Ulrik Skre Fjordholm

• Fjordholm, U.S. & Mæhlen, Ola Isaac Høgåsen (2025). Transport equations for Osgood velocity fields. Journal of Differential Equations. ISSN 0022-0396. 443. doi: 10.1016/j.jde.2025.113566. Full text in Research Archive Show summary

  We consider the transport equation with a velocity field satisfying the Osgood condition. The weak formulation is not meaningful in the usual Lebesgue sense, meaning that the usual DiPerna–Lions treatment of the problem is not applicable (in particular, the divergence of the velocity might be unbounded). Instead, we use Riemann–Stieltjes integration to interpret the weak formulation, leading to a well-posedness theory in regimes not covered by existing works. The most general results are for the one-dimensional problem, with generalisations to multiple dimensions in the particular case of log-Lipschitz velocities.

• Fjordholm, Ulrik Skre; Karlsen, Kenneth Aksel Hvistendahl & Pang, Ho Cheung (2025). Convergent Finite Difference Schemes for Stochastic Transport Equations. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 63(1), p. 149–192. doi: 10.1137/23M159946X. Full text in Research Archive Show summary

  We present difference schemes for stochastic transport equations with low-regularity velocity fields. We establish 𝐿2 stability and convergence of the difference approximations under conditions that are less strict than those required for deterministic transport equations. The 𝐿2 estimate, crucial for the analysis, is obtained through a discrete duality argument and a comprehensive examination of a class of backward parabolic difference schemes.

• Fjordholm, Ulrik Skre; Mishra, Siddhartha & Weber, Franziska (2024). On the Vanishing Viscosity Limit of Statistical Solutions of the Incompressible Navier–Stokes Equations. SIAM Journal on Mathematical Analysis. ISSN 0036-1410. 56(4), p. 5099–5143. doi: 10.1137/23M1566261. Full text in Research Archive Show summary

  We study statistical solutions of the incompressible Navier–Stokes equation and their vanishing viscosity limit. We show that a formulation using correlation measures as in [U. S. Fjordholm, S. Lanthaler, and S. Mishra, Arch. Ration. Mech. Anal., 226 (2017), pp. 809–849] and moment equations is equivalent to statistical solutions in the Foiaş–Prodi sense. Under the assumption of weak scaling, a weaker version of Kolmogorov’s self-similarity at small scales hypothesis that allows for intermittency corrections, we show that the limit is a statistical solution of the incompressible Euler equations. To pass to the limit, we derive a Kármán–Howarth–Monin relation for statistical solutions and combine it with the weak scaling assumption and a compactness theorem for correlation measures from [U. S. Fjordholm et al., Math. Models Methods Appl. Sci., 30 (2020), pp. 539–609].

• Fjordholm, Ulrik Skre; Mæhlen, Ola Isaac Høgåsen & Ørke, Magnus Christie (2024). The particle paths of hyperbolic conservation laws. Mathematical Models and Methods in Applied Sciences. ISSN 0218-2025. 34(7), p. 1205–1234. doi: 10.1142/S0218202524500209. Full text in Research Archive Show summary

  Nonlinear scalar conservation laws are traditionally viewed as transport equations. We take instead the viewpoint of these PDEs as continuity equations with an implicitly defined velocity field. We show that a weak solution is the entropy solution if and only if the ODE corresponding to its velocity field is well-posed. We also show that the flow of the ODE is 1/2-Hölder regular. Finally, we give several examples showing that our results are sharp, and we provide explicit computations in the case of a Riemann problem.

• Musch, Markus; Fjordholm, Ulrik Skre & Risebro, Nils Henrik (2022). Well-posedness theory for nonlinear scalar conservation laws on networks. Networks and Heterogeneous Media. ISSN 1556-1801. 17(1), p. 101–128. doi: 10.3934/nhm.2021025. Full text in Research Archive Show summary

  We consider nonlinear scalar conservation laws posed on a network. We define an entropy condition for scalar conservation laws on networks and establish $L^1$ stability, and thus uniqueness, for weak solutions satisfying the entropy condition. We apply standard finite volume methods and show stability and convergence to the unique entropy solution, thus establishing existence of a solution in the process. Both our existence and stability/uniqueness theory is centred around families of stationary states for the equation. In one important case – for monotone fluxes with an upwind difference scheme – we show that the set of (discrete) stationary solutions is indeed sufficiently large to suit our general theory. We demonstrate the method's properties through several numerical experiments.

• Fjordholm, Ulrik Skre & Lye, Kjetil Olsen (2022). Convergence Rates of Monotone Schemes for Conservation Laws for Data with Unbounded Total Variation. Journal of Scientific Computing. ISSN 0885-7474. 91. doi: 10.1007/s10915-022-01806-x. Full text in Research Archive Show summary

  We prove convergence rates of monotone schemes for conservation laws for Hölder continuous initial data with unbounded total variation, provided that the Hölder exponent of the initial data is greater than 12/. For strictly Lip+ stable monotone schemes, we prove convergence for any positive Hölder exponent. Numerical experiments are presented which verify the theory.

• Fjordholm, Ulrik Skre; Musch, Markus & Risebro, Nils Henrik (2022). Well-Posedness and Convergence of a Finite Volume Method for Conservation Laws on Networks. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 60(2), p. 606–630. doi: 10.1137/21M145001X. Full text in Research Archive Show summary

  We continue the analysis of nonlinear conservation laws on networks initiated in [M. Musch, U. S. Fjordholm, and N. H. Risebro, Netw. Heterog. Media, 17 (2022), pp. 101--128] by extending our analysis to a large class of flux functions which must be neither monotone nor convex/concave. We utilize the framework laid down in [M. Musch, U. S. Fjordholm, and N. H. Risebro, Netw. Heterog. Media, 17 (2022), pp. 101--128] and prove existence and uniqueness within a natural class of entropy solutions via the convergence of an explicit finite volume method. In particular, this leads to the existence of a semigroup of solutions. The theoretical results are supported with numerical experiments including an experimental order of convergence.

• Fjordholm, Ulrik Skre & Ruf, Adrian Montgomery (2021). Second-order accurate TVD numerical methods for nonlocal nonlinear conservation laws. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 59(3), p. 1167–1194. doi: 10.1137/20M1360979. Full text in Research Archive Show summary

  We present a second-order accurate numerical method for a class of nonlocal nonlinear conservation laws called the “nonlocal pair-interaction model,” which was recently introduced by Du, Huang, and LeFloch [SIAM J. Numer. Anal., 55 (2017), pp. 2465--2489]. Our numerical method uses second-order accurate reconstruction-based schemes for local conservation laws in conjunction with appropriate numerical integration. We show that the resulting method is total variation diminishing (TVD) and converges towards a weak solution. In fact, in contrast to local conservation laws, our second-order reconstruction-based method converges towards the unique entropy solution provided that the nonlocal interaction kernel satisfies a certain growth condition near zero. Furthermore, as the nonlocal horizon parameter in our method approaches zero we recover a well-known second-order method for local conservation laws. In addition, we answer several questions from the paper by Du, Huang, and LeFloch [SIAM J. Numer. Anal., 55 (2017), pp. 2465--2489] concerning regularity of solutions. In particular, we prove that any discontinuity present in a weak solution must be stationary and that if the interaction kernel satisfies a certain growth condition, then weak solutions are unique. We present a series of numerical experiments in which we investigate the accuracy of our second-order scheme, demonstrate shock formation in the nonlocal pair-interaction model, and examine how the regularity of the solution depends on the choice of flux function.

• Chatterjee, Neelabja & Fjordholm, Ulrik Skre (2020). A convergent finite volume method for the Kuramoto equation and related nonlocal conservation laws. IMA Journal of Numerical Analysis. ISSN 0272-4979. 40(1), p. 405–421. doi: 10.1093/imanum/dry074. Full text in Research Archive
• Chatterjee, Neelabja & Fjordholm, Ulrik Skre (2020). Convergence of second-order, entropy stable methods for multi-dimensional conservation laws. Modélisation mathématique et analyse numérique. ISSN 0764-583X. 54(4), p. 1415–1428. doi: 10.1051/m2an/2019090. Full text in Research Archive Show summary

  High-order accurate, entropy stable numerical methods for hyperbolic conservation laws have attracted much interest over the last decade, but only a few rigorous convergence results are available, particularly in multiple space dimensions. In this paper we show how the entropy stability of one such method, which is semi-discrete in time, yields a (weak) bound on oscillations. Under the assumption of L ∞ -boundedness of the approximations we use compensated compactness to prove convergence to a weak solution satisfying at least one entropy condition.

• Fjordholm, Ulrik Skre; Lye, Kjetil; Mishra, Siddhartha & Weber, Franziska (2020). Statistical solutions of hyperbolic systems of conservation laws: Numerical approximation. Mathematical Models and Methods in Applied Sciences. ISSN 0218-2025. 30(3), p. 539–609. doi: 10.1142/S0218202520500141. Full text in Research Archive Show summary

  Statistical solutions are time-parameterized probability measures on spaces of integrable functions, which have been proposed recently as a framework for global solutions and uncertainty quantification for multi-dimensional hyperbolic system of conservation laws. By combining high-resolution finite volume methods with a Monte Carlo sampling procedure, we present a numerical algorithm to approximate statistical solutions. Under verifiable assumptions on the finite volume method, we prove that the approximations, generated by the proposed algorithm, converge in an appropriate topology to a statistical solution. Numerical experiments illustrating the convergence theory and revealing interesting properties of statistical solutions are also presented.

• Carrillo, José A; Fjordholm, Ulrik Skre & Solem, Susanne (2020). A second-order numerical method for the aggregation equations. Mathematics of Computation. ISSN 0025-5718. 90, p. 103–139. doi: 10.1090/mcom/3563. Full text in Research Archive Show summary

  Abstract: Inspired by so-called TVD limiter-based second-order schemes for hyperbolic conservation laws, we develop a formally second-order accurate numerical method for multi-dimensional aggregation equations. The method allows for simulations to be continued after the first blow-up time of the solution. In the case of symmetric, $ \lambda $-convex potentials with a possible Lipschitz singularity at the origin, we prove that the method converges in the Monge-Kantorovich distance towards the unique gradient flow solution. Several numerical experiments are presented to validate the second-order convergence rate and to explore the performance of the scheme.

• Fjordholm, Ulrik Skre (2018). Sharp uniqueness conditions for one-dimensional, autonomous ordinary differential equations. Comptes rendus. Mathematique. ISSN 1631-073X. 356(9), p. 916–921. doi: 10.1016/j.crma.2018.07.008. Full text in Research Archive Show summary

  We give two conditions that are necessary and sufficient for the uniqueness of Filippov solutions to scalar, autonomous ordinary differential equations with discontinuous velocity fields. When only one of the two conditions is satisfied, we give a natural selection criterion that guarantees uniqueness of the solution.

• Chatterjee, Neelabja & Fjordholm, Ulrik Skre (2018). A convergent finite volume method for the Kuramoto equation and related nonlocal conservation laws. IMA Journal of Numerical Analysis. ISSN 0272-4979. 54(4), p. 1415–1428. doi: 10.1093/imanum/dry074. Full text in Research Archive Show summary

  We derive and study a Lax–Friedrichs-type finite volume method for a large class of nonlocal continuity equations in multiple dimensions. We prove that the method converges weakly to the measure-valued solution and converges strongly if the initial data is of bounded variation. Several numerical examples for the kinetic Kuramoto equation are provided, demonstrating that the method works well for both regular and singular data.

• Fjordholm, Ulrik Skre; Lye, Kjetil & Mishra, Siddhartha (2018). Numerical Approximation of Statistical Solutions of Scalar Conservation Laws. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 56(5), p. 2989–3009. doi: 10.1137/17M1154874. Full text in Research Archive Show summary

  We propose efficient numerical algorithms for approximating statistical solutions of scalar conservation laws. The proposed algorithms combine finite volume spatio-temporal approximations with Monte Carlo and multilevel Monte Carlo discretizations of the probability space. Both sets of methods are proved to converge to the entropy statistical solution. We also prove that there is a considerable gain in efficiency resulting from the multilevel Monte Carlo method over the standard Monte Carlo method. Numerical experiments illustrating the ability of both methods to accurately compute multipoint statistical quantities of interest are also presented.

• Fjordholm, Ulrik Skre; Lanthaler, Samuel & Mishra, Siddhartha (2017). Statistical solutions of hyperbolic conservation laws I: Foundations. Archive for Rational Mechanics and Analysis. ISSN 0003-9527. 226(2), p. 809–849. doi: 10.1007/s00205-017-1145-9. Full text in Research Archive
• Fjordholm, Ulrik Skre & Wiedemann, Emil (2017). Statistical solutions and Onsager’s conjecture. Physica D : Non-linear phenomena. ISSN 0167-2789. 376-377, p. 259–265. doi: 10.1016/j.physd.2017.10.009. Full text in Research Archive Show summary

  We prove a version of Onsager’s conjecture on the conservation of energy for the incompressible Euler equations in the context of statistical solutions, as introduced recently by Fjordholm et al. (2017). As a byproduct, we also obtain an alternative proof for the conservative direction of Onsager’s conjecture for weak solutions, under a weaker Besov-type regularity assumption than previously known.

• Ray, Deep; Chandrashekar, Praveen; Fjordholm, Ulrik Skre & Mishra, Siddhartha (2016). Entropy Stable Scheme on Two-Dimensional Unstructured Grids for Euler Equations. Communications in Computational Physics. ISSN 1815-2406. 19(5), p. 1111–1140. doi: 10.4208/cicp.scpde14.43s. Full text in Research Archive
• Fjordholm, Ulrik Skre & Solem, Susanne (2016). Second-order convergence of monotone schemes for conservation laws. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 54(3), p. 1920–1945. doi: 10.1137/16M1059138. Full text in Research Archive
• Zakerzadeh, Hamed & Fjordholm, Ulrik Skre (2016). High-order accurate, fully discrete entropy stable schemes for scalar conservation laws. IMA Journal of Numerical Analysis. ISSN 0272-4979. 36(2), p. 633–654. doi: 10.1093/imanum/drv020. Full text in Research Archive
• Fjordholm, Ulrik Skre (2016). Stability properties of the ENO method. In Abgrall, Rémi & Shu, Chi-Wang (Ed.), Handbook of Numerical Methods for Hyperbolic Problems Applied and Modern Issues. Elsevier. ISSN 9780444639103. doi: 10.1016/bs.hna.2016.09.004. Full text in Research Archive
• Fjordholm, Ulrik Skre; Mishra, Siddhartha & Tadmor, Eitan (2016). On the computation of measure-valued solutions. Acta Numerica. ISSN 0962-4929. 25, p. 567–679. doi: 10.1017/S0962492916000088. Full text in Research Archive
• Fjordholm, Ulrik Skre & Ray, Deep (2015). A Sign Preserving WENO Reconstruction Method. Journal of Scientific Computing. ISSN 0885-7474. doi: 10.1007/s10915-015-0128-y. Full text in Research Archive
• Fjordholm, Ulrik Skre; Käppeli, Roger; Mishra, Siddhartha & Tadmor, Eitan (2015). Construction of Approximate Entropy Measure-Valued Solutions for Hyperbolic Systems of Conservation Laws. Foundations of Computational Mathematics. ISSN 1615-3375. p. 1–65. doi: 10.1007/s10208-015-9299-z. Full text in Research Archive

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• Solem, Susanne; Fjordholm, Ulrik Skre & Carrillo, José A (2018). A second-order numerical method for the aggregation equations. Full text in Research Archive
• Fjordholm, Ulrik Skre (2016). Statistical solutions of hyperbolic conservation laws. Full text in Research Archive
• Fjordholm, Ulrik Skre (2016). Statistical solutions of hyperbolic conservation laws. Full text in Research Archive
• Fjordholm, Ulrik Skre (2016). Statistical solutions of hyperbolic conservation laws. Full text in Research Archive
• Fjordholm, Ulrik Skre (2016). Statistical solutions of hyperbolic conservation laws. Full text in Research Archive

View all works in NVA