in numerical examples. In addition, the numerical examples show that the front tracking method is able to preserve steady–state solutions (or achieving them in the long time limit) with good accuracy. 1.

### Citations

156 |
First order quasi-linear equations in several independent variables
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(Show Context)
Citation Context ... for all test functions ' 2 C10 (R [0; T )). However, weak solutions are not unique, and the unique entropy solution to (1.3) is dened as the weak solution satisfying the Kruzkov entropy condition =-=[48]-=- (3.2) Z T 0 Z R ju cj't + sign (u c) (f(u) f(c))'x + sign (u c) k0(x)g(u)' dxdt + Z R ju0 cj'(x; 0) dx 0; for all nonnegative ' 2 C10 (R [0; T )) and for all constants c 2 R. Actually, (3.2)... |

84 |
A well-balanced scheme for the numerical processing of source terms in hyperbolic equations
- Greenberg, LeRoux
- 1996
(Show Context)
Citation Context ...d so by proving convergence of a Godunov scheme with a generalized Riemann solver (in the spirit of Liu) using steady–state solutions to incorporate the geometrical source term. Greenberg and Le Roux =-=[25]-=- (see also [24, 21, 26]) introduced the so-called well–balanced Godunov schemes for scalar equations. These schemes are also based on appropriately modifying the Riemann problem to account for the sou... |

78 | Balancing source terms and flux gradients in high resolution Godunov methods
- LeVeque
- 1998
(Show Context)
Citation Context ... such a way that the delicate balance in (1.4) is respected. The well–balanced schemes have been extended to systems by Gosse in [22, 23]. Another (but related) Godunov scheme can be found in LeVeque =-=[51]-=-, which is able to capture the steady–state relation (1.4) by building it into the flux calculations. The balance (1.4) is achieved in his quasi-steady method by introducing additional Riemann problem... |

73 |
Polygonal approximations of solutions of the initial value problem for a conservation law
- Dafermos
- 1972
(Show Context)
Citation Context ... of this introduction is seems appropriate to put the front tracking method into a historical perspective. Front tracking for scalar conservation laws (without source term) was introduced by Dafermos =-=[11]-=- as a mathematical tool for constructing and deriving properties of entropy solutions. It was later proved by Holden, Holden, and Høegh–Krohn [30, 31] that Dafermos’ construction procedure was well-de... |

67 |
order quasilinear equations in several independent variables
- First
(Show Context)
Citation Context ...r all test functions ϕ ∈ C∞ 0 (R × [0, T )). However, weak solutions are not unique, and the unique entropy solution to (1.3) is defined as the weak solution satisfying the Kruˇzkov entropy condition =-=[48]-=- ∫ T ∫ |u − c| ϕt + sign (u − c) (f(u) − f(c))ϕx + sign (u − c) k (3.2) 0 R ′ (x)g(u)ϕ dxdt ∫ + |u0 − c| ϕ(x, 0) dx ≥ 0, R for all nonnegative ϕ ∈ C∞ 0 (R × [0, T )) and for all constants c ∈ R. Actua... |

58 |
Upwind methods for hyperbolic conservation laws with source terms. Computers Fluids
- Bermudez, Vazquez
- 1994
(Show Context)
Citation Context ...ng a TVD scheme. Roe [65] suggested to apply high-resolution upwind schemes to a modified flux that includes the source term (“upwinding of the source term”). In the same spirit, Bermúdez and Vazquez =-=[4]-=- (see also [3, 71, 37, 16, 9, 59, 60] for related papers) extended some upwind (flux–difference splitting and flux–vector splitting) schemes to hyperbolic systems of conservation laws with source term... |

58 |
Convex conservation laws with discontinuous coecients. Existence, uniqueness and asymptotic behavior
- Klingenberg, Risebro
- 1995
(Show Context)
Citation Context ...ily uniformly bounded, hence showing that such schemes converge is more complicated and does not follow the standard route, see, e.g., Temple [69], Gimse and Risebro [18], and Klingenberg and Risebro =-=[47]-=-. The suggested front tracking method is a numerical scheme where the solution of the Riemann problem to (1.5) is approximated by a piecewise constant (in x/t) function, whose discontinuities we call ... |

58 |
A kinetic scheme for the Saint-Venant system with a source. Calcolo 38
- Perthame, Simeoni
- 2001
(Show Context)
Citation Context ...S 3 solver that preserves all steady–states. Kinetic equilibrium schemes (with reflections) are suggested by Botchorishvili, Perthame, and Vasseur [5] for scalar equations and by Perthame and Simeoni =-=[61]-=- for the shallow water equations with bottom topography. Gas-kinetic schemes that account for the source terms in the flux calculation across a cell interface are studied in Xu [72] and Tang, Tang, an... |

57 |
A study of numerical methods for hyperbolic conservation laws with sti source terms
- LeVeque, Yee
- 1990
(Show Context)
Citation Context ...he EC as contract HPRN-CT-2002-00282. K. H. Karlsen is grateful to Hermano Frid for discussions. 12 KARLSEN, RISEBRO, AND TOWERS that the ordinary differential equation is stiff (see LeVeque and Yee =-=[52]-=-), an exception being stiff source terms of dissipative nature as in Tang [68], see also Helzel, LeVeque, and Warnecke [28] for a modified splitting method for approximating detonation waves. Unaccept... |

54 | Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. Fluids 32
- Gallouët, Hérard, et al.
- 2003
(Show Context)
Citation Context ...quations with bottom topography, which is based on the Harten–Lax–van Leer (HLL) approximate Riemann solver and the surface gradient method. For the shallow water system, Gallouët, Hérard, and Seguin =-=[15]-=- present a well–balanced Godunov scheme based on a very simple (linearized) approximate RiemannFRONT TRACKING FOR SCALAR BALANCE EQUATIONS 3 solver that preserves all steady–states. Kinetic equilibri... |

51 | Global solution of the Cauchy problem for a class of 22 nonstrictly hyperbolic conservation laws
- Temple
- 1982
(Show Context)
Citation Context ...otal variation of numerical solutions do not apply here. Instead compactness of a sequence of front tracking solutions is achieved using a variant of the singular mapping technique invented by Temple =-=[69]-=-. The front tracking method has no CFL–condition associated with it, and it does not discriminate between stiff and non-stiff source terms. This makes it an attractive approach for stiff problems, as ... |

50 |
One-sided difference approximations for nonlinear conservation laws
- Engquist, Osher
- 1981
(Show Context)
Citation Context ...we use, as before, (4.1) the flux function f(u) = 4u(1 − u) and the source term g(u) = u. To compare the front tracking computations with a reference solution we use the standard EngquistOsher scheme =-=[13]-=- defined by u n+1 j where the Engquist-Osher flux f EO reads = un j − λ ( f EO (u n j+1, u n j ) − f EO (u n j , u n j−1) ) + u n j k ′ (xj) ∆t, f EO (u, v) = f(max{1/2, u}) + f(min{1/2, v}), □FRONT ... |

47 | Equilibrium schemes for scalar conservation laws with sti sources
- Botchorishvili, Perthame, et al.
(Show Context)
Citation Context ...te RiemannFRONT TRACKING FOR SCALAR BALANCE EQUATIONS 3 solver that preserves all steady–states. Kinetic equilibrium schemes (with reflections) are suggested by Botchorishvili, Perthame, and Vasseur =-=[5]-=- for scalar equations and by Perthame and Simeoni [61] for the shallow water equations with bottom topography. Gas-kinetic schemes that account for the source terms in the flux calculation across a ce... |

45 |
Global solutions to systems of conservation laws by wave-front tracking
- Bressan
- 1992
(Show Context)
Citation Context ...n, and Høegh–Krohn [30, 31] that Dafermos’ construction procedure was well-defined and could be viewed as a convergent numerical method. Front tracking was extended to systems of equations by Bressan =-=[7]-=-, DiPerna [12], and Risebro [62], who used the method to give an alternative proof of Glimm’s famous existence result for hyperbolic systems. Recently a modification of the front tracking method was u... |

45 |
Solution of the Cauchy problem for a conservation law with a discontinuous function
- Gimse, Risebro
- 1992
(Show Context)
Citation Context ... or front tracking is not necessarily uniformly bounded, hence showing that such schemes converge is more complicated and does not follow the standard route, see, e.g., Temple [69], Gimse and Risebro =-=[18]-=-, and Klingenberg and Risebro [47]. The suggested front tracking method is a numerical scheme where the solution of the Riemann problem to (1.5) is approximated by a piecewise constant (in x/t) functi... |

38 |
A numerical method for first order nonlinear scalar hyperbolic conservation laws in one dimension
- Holden, Holden, et al.
- 1993
(Show Context)
Citation Context ...laws (without source term) was introduced by Dafermos [11] as a mathematical tool for constructing and deriving properties of entropy solutions. It was later proved by Holden, Holden, and Høegh–Krohn =-=[30, 31]-=- that Dafermos’ construction procedure was well-defined and could be viewed as a convergent numerical method. Front tracking was extended to systems of equations by Bressan [7], DiPerna [12], and Rise... |

37 |
A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms
- Gosse
(Show Context)
Citation Context ...riately modifying the Riemann problem to account for the source term in such a way that the delicate balance in (1.4) is respected. The well–balanced schemes have been extended to systems by Gosse in =-=[22, 23]-=-. Another (but related) Godunov scheme can be found in LeVeque [51], which is able to capture the steady–state relation (1.4) by building it into the flux calculations. The balance (1.4) is achieved i... |

36 |
On the Relation Between the Upwind-Differencing Schemes of Godunov, Enguist-Osher and Roe
- Leer
- 1985
(Show Context)
Citation Context ...lations of transonic nozzle flow (including the case of nonlinear resonance). In their paper, they also discuss scalar conservation laws with source term. Glimm, Marshall, and Plohr [20] and van Leer =-=[70]-=- devised second order schemes for such gas flow using the idea of Liu. Isaacson and Temple [39] analyzed a Godunov scheme for a general class of (resonant) scalar balance equations using a generalized... |

35 | Numerical solution of reservoir flow models based on large time step operator splitting algorithms
- Espedal, Karlsen
- 1998
(Show Context)
Citation Context ...7] (to mention only a few relevant references). Front tracking for Hamilton-Jacobi equations is the topic in Karlsen and Risebro [45, 46]. For front tracking applied to viscous conservation laws, see =-=[14]-=- and the references cited therein. An oil reservoir simulator based on the front tracking idea is described in [6]. We refer to the book [35] by Holden and Risebro for a general introduction to front ... |

34 |
Central-upwind schemes for the Saint-Venant system
- Kurganov, Levy
- 2002
(Show Context)
Citation Context ... interface values instead of the cell–averages to process the source. Central–upwind schemes with a good treatment of the source terms in the shallow water equations can be found in Kurganov and Levy =-=[49]-=-. The goal of this paper is to devise a front tracking method for the scalar balance equation (1.1) which is able to cope with stiff sources as well as capturing steady–state solutions (1.4). The star... |

33 |
A steady-state capturing method for hyperbolic systems with geometrical source terms
- Jin
(Show Context)
Citation Context ...ation property but that the extensions of the flux–vector splitting schemes do not. Another simple method for processing source terms so that (1.4) is respected with high accuracy can be found in Jin =-=[42]-=-. His approach applies to Godunov and Roe type upwind schemes, but does not require any modification of the Riemann solver, since he is using the interface values instead of the cell–averages to proce... |

33 |
A front-tracking alternative to the random choice method
- Risebro
- 1993
(Show Context)
Citation Context ...t Dafermos’ construction procedure was well-defined and could be viewed as a convergent numerical method. Front tracking was extended to systems of equations by Bressan [7], DiPerna [12], and Risebro =-=[62]-=-, who used the method to give an alternative proof of Glimm’s famous existence result for hyperbolic systems. Recently a modification of the front tracking method was used by Bressan and his collabora... |

30 |
Riemann problems with a discontinuous flux function
- Gimse, Risebro
- 1990
(Show Context)
Citation Context ...resonant case f ′ ̸= 0. He uses a framework of non-conservative products to make sense to the solution of the Riemann problem. We next detail the solution of the Riemann problem (2.1) (here we follow =-=[39, 57, 17]-=- closely). Our prototype for this problem is the case where (2.4) f(u) = 4u(1 − u) and g(u) = u, so that u∗ in (1.7) is zero and u ∗ in (1.6) is 1/2. In this case we assume that ul and ur are positive... |

30 |
Quasilinear hyperbolic systems
- Liu
- 1979
(Show Context)
Citation Context ...law step. Many authors have designed unsplit schemes based on the idea of modifying the Riemann problem to account for the source term in such a way that the steady–state equations are satisfied. Liu =-=[54, 55]-=- pioneered this idea of using a generalized Riemann problem in the context of quasione dimensional (area varying) gas flow. He used the generalized Riemann problems to build a random choice scheme bas... |

28 | A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms
- Gosse
(Show Context)
Citation Context ...riately modifying the Riemann problem to account for the source term in such a way that the delicate balance in (1.4) is respected. The well–balanced schemes have been extended to systems by Gosse in =-=[22, 23]-=-. Another (but related) Godunov scheme can be found in LeVeque [51], which is able to capture the steady–state relation (1.4) by building it into the flux calculations. The balance (1.4) is achieved i... |

28 |
Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry
- Vazquez-Cendon
- 1999
(Show Context)
Citation Context ...e. Roe [65] suggested to apply high-resolution upwind schemes to a modified flux that includes the source term (“upwinding of the source term”). In the same spirit, Bermúdez and Vazquez [4] (see also =-=[3, 71, 37, 16, 9, 59, 60]-=- for related papers) extended some upwind (flux–difference splitting and flux–vector splitting) schemes to hyperbolic systems of conservation laws with source terms (shallow water equations). To ensur... |

27 |
Upwind differencing schemes for hyperbolic conservation laws with source terms
- Roe
- 1986
(Show Context)
Citation Context ...mogenous conservation laws is not appropriate for nonhomogeneous conservation laws, and he suggested to transform the nonhomogeneous problem to a homogeneous problem before applying a TVD scheme. Roe =-=[65]-=- suggested to apply high-resolution upwind schemes to a modified flux that includes the source term (“upwinding of the source term”). In the same spirit, Bermúdez and Vazquez [4] (see also [3, 71, 37,... |

25 |
Hyperbolic Systems of Conservation Laws, volume 20 of Oxford
- Bressan
- 2000
(Show Context)
Citation Context ... Recently a modification of the front tracking method was used by Bressan and his collaborators to prove stability and uniqueness of weak solutions of strictly hyperbolic systems of conservation laws =-=[8]-=-. The front tracking method was used by Risebro and Tveito [63, 64] to numerically solve the Euler equations of gas dynamics and a non-strictly hyperbolic system modeling polymer flow. Holden and Rise... |

25 |
A method of fractional steps for scalar conservation laws without the CFL condition
- Holden, Risebro
- 1993
(Show Context)
Citation Context ...e front tracking method was used by Risebro and Tveito [63, 64] to numerically solve the Euler equations of gas dynamics and a non-strictly hyperbolic system modeling polymer flow. Holden and Risebro =-=[33]-=- extended the scalar front tracking method to multi-dimensional scalar conservation laws by means of dimensional splitting, see also [43]. ItFRONT TRACKING FOR SCALAR BALANCE EQUATIONS 5 should be no... |

24 |
Global existence of solutions to nonlinear hyperbolic systems of conservation laws
- DiPerna
- 1976
(Show Context)
Citation Context ...Krohn [30, 31] that Dafermos’ construction procedure was well-defined and could be viewed as a convergent numerical method. Front tracking was extended to systems of equations by Bressan [7], DiPerna =-=[12]-=-, and Risebro [62], who used the method to give an alternative proof of Glimm’s famous existence result for hyperbolic systems. Recently a modification of the front tracking method was used by Bressan... |

24 |
Convergence of the 22 Godunov method for a general resonant nonlinear balance law
- Isaacson, Temple
- 1995
(Show Context)
Citation Context ...hey also discuss scalar conservation laws with source term. Glimm, Marshall, and Plohr [20] and van Leer [70] devised second order schemes for such gas flow using the idea of Liu. Isaacson and Temple =-=[39]-=- analyzed a Godunov scheme for a general class of (resonant) scalar balance equations using a generalized Riemann problem (see also the discussion about resonance later on in this introduction). Chen ... |

23 | A Q-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system
- Castro, Macas, et al.
(Show Context)
Citation Context ...e. Roe [65] suggested to apply high-resolution upwind schemes to a modified flux that includes the source term (“upwinding of the source term”). In the same spirit, Bermúdez and Vazquez [4] (see also =-=[3, 71, 37, 16, 9, 59, 60]-=- for related papers) extended some upwind (flux–difference splitting and flux–vector splitting) schemes to hyperbolic systems of conservation laws with source terms (shallow water equations). To ensur... |

23 | Global solutions to the compressible Euler equations with geometrical structure
- Chen, Glimm
- 1996
(Show Context)
Citation Context ...a Godunov scheme for a general class of (resonant) scalar balance equations using a generalized Riemann problem (see also the discussion about resonance later on in this introduction). Chen and Glimm =-=[10]-=- demonstrated the existence of global solutions to the Euler equations of compressible isentropic gas dynamics with geometric structure. They did so by proving convergence of a Godunov scheme with a g... |

21 |
Global BV entropy solutions and uniqueness for hyperbolic systems of balance laws
- Amadori, Gosse, et al.
- 2002
(Show Context)
Citation Context ...hat the initial data u0(x) take values in the interval (u∗, M) for some finite constant M: (1.8) u0(x) ∈ (u∗, M), ∀x ∈ R. For the nonresonant (or strictly hyperbolic) case, Amadori, Gosse, and Guerra =-=[2]-=- established recently the existence and uniqueness of BV solutions to the Cauchy problem for hyperbolic systems of equations. Their existence proof was based on front tracking (different from ours), b... |

21 |
Analysis and approximation of conservation laws with source terms
- Greenberg, LeRoux, et al.
- 1997
(Show Context)
Citation Context ... convergence of a Godunov scheme with a generalized Riemann solver (in the spirit of Liu) using steady–state solutions to incorporate the geometrical source term. Greenberg and Le Roux [25] (see also =-=[24, 21, 26]-=-) introduced the so-called well–balanced Godunov schemes for scalar equations. These schemes are also based on appropriately modifying the Riemann problem to account for the source term in such a way ... |

21 |
Front tracking applied to a nonstrictly hyperbolic system of conservation laws
- Risebro, Tveito
- 1991
(Show Context)
Citation Context ... by Bressan and his collaborators to prove stability and uniqueness of weak solutions of strictly hyperbolic systems of conservation laws [8]. The front tracking method was used by Risebro and Tveito =-=[63, 64]-=- to numerically solve the Euler equations of gas dynamics and a non-strictly hyperbolic system modeling polymer flow. Holden and Risebro [33] extended the scalar front tracking method to multi-dimensi... |

18 |
Nonlinear resonance for quasilinear hyperbolic equation
- Liu
- 1987
(Show Context)
Citation Context ...law step. Many authors have designed unsplit schemes based on the idea of modifying the Riemann problem to account for the source term in such a way that the steady–state equations are satisfied. Liu =-=[54, 55]-=- pioneered this idea of using a generalized Riemann problem in the context of quasione dimensional (area varying) gas flow. He used the generalized Riemann problems to build a random choice scheme bas... |

17 |
The asymptotic analysis of wave interactions and numerical calculations of transonic nozzle flow”, Adv
- Glaz, Liu
- 1984
(Show Context)
Citation Context ...tionary data” instead of “piecewise constant data”. The random choice scheme was then used to prove the global existence of solutions for the nonhomogeneous system of gas flow equations. Glaz and Liu =-=[19]-=- used the generalized random choice scheme for numerical calculations of transonic nozzle flow (including the case of nonlinear resonance). In their paper, they also discuss scalar conservation laws w... |

16 |
A modi fractional step method for the accurate approximation of detonation waves
- Helzel, LeVeque, et al.
(Show Context)
Citation Context ...TOWERS that the ordinary differential equation is stiff (see LeVeque and Yee [52]), an exception being stiff source terms of dissipative nature as in Tang [68], see also Helzel, LeVeque, and Warnecke =-=[28]-=- for a modified splitting method for approximating detonation waves. Unacceptable errors occur in operator splitting methods also when seeking to preserve steady– state solutions or reaching them in t... |

16 | Dimensional splitting with front tracking and adaptive grid re
- Lie, Haugse, et al.
- 1998
(Show Context)
Citation Context ...hod, which implies that the method is fast compared with conventional difference schemes. Multi-dimensional scalar computations using CFL–numbers as high as 10 − 20 have been reported, see Lie et al. =-=[53]-=-. Computational results for multi-dimensional systems (using CFL–numbers 2 − 5) can be found in Holden et al. [32, 29] and Haugse, Karlsen, Lie, and Natvig [27]. A front tracking method for conservati... |

15 |
On the convergence of operator splitting applied to conservation laws with source terms
- Langseth, Tveito, et al.
- 1996
(Show Context)
Citation Context ...e step ∆t, using the result of one computation as the initial data for the other. In this way, one can use specialized methods for the conservation law and for the ordinary differential equation, see =-=[50, 29, 34, 44, 40]-=- for some applications of this strategy in the context of front tracking methods. In some cases this approach works well (see Langseth, Tveito, and Winther [50] for a favorable error estimate for oper... |

15 |
A front tracking method for conservation laws in one dimension
- Risebro, Tveito
- 1992
(Show Context)
Citation Context ... by Bressan and his collaborators to prove stability and uniqueness of weak solutions of strictly hyperbolic systems of conservation laws [8]. The front tracking method was used by Risebro and Tveito =-=[63, 64]-=- to numerically solve the Euler equations of gas dynamics and a non-strictly hyperbolic system modeling polymer flow. Holden and Risebro [33] extended the scalar front tracking method to multi-dimensi... |

14 |
A generalized Riemann problem for quasi{ one{dimensioanl gas
- Glimm, Marshall, et al.
- 1984
(Show Context)
Citation Context ...or numerical calculations of transonic nozzle flow (including the case of nonlinear resonance). In their paper, they also discuss scalar conservation laws with source term. Glimm, Marshall, and Plohr =-=[20]-=- and van Leer [70] devised second order schemes for such gas flow using the idea of Liu. Isaacson and Temple [39] analyzed a Godunov scheme for a general class of (resonant) scalar balance equations u... |

14 | On the convergence rate of operator splitting for Hamilton-Jacobi equations with source terms - Jakobsen, Karlsen, et al. |

13 |
FRONTLINE and FRONTSIM: two full scale, two-phase, black oil reservoir simulators based on front tracking
- Bratvedt, Bratvedt, et al.
- 1993
(Show Context)
Citation Context ...n and Risebro [45, 46]. For front tracking applied to viscous conservation laws, see [14] and the references cited therein. An oil reservoir simulator based on the front tracking idea is described in =-=[6]-=-. We refer to the book [35] by Holden and Risebro for a general introduction to front tracking methods. The rest of this paper is organized as follows, in Section 2 we review the solution of the Riema... |

13 |
Front tracking for hyperbolic conservation laws, volume 152 of Applied Mathematical Sciences
- Holden, Risebro
- 2002
(Show Context)
Citation Context ...r front tracking applied to viscous conservation laws, see [14] and the references cited therein. An oil reservoir simulator based on the front tracking idea is described in [6]. We refer to the book =-=[35]-=- by Holden and Risebro for a general introduction to front tracking methods. The rest of this paper is organized as follows, in Section 2 we review the solution of the Riemann problem and show that th... |

13 | An unconditionally stable splitting scheme for a class of nonlinear parabolic equations
- Karlsen, Lie
- 1999
(Show Context)
Citation Context ...e step ∆t, using the result of one computation as the initial data for the other. In this way, one can use specialized methods for the conservation law and for the ordinary differential equation, see =-=[50, 29, 34, 44, 40]-=- for some applications of this strategy in the context of front tracking methods. In some cases this approach works well (see Langseth, Tveito, and Winther [50] for a favorable error estimate for oper... |

11 | Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes
- Bermudez, Dervieux, et al.
- 1998
(Show Context)
Citation Context ...e. Roe [65] suggested to apply high-resolution upwind schemes to a modified flux that includes the source term (“upwinding of the source term”). In the same spirit, Bermúdez and Vazquez [4] (see also =-=[3, 71, 37, 16, 9, 59, 60]-=- for related papers) extended some upwind (flux–difference splitting and flux–vector splitting) schemes to hyperbolic systems of conservation laws with source terms (shallow water equations). To ensur... |

11 |
Numerical solution of reservoir models based on large time step operator splitting algorithms
- Espedal, Karlsen
- 1998
(Show Context)
Citation Context ...7] (to mention only a few relevant references). Front tracking for Hamilton-Jacobi equations is the topic in Karlsen and Risebro [45, 46]. For front tracking applied to viscous conservation laws, see =-=[14]-=- and the references cited therein. An oil reservoir simulator based on the front tracking idea is described in [6]. We refer to the book [35] by Holden and Risebro for a general introduction to front ... |

11 |
On the relation between the upwind-dierencing schemes of Godunov, Engquist-Osher and Roe
- Leer
- 1984
(Show Context)
Citation Context ...culations of transonic nozzlesow (including the case of nonlinear resonance). In their paper, they also discuss scalar conservation laws with source term. Glimm, Marshall, and Plohr [20] and van Leer =-=[70]-=- devised second order schemes for such gassow using the idea of Liu. Isaacson and Temple [39] analyzed a Godunov scheme for a general class of (resonant) scalar balance equations using a generalized R... |