Publications on front tracking and operator splitting methods

Hyperbolic Conservation Laws

The front tracking method for scalar problems:

H. Holden, L. Holden, and R. Høegh-Krohn. A numerical method for first order nonlinear scalar conservation laws in one dimension. Comput. Math. Applic. 15, pp. 595-602, 1988.
H. Holden and L. Holden. On scalar conservation laws in one-dimension. In S. Albeverio, J. Fenstad, H. Holden, and T. Lindstrøm, editors, Ideas and Methods in Mathematics and Physics, pp. 480-509. Cambridge University Press, Cambridge, 1988.
K.-A. Lie. Front tracking for one-dimensional quasilinear hyperbolic equations with variable coefficients. Manuscript, 1999. (ps.gz)
K. Hvistendahl Karlsen, K.-A. Lie, and N. H. Risebro. A front tracking method for conservation laws with boundary conditions. In Hyperbolic problems: theory, numerics, applications (Seventh international conference in Zürich, 1998), Eds. M. Fey and R. Jeltsch, ISNM Vol. 129, pp. 493-502, Birkhäuser, 1999. (ps.gz)

The front tracking method for systems:

N. H. Risebro. A front tracking alternative to the random choice method. Proc. Amer. Math. Soc., vol. 117, no. 4 (1993), pp. 1125-1139.
N. H. Risebro and A. Tveito. A front tracking method for conservation laws in one dimension. J. Comp. Phys., vol. 101, no. 1, pp. 130-139, 1991.
N. H. Risebro and A. Tveito. Front tracking applied to a non-strictly hyperbolic system of conservation laws. SIAM J. Sci. Stat. Comput., vol. 12, no. 6, pp. 1401-1419, 1991.
J. O. Langseth, N. H. Risebro, and A. Tveito. A conservative front tracking scheme for 1D hyperbolic conservation laws. In Proc. 4th Internat. Conf. Hyperbolic Problems, Taormina-Italy, pp. 385-392, 1992.

Front tracking with discontinuous flux/coefficients:

T. Gimse. Conservation laws with discontinuous flux functions. SIAM J. Math. Anal., vol. 24, no. 2, pp. 279--289, 1993.
T. Gimse and N. H. Risebro. Riemann problems with a discontinuous flux function. In Proc. 3rd Internat. Conf. Hyperbolic Problems, pp. 488-502, Uppsala, 1991. Studentlitteratur.
T. Gimse and N. H. Risebro. Solution of the Cauchy problem for a conservation law with a discontinuous flux function. SIAM J. Math. Anal., vol. 23, no. 3, pp. 635-648, 1992.
C. Klingenberg and N. H. Risebro. Convex conservation laws with discontinuous coefficients. Existence, uniqueness and asymptotic behavior. Comm. Partial Differential Equations, vol. 20, no. 11-12, 1959--1990, 1995.

Front tracking in multidimensions:

H. Holden and N. H. Risebro. A method for fractional steps for scalar conservation laws without the CFL condition Math. Comput., vol. 60, no. 201, pp. 221-232, 1993.
K.-A. Lie, V. Haugse, and K. Hvistendahl Karlsen. Dimensional splitting with front tracking and adaptive local grid refinement. Numer. Methods Partial Differential Equations, vol. 14, no. 5, pp. 627-648, 1998. (ps.gz)
K.-A. Lie. A dimensional splitting method for quasilinear hyperbolic equations with variable coefficients. BIT, vol. 39, no. 4, 1999. (ps.gz)
H. Holden, K.-A. Lie, and N. H. Risebro. An unconditionally stable method for the Euler equations. J. Comput. Phys., vol. 150, no. 1, pp. 76-96, 1999. (ps.gz)
R. Holdahl, H. Holden, and K.-A. Lie. Unconditionally stable splitting methods for the shallow water equations. BIT, vol. 39, no. 3, pp. 451-472, 1999. (ps.gz)
V. Haugse, K. H. Karlsen, K.-A. Lie, and J. R. Natvig. Numerical solution of the polymer system by front tracking. Preprint, 1999, University of Bergen, Bergen, Norway. (ps.gz)

Operator splitting for inhomogeneous equations

H. Holden and N. H. Risebro. Conservation laws with a random source. Appl. Math. Optim., vol. 36, pp. 229-241, 1997. (ps)
J. O. Langseth, A. Tveito, and R. Winther. On the convergence of operator splitting applied to conservation laws with source terms. SIAM J. Numer. Anal., vol. 33, no. 3, pp. 843-863, 1996.

Implementation of front tracking:

J. O. Langseth. On an implementation of a front tracking method for hyperbolic conservation laws. Advances in Engineering Software, vol. 26, no. 1, pp. 43-63, 1996.

Applications:

F. Bratvedt, K. Bratvedt, C. F. Bucholz, T. Gimse, H. Holden, L. Holden and N. H. Risebro. Frontline and Frontsim, two full scale, two-phase, black oil reservoir simulators based on front tracking Surv. Math. Indu., vol. 3, pp. 185-215, 1993.
F. Bratvedt, K. Bratvedt, C. F. Buchholz, T. Gimse, H. Holden, L. Holden, R. Olufsen, and N. H. Risebro. Three-dimensional reservoir simulation based on front tracking. In North Sea Oil and Gas Reservoirs - III, Kluwer Academic Publishers, 1994.
T. Gimse, H. Holden, and N. H. Risebro. Reservoir simulation by front tracking. In Hyperbolic problems: theory, numerics, applications (Stony Brook, NY, 1994), pp. 52--62, World Sci. Publishing, River Edge, NJ, 1996.
T. Gimse. A numerical method for a system of equations modelling one-dimensional three-phase flow in a porous medium. In Nonlinear hyperbolic equations---theory, computation methods, and applications (Aachen, 1988), p. 159--168, Notes Numer. Fluid Mech., 24, Vieweg, Braunschweig, 1989.
H. Holden and N. H. Risebro. Stochastic properties of the scalar Buckley-Leverett equation. SIAM J. Appl. Math., vol. 51, no. 5, pp. 1472-1488, 1991.
H. Holden and N. H. Risebro. Riemann problems with a kink. SIAM J. Math. Anal., vol 30, no. 3, pp. 497 - 515. (ps)

Convection-Diffusion Equations

Operator splitting:

K. H. Karlsen and N. H. Risebro. An operator splitting method for nonlinear convection-diffusion equations. Numer. Math., vol. 77, no. 3, pp. 365-382, 1997. (ps)
M. S. Espedal and K. H. Karlsen. Numerical solution of reservoir flow models based on large time step operator splitting algorithms. To appear in Lecture Notes in Mathematics, Springer. (ps)

Corrected operator splitting:

K. H. Karlsen and N. H. Risebro. Corrected operator splitting for nonlinear parabolic equations. To appear in SIAM J. Numer. Anal. (ps)
K. H. Karlsen, K. Brusdal, H. K. Dahle, S. Evje, and K.-A. Lie. The corrected operator splitting approach applied to an advection-diffusion problem. Comput. Methods Appl. Mech. Engrg., vol 167, no. 3-4, pp. 239-260, 1998. (ps.gz)
K. H. Karlsen and K.-A. Lie. An unconditionally stable splitting for a class of nonlinear parabolic equations. To appear in IMA J. Numer. Anal. (ps.gz)
K. H. Karlsen, K.-A. Lie, N. H. Risebro and J. Frøyen. A front tracking approach to a two-phase fluid flow model with capillary forces. In Situ, vol. 22,no. 1, pp. 59-89, 1998. (ps.gz)

Mixed hyperbolic-parabolic equations:

S. Evje and K. H. Karlsen. Viscous splitting approximation of mixed hyperbolic-parabolic convection-diffusion equations. To appear in Numer. Math. (ps)
S. Evje, K. H. Karlsen, K.-A. Lie, and N. H. Risebro. Front tracking and operator splitting for nonlinear degenerate convection-diffusion equations. To appear in Parallel solution of partial differential equations, eds., P. Bjørstad and M. Luskin, ISNM Vol. 129, pp. 493-502, Birkhäuser, 1999. (ps.gz)

Applications:

R. Bürger, S. Evje, K. Hvistendahl Karlsen, and K.-A. Lie. Numerical methods for the simulation of the settling of flocculated suspensions. Preprint 99/03, Sonderforschungsbereich 404, University of Stuttgart, Stuttgart, Germany. (ps.gz)
K. Brusdal, H. K. Dahle, K. Hvistendahl Karlsen, and T. Mannseth. A study of the modelling error in two operator splitting algorithms for porous media flow. Comp. Geosciences, vol. 2, pp. 23-36, 1998, (ps)

Knut-Andreas Lie <andreas@math.ntnu.no>
Last modified: Thu Jun 24 13:16:51 1999