Nils Henrik Risebro

• Holden, Helge & Risebro, Nils Henrik (2018). Continuum limit of Follow-the-Leader models – a short proof. Discrete and Continuous Dynamical Systems. Series A.  ISSN 1078-0947.  38(2), s 715- 722 . doi: 10.3934/dcds.2018031 Fulltekst i vitenarkiv.
• Holden, Helge & Risebro, Nils Henrik (2018). Follow-the-leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow. Networks and Heterogeneous Media.  ISSN 1556-1801.  13(3), s 409- 421 . doi: 10.3934/nhm.2018018
• Coclite, Giuseppe Maria; Ridder, Johanna & Risebro, Nils Henrik (2017). A Convergent Finite Difference Scheme for the Variational Heat Equation. Zeitschrift für Angewandte Mathematik und Physik.  ISSN 0044-2275.  68(6) . doi: 10.1007/s00033-017-0871-z
• Coclite, Giuseppe Maria; Ridder, Johanna & Risebro, Nils Henrik (2017). A convergent finite difference scheme for the Ostrovsky-Hunter equation on a bounded domain. BIT Numerical Mathematics.  ISSN 0006-3835.  57(1), s 93- 122 . doi: 10.1007/s10543-016-0625-x
• Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik & Storrøsten, Erlend Briseid (2017). Practical Convergence Rates for Degenerate Parabolic Equations, In Laurent Gosse & Roberto Natalini (ed.),  Innovative Algorithms and Analysis.  Springer.  ISBN 978-3-319-49261-2.  kapittel.  s 243 - 263
• Koley, Ujjwal; Risebro, Nils Henrik; Schwab, Christoph & Weber, Franziska Leonie (2017). A multilevel Monte Carlo finite difference method for random scalar degenerate convection–diffusion equations. Journal of Hyperbolic Differential Equations.  ISSN 0219-8916.  14(3), s 415- 454 . doi: 10.1142/S021989161750014X Fulltekst i vitenarkiv.
• Risebro, Nils Henrik; Schwab, Christoph & Weber, Franziska (2017). Correction to: Multilevel Monte Carlo front-tracking for random scalar conservation laws (BIT Numerical Mathematics, (2016), 56, 1, (263-292), 10.1007/s10543-015-0550-4). BIT Numerical Mathematics.  ISSN 0006-3835.  58(1), s 247- 255 . doi: 10.1007/s10543-017-0670-0
• Coclite, Giuseppe Maria; Reichmann, Oleg & Risebro, Nils Henrik (2016). A convergent difference scheme for a class of partial integro-differential equations modeling pricing under uncertainty. SIAM Journal on Numerical Analysis.  ISSN 0036-1429.  54(2), s 588- 605 . doi: 10.1137/15M1025530
• Dutta, Rajib; Holden, Helge; Koley, Ujjwal & Risebro, Nils Henrik (2016). Convergence of finite difference schemes for the Benjamin–Ono equation. Numerische Mathematik.  ISSN 0029-599X.  134(2), s 249- 274 . doi: 10.1007/s00211-015-0778-6
• Dutta, Rajib & Risebro, Nils Henrik (2016). A note on the convergence of a crank-nicolson scheme for the KDV equation. International Journal of Numerical Analysis & Modeling.  ISSN 1705-5105.  13(5), s 657- 675
• Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik & Storrøsten, Erlend Briseid (2016). On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions. Mathematical Modelling and Numerical Analysis.  ISSN 0764-583X.  50(2), s 499- 539 . doi: 10.1051/m2an/2015057
• Mishra, Siddhartha; Risebro, Nils Henrik; Schwab, Christoph & Tokareva, Svetlana (2016). Numerical solution of scalar conservation laws with random flux functions. SIAM/ASA Journal on Uncertainty Quantification (JUQ).  ISSN 2166-2525.  4(1), s 552- 591 . doi: 10.1137/120896967
• Risebro, Nils Henrik; Schwab, Christoph & Weber, Franziska Leonie (2016). Multilevel Monte Carlo front-tracking for random scalar conservation laws. BIT Numerical Mathematics.  ISSN 0006-3835.  56(1), s 263- 292 . doi: 10.1007/s10543-015-0550-4
• Dutta, Rajib; Holden, Helge; Koley, Ujjwal & Risebro, Nils Henrik (2015). Operator splitting for the Benjamin-Ono equation. Journal of Differential Equations.  ISSN 0022-0396.  259(11), s 6694- 6717 . doi: 10.1016/j.jde.2015.08.002 Fulltekst i vitenarkiv.
• Dutta, Rajib; Koley, Ujjwal & Risebro, Nils Henrik (2015). Convergence of a higher order scheme for the korteweg-de vries equation. SIAM Journal on Numerical Analysis.  ISSN 0036-1429.  53(4), s 1963- 1983 . doi: 10.1137/140982532
• Holden, Helge; Koley, Ujjwal & Risebro, Nils Henrik (2015). Convergence of a fully discrete finite difference scheme for the Korteweg-de Vries equation. IMA Journal of Numerical Analysis.  ISSN 0272-4979.  35(3), s 1047- 1077 . doi: 10.1093/imanum/dru040 Fulltekst i vitenarkiv.
• Agrell, Christian & Risebro, Nils Henrik (2014). Convergence of a relaxation scheme for a 2 × 2 triangular system of conservation laws. International Journal of Numerical Analysis & Modeling.  ISSN 1705-5105.  11(1), s 148- 171
• Coclite, Giuseppe Maria; Mishra, Siddhartha; Risebro, Nils Henrik & Weber, Franziska Leonie (2014). Analysis and numerical approximation of Brinkman regularization of two-phase flows in porous media. Computational Geosciences.  ISSN 1420-0597.  18(5), s 637- 659 . doi: 10.1007/s10596-014-9410-6
• Fuchs, Franz Georg; McMurry, Andrew David; Risebro, Nils Henrik & Mishra, Siddhartha (2014). Explicit and implicit finite volume schemes for radiation MHD and the effects of radiation on wave propagation in stratified atmospheres, In Fabio Ancona; Alberto Bressan; Pierangelo Marcati & Andrea Marson (ed.),  Hyperbolic Problems: Theory, Numerics, Applications. Proceedings of the Fourteenth International Conference on Hyperbolic Problems held in Padova, June 25-29, 2012.  American Institute of Mathematical Sciences.  ISBN 978-1-60133-017-8.  artikkel.  s 41 - 59 Vis sammendrag

  We present a radiation MHD model based on moments (M1) of the radiative transport equation. This M1 model is approximated numerically by robust finite volume schemes. We compare explicit and semi-implicit schemes and show how radiation affects wave propagation in stratified atmospheres.

• Holden, Helge; Priuli, Fabio Simone & Risebro, Nils Henrik (2014). On an inverse problem for scalar conservation laws. Inverse Problems.  ISSN 0266-5611.  30(3) . doi: 10.1088/0266-5611/30/3/035015
• Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik & Storrøsten, Erlend Briseid (2014). L1 error estimates for difference approximations of degenerate convection-diffusion equations. Mathematics of Computation.  ISSN 0025-5718.  83(290), s 2717- 2762 . doi: 10.1090/S0025-5718-2014-02818-4
• Frid, Hermano; Risebro, Nils Henrik & Sande, Hilde (2013). SPATIALLY PERIODIC SOLUTIONS FOR GAS FLOWS WITH PRESSURE DEPENDING ON A VARIABLE COEFFICIENT. Journal of Hyperbolic Differential Equations.  ISSN 0219-8916.  10(1), s 129- 148 . doi: 10.1142/S0219891613500057
• Holden, Helge; Lubich, Christian & Risebro, Nils Henrik (2013). Operator splitting for partial differential equations with Burgers nonlinearity. Mathematics of Computation.  ISSN 0025-5718.  82, s 173- 185 . doi: 10.1090/S0025-5718-2012-02624-X
• Koley, Ujjwal & Risebro, Nils Henrik (2013). Finite difference schemes for the symmetric Keyfitz-Kranzer system. Zeitschrift für Angewandte Mathematik und Physik.  ISSN 0044-2275.  64(4), s 1057- 1085 . doi: 10.1007/s00033-012-0292-y
• Risebro, Nils Henrik & Weber, Franziska Leonie (2013). A note on front tracking for the Keyfitz-Kranzer system. Journal of Mathematical Analysis and Applications.  ISSN 0022-247X.  407(2), s 190- 199 . doi: 10.1016/j.jmaa.2013.05.026
• Coclite, Giuseppe Maria; Karlsen, Kenneth Hvistendahl; Mishra, Siddhartha & Risebro, Nils Henrik (2012). A HYPERBOLIC-ELLIPTIC MODEL OF TWO-PHASE FLOW IN POROUS MEDIA EXISTENCE OF ENTROPY SOLUTIONS. International Journal of Numerical Analysis & Modeling.  ISSN 1705-5105.  9(3), s 562- 583
• Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik & Koley, Ujjwal (2012). An error estimate for the finite difference approximation to degenerate convection-diffusion equations. Numerische Mathematik.  ISSN 0029-599X.  121(2), s 367- 395 . doi: 10.1007/s00211-011-0433-9
• Koley, Ujjwal; Mishra, Siddhartha; Risebro, Nils Henrik & Svärd, Magnus (2012). Higher-order finite difference schemes for the magnetic induction equations with resistivity. IMA Journal of Numerical Analysis.  ISSN 0272-4979.  32(3), s 1173- 1193 . doi: 10.1093/imanum/drq030
• Andreianov, Boris; Karlsen, Kenneth Hvistendahl & Risebro, Nils Henrik (2011). A Theory of L (1)-Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux. Archive for Rational Mechanics and Analysis.  ISSN 0003-9527.  201(1), s 27- 86 . doi: 10.1007/s00205-010-0389-4
• Fuchs, Franz Georg; McMurry, Andrew David; Mishra, Siddhartha; Risebro, Nils Henrik & Waagan, Knut (2011). Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations. Communications in Computational Physics.  ISSN 1815-2406.  9(2), s 324- 362 . doi: 10.4208/cicp.171109.070510a
• Holden, Helge; Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik & Tao, T (2011). OPERATOR SPLITTING FOR THE KdV EQUATION. Mathematics of Computation.  ISSN 0025-5718.  80(274), s 821- 846 . doi: 10.1090/S0025-5718-2010-02402-0
• Andreianov, Boris; Karlsen, Kenneth Hvistendahl & Risebro, Nils Henrik (2010). ON VANISHING VISCOSITY APPROXIMATION OF CONSERVATION LAWS WITH DISCONTINUOUS FLUX. Networks and Heterogeneous Media.  ISSN 1556-1801.  5(3), s 617- 633 . doi: 10.3934/nhm.2010.5.617
• Coclite, Giuseppe Maria; Mishra, Siddhartha & Risebro, Nils Henrik (2010). CONVERGENCE OF AN ENGQUIST-OSHER SCHEME FOR A MULTI-DIMENSIONAL TRIANGULAR SYSTEM OF CONSERVATION LAWS. Mathematics of Computation.  ISSN 0025-5718.  79(269), s 71- 94 . doi: 10.1090/S0025-5718-09-02251-0 Vis sammendrag

  (utgivelses år må dobbelsjekkes før posten kan godkjennes)

• Fuchs, Franz Georg; McMurry, Andrew David; Mishra, Siddhartha; Risebro, Nils Henrik & Waagan, Knut (2010). Finite Volume Methods for Wave Propagation in Stratified Magneto-Atmospheres. Communications in Computational Physics.  ISSN 1815-2406.  7(3), s 403- 638 . doi: 10.4208/cicp.2009.08.154
• Fuchs, Franz Georg; McMurry, Andrew David; Mishra, Siddhartha; Risebro, Nils Henrik & Waagan, Knut (2010). High order well-balanced finite volume schemes for simulating wave propagation in stratified magnetic atmospheres. Journal of Computational Physics.  ISSN 0021-9991.  229(11), s 4033- 4058 . doi: 10.1016/j.jcp.2010.01.038
• Holden, Helge; Risebro, Nils Henrik & Sande, Hilde (2010). Front tracking for a model of immiscible gas flow with large data. BIT Numerical Mathematics.  ISSN 0006-3835.  50(2), s 331- 376 . doi: 10.1007/s10543-010-0264-6
• Fuchs, Franz Georg; Mishra, Siddhartha & Risebro, Nils Henrik (2009). Splitting based finite volume schemes for ideal MHD equations. Journal of Computational Physics.  ISSN 0021-9991.  228(3), s 641- 660 . doi: 10.1016/j.jcp.2008.09.027
• Holden, Helge; Karlsen, KH & Risebro, N.H. (2009). A convergent finite-difference method for a nonlinear variational wave equation. IMA Journal of Numerical Analysis.  ISSN 0272-4979.  29(3), s 539- 572 . doi: 10.1093/imanum/drn026
• Holden, Helge; Risebro, Nils H. & Sande, Hilde (2009). THE SOLUTION OF THE CAUCHY PROBLEM WITH LARGE DATA FOR A MODEL OF A MIXTURE OF GASES. Journal of Hyperbolic Differential Equations.  ISSN 0219-8916.  6(1), s 25- 106
• Holden, Helge; Risebro, Nils Henrik & Sande, Hilde (2009). Convergence of front tracking and the Glimm scheme for a model of the flow of immiscible gases, In Eitan Tadmor (ed.),  Hyperbolic Problems: Theory, Numerics and Applications.  American Mathematical Society (AMS).  ISBN 978-0-8218-4728-2.  kapittel.  s 653 - 662
• Holden, Helge; Risebro, Nils Henrik & Sande, Hilde (2009). Convergens of front tracking and the Glimm scheme for a model of the flow of immiscible gases, In Eitan Tadmor; Jian-Guo Liu & Athanasios E. Tzavaras (ed.),  Hyperbolic Problems: Theory, Numerics and Applications.  American Mathematical Society (AMS).  ISBN 978-0-8218-4728-2.  Kapitel.  s 653 - 662
• Karlsen, Kenneth Hvistendahl; Fuchs, Franz Georg; Mishra, Siddhartha & Risebro, Nils Henrik (2009). Stable upwind schemes for the magnetic induction equation. Mathematical Modelling and Numerical Analysis.  ISSN 0764-583X.  43(5), s 825- 852 . doi: 10.1051/m2an/2009006
• Karlsen, Kenneth Hvistendahl; Mishra, Siddhartha & Risebro, Nils Henrik (2009). Convergence of finite volume schemes for triangular systems of conservation laws. Numerische Mathematik.  ISSN 0029-599X.  111(4), s 559- 589 . doi: 10.1007/s00211-008-0199-x
• Karlsen, Kenneth Hvistendahl; Mishra, Siddhartha & Risebro, Nils Henrik (2009). Semi-Godunov schemes for multiphase flows in porous media. Applied Numerical Mathematics.  ISSN 0168-9274.  59(9), s 2322- 2336 . doi: 10.1016/j.apnum.2008.12.019 Vis sammendrag

  We describe a class of finite volume schemes for 2×2 systems of conservations laws based on a “local” decomposition of the system into a series of single conservation laws but with discontinuous coefficients. The resulting schemes are based on Godunov type solvers of the reduced equations. These schemes are very easy to implement since they do not use detailed information about the eigenstructure of the full system. We illustrate the efficiency of the schemes on a variety of numerical experiments focusing on three-phase flows in porous media and show that they are robust and approximate the flow very well, even in the presence of gravity.

• Karlsen, Kenneth Hvistendahl; Mishra, Siddhartha & Risebro, Nils Henrik (2009). WELL-BALANCED SCHEMES FOR CONSERVATION LAWS WITH SOURCE TERMS BASED ON A LOCAL DISCONTINUOUS FLUX FORMULATION. Mathematics of Computation.  ISSN 0025-5718.  78(265), s 55- 78
• Koley, Ujjwal; Mishra, Siddhartha; Risebro, Nils Henrik & Svärd, Magnus (2009). Higher order finite difference schemes for the magnetic induction equations. BIT Numerical Mathematics.  ISSN 0006-3835.  49(2), s 375- 395 . doi: 10.1007/s10543-009-0219-y
• Risebro, Nils Henrik (2009). An introduction to scalar conservation laws with spatially discontinuous flux functions, In Tarmo Soomere (ed.),  Applied wave mathematics.  Springer.  ISBN 978-3-642-00584-8.  Mathematical methods.  s 395 - 464
• Risebro, Nils Henrik; Karlsen, Kenneth Hvistendahl & Coclite, Giuseppe Maria (2009). Convergence of vanishing viscos- ity approximations of 2x2 triangular systems of multi-dimensional conservation laws. Bollettino dell'Unione Matematica Italiana, Sezione B..  ISSN 0392-4041.  9(2), s 275- 284
• Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik & Coclite, Giuseppe Maria (2008). A convergent finite difference scheme for the Camassa-Holm equation with general H^1 initial data. SIAM Journal on Numerical Analysis.  ISSN 0036-1429.  46, s 1554- 1579
• Risebro, Nils Henrik; Karlsen, Kenneth Hvistendahl & Coclite, Giuseppe Maria (2008). An explicit finite difference scheme for the Camassa-Holm equation. Advances in Difference Equations.  ISSN 1687-1839.  13(7-8), s 681- 732
• Coclite, GM; Karlsen, Kenneth Hvistendahl & Risebro, Nils Henrik (2008). Numerical schemes for computing discontinuous solutions of the Degasperis-Procesi equation. IMA Journal of Numerical Analysis.  ISSN 0272-4979.  28 . doi: 10.1093/imanum/drm003
• Karlsen, Kenneth Hvistendahl; Mishra, Siddhartha & Risebro, Nils Henrik (2008). A large-time-stepping scheme for balance equations. Journal of Engineering Mathematics.  ISSN 0022-0833.  60, s 351- 363 . doi: 10.1007/s10665-007-9164-4
• Karlsen, Kenneth Hvistendahl; Mishra, Siddhartha & Risebro, Nils Henrik (2008). Semi-Godunov schemes for general triangular systems of conservation laws. Journal of Engineering Mathematics.  ISSN 0022-0833.  60 . doi: 10.1007/s10665-007-9163-5
• Risebro, Nils Henrik; Karlsen, Kenneth Hvistendahl & Holden, Helge (2008). A convergent finite difference method for a variational wave equation. IMA Journal of Numerical Analysis.  ISSN 0272-4979.  s 1- 34 . doi: 10.1093/imanum/drn026
• Coclite, Giuseppe Maria & Risebro, Nils Henrik (2007). Viscosity solutions of Hamilton-Jacobi equations with discontinuous coefficients. Journal of Hyperbolic Differential Equations.  ISSN 0219-8916.  4(4), s 771- 795
• Holden, Helge; Karlsen, Kenneth Hvistendahl & Risebro, Nils Henrik (2007). Convergent difference schemes for the Hunter-Saxton equation. Mathematics of Computation.  ISSN 0025-5718.  76, s 699- 744
• Burger, R; Karlsen, Kenneth Hvistendahl & Risebro, Nils Henrik (2005). A relaxation scheme for continuous sedimentation in ideal clarifier-thickener units. Computers and Mathematics with Applications.  ISSN 0898-1221.  50(7), s 687- 704
• Coclite, Giuseppe Maria & Risebro, Nils Henrik (2005). Conservation laws with time dependent discontinuous coefficients. SIAM Journal on Mathematical Analysis.  ISSN 0036-1410.  36
• Burger, Raimund; Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik & Towers, John D. (2004). Monotone difference approximations for the simulation of clarifier-thickener units. Computing and Visualization in Science.  ISSN 1432-9360.  6, s 83- 91
• Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik & Towers, John D. (2004). Front tracking for scalar balance equations. Journal of Hyperbolic Differential Equations.  ISSN 0219-8916.  1(1), s 115- 148
• Burger, R; Karlsen, Kenneth Hvistendahl; Risebro, NH & Towers, JD (2004). Numerical methods for the simulation of continuous sedimentation in ideal clarifier-thickener units. International Journal of Mineral Processing.  ISSN 0301-7516.  73, s 209- 228
• Burger, R; Karlsen, Kenneth Hvistendahl; Risebro, NH & Towers, JD (2004). Well-posedness in BVt and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units. Numerische Mathematik.  ISSN 0029-599X.  97, s 25- 65
• Karlsen, Kenneth Hvistendahl; Klingenberg, C & Risebro, NH (2004). A relaxation scheme for conservation laws with a discontinuous coefficient. Mathematics of Computation.  ISSN 0025-5718.  73, s 1235- 1259
• Burger, Raimund; Karlsen, Kenneth Hvistendahl; Klingenberg, Christian & Risebro, Nils Henrik (2003). A front tracking approach to a model of continuous sedimentation in ideal clarifier-thickener units. Nonlinear Analysis: Real world applications.  ISSN 1468-1218.  4(3), s 457- 481
• Burger, Raimund; Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik & Towers, John D. (2003). On a model for continuous sedimentation in vessels with discontinuously varying cross-sectional area. Proceedings / Hyperbolic Problems: Theory, Numerics, Applications.  s 397- 406
• Holden, Helge; Karlsen, Kenneth Hvistendahl & Risebro, Nils Henrik (2003). On uniqueness and existence of entropy solutions of weakly coupled systems of nonlinear degenerate parabolic equations. Electronic Journal of Differential Equations.  ISSN 1072-6691.  46, s 1- 31
• Holden, Helge; Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik & Risebro, Nils Henrik (2003). On uniqueness and existence of entropy solutions of weakly coupled systems of nonlinear degenerate parabolic systems. Electronic Journal of Differential Equations.  ISSN 1072-6691.  2003(46), s 1- 31
• Karlsen, Kenneth Hvistendahl; Klingenberg, Christian & Risebro, Nils Henrik (2003). Relaxation schemes for conservation laws with discontinuous coefficients. Proceedings / Hyperbolic Problems: Theory, Numerics, Applications.  s 611- 620
• Karlsen, Kenneth Hvistendahl & Risebro, Nils Henrik (2003). On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete and Continuous Dynamical Systems. Series A.  ISSN 1078-0947.  9(5), s 1081- 1104
• Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik & Towers, John D. (2003). L^1 stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients. Skrifter.  ISSN 0368-6310.  13, s 1- 49
• Risebro, Nils Henrik (2003). A front tracking approach to a model of continuous sedimentation in ideal clarifier-thickener units. Nonlinear Analysis: Real world applications.  ISSN 1468-1218.  4(3), s 457- 481
• Risebro, Nils Henrik (2003). $L^1$ stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients. Skrifter.  ISSN 0368-6310.  3
• Risebro, Nils Henrik (2003). On a model for continuous sedimentation in vessels with discontinuously varying cross-sectional area. Hyperbolic Problems: Theory, Numerics, Applications..  s 397- 406
• Risebro, Nils Henrik (2003). Relaxation schemes for conservation laws with discontinuous coefficients. Hyperbolic Problems: Theory, Numerics, Applications..  s 611- 620
• Risebro, Nils Henrik (2003). When are HJB-equations for control problems with stochastic delay equations finite dimensional?. Stochastic Analysis and Applications.  ISSN 0736-2994.  21(3), s 643- 671
• Holden, Helge; Risebro, Nils Henrik & Hanche-Olsen, Harald (2002). The Riemann problem for an elastic string with a linear Hooke's law. Quarterly of Applied Mathematics.  ISSN 0033-569X.  60(4), s 695- 705
• Risebro, Nils Henrik (2002). On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient. Electronic Journal of Differential Equations.  ISSN 1072-6691.  (93), s 1- 23
• Risebro, Nils Henrik & Karlsen, Kenneth Hvistendahl (2002). A note on front tracking and the equivalence between viscosity solutions of Hamilton-Jacobi equations and entropy solutions of scalar conservation laws. Nonlinear Analysis.  ISSN 0362-546X.  50, s 455- 469 Vis sammendrag

  We directly show the equivalence between these two solution concepts

• Risebro, Nils Henrik & Karlsen, Kenneth Hvistendahl (2002). Unconditionally stable methods for Hamilton-Jacobi equations. Journal of Computational Physics.  ISSN 0021-9991.  180, s 710- 735
• Risebro, Nils Henrik; Karlsen, Kenneth Hvistendahl & Towers, John D. (2002). On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient. Electronic Journal of Differential Equations.  ISSN 1072-6691.  93, s 1- 23
• Risebro, Nils Henrik; Karlsen, Kenneth Hvistendahl & Towers, John D. (2002). Upwind difference approximations for degenerate parabolic convection diffusion equations with a discontinuous coefficient. IMA Journal of Numerical Analysis.  ISSN 0272-4979.  22, s 623- 664
• Jakobsen, Espen Robstad; Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik & Risebro, Nils Henrik (2001). On the convergence rate of operator splitting for Hamilton-Jacobi equations with source terms. SIAM Journal on Numerical Analysis.  ISSN 0036-1429.  39(2), s 499- 518
• Jacobsen, E.R.; Karlsen, K. & Risebro, Nils Henrik (2001). On the convergence rate of operator splitting for Hamilton-Jacobi equations with source terms. SIAM Journal on Numerical Analysis.  ISSN 0036-1429.  39(2), s 499- 518
• Karlsen, K. & Risebro, Nils Henrik (2001). Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients. Mathematical Modelling and Numerical Analysis.  ISSN 0764-583X.  35(2), s 239- 270
• Klingenberg, Christian & Risebro, Nils Henrik (2001). Stability of a Resonant System of Conservation Laws Modeling Polymer Flow with Gravitation. Journal of Differential Equations.  ISSN 0022-0396.  170, s 344- 380
• Evje, Steinar; Karlsen, Kenneth Hvistendahl; Lie, Knut-Andreas & Risebro, Nils Henrik (2000). Front tracking and operator splitting for nonlinear degenerate convection-diffusion equations. IMA Journal of Applied Mathematics.  ISSN 0272-4960.  120, s 209- 227
• Karlsen, Kenneth Hvistendahl; Lie, Knut-Andreas & Risebro, Nils Henrik (2000). A fast marching method for reservoir simulation. Computational Geosciences.  ISSN 1420-0597.  4, s 185- 206 Vis sammendrag

  We present a fast marching level set method for reservoir simulation based on a fractional flow formulation of two-phase, incompressible, immiscible flow in two or three space dimensions. The method uses a fast marching approach and is therefore considerably faster than conventional finite difference methods. The fast marching approach compares favorably with a front tracking method what regards both efficiency and accuracy. In addition, it maintains the advantage of being able to handle changing topologies of the front structure.

• Karlsen, Kenneth Hvistendahl & Risebro, Nils Henrik (2000). Corrected operator splitting for convection diffusion equations. SIAM Journal on Numerical Analysis.  ISSN 0036-1429.  37(3), s 980- 1003 Vis sammendrag

  We present a corrected operator splitting (COS) method for solving nonlinear parabolic equations of convection-diffusion type. The main feature of this method is the ability to correctly resolve nonlinear shock fronts for large time steps, as opposed to standard operator splitting (OS) which fails to do so. COS is based on solving a conservation law for modeling convection, a heat type equation for modeling diffusion, and finally a certain ``residual'' conservation law for necessary correction. The residual equation represents the entropy loss generated in the hyperbolic (convection) step. In OS the entropy loss manifests itself in the form of too wide shock fronts. The purpose of the correction step in COS is to counterbalance the entropy loss so that correct width of nonlinear shock fronts is ensured. The polygonal method of Dafermos constitutes an important part of our solution strategy. It is shown that COS generates a compact sequence of approximate solutions which converges to the solution of the problem. Finally, some numerical examples are presented where we compare OS and COS methods with respect to accuracy.

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• Gesztesy, Fritz; Hanche-Olsen, Harald; Jakobsen, Espen Robstad; Lyubarskii, Yurii; Risebro, Nils Henrik & Seip, Kristian (2018). Non-linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis: The Helge Holden Anniversary Volume. European Mathematical Society Publishing House.  ISBN 978-3-03719-186-6.  486 s.
• Holden, Helge & Risebro, Nils Henrik (2015). Front Tracking for Hyperbolic Conservation Laws. Springer Science+Business Media B.V..  ISBN 978-3-662-47506-5.  515 s.
• Holden, Helge & Risebro, Nils Henrik (2011). Front Tracking for Hyperbolic Conservation Laws. Springer Science+Business Media B.V..  ISBN 9783642239106.  361 s.
• Holden, Helge; Karlsen, Kenneth Hvistendahl; Lie, Knut-Andreas & Risebro, Nils Henrik (2010). Splitting Methods for Partial Differential Equations with Rough Solutions. Analysis and Matlab Programs. European Mathematical Society Publishing House.  ISBN 978-3-03719-078-4.  226 s.
• Holden, Helge & Risebro, Nils Henrik (2002). Front Tracking for Hyperbolic Conservation Laws. Springer.  ISBN 3-540-43289-2.  380 s.

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• Rasmussen, Atgeirr Flø; Floater, Michael S. & Risebro, Nils Henrik (2006). Reparametrization of surfaces using PDE methods.
• Rasmussen, Atgeirr Flø; Floater, Michael S. & Risebro, Nils Henrik (2006). Reparametrization of surfaces using PDE methods.
• Rasmussen, Atgeirr Flø; Floater, Michael S. & Risebro, Nils Henrik (2006). Reparametrization of surfaces using PDE methods.
• Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik & Towers, John D. (2003). Front tracking for scalar balance equations.
• Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik & Towers, John D. (2003). L1 stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients.
• Jakobsen, Espen Robstad; Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik & Risebro, Nils Henrik (2001). On the convergence rate of operator splitting for weakly coupled systems of Hamilton-Jacobi equations.
• Karlsen, Kenneth Hvistendahl; Lie, K. A.; Nativig, J. & Risebro, Nils Henrik (2000). A fast marching method for 3D reservoir simulation.

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