116 citations · 308 across the 21 of their papers we have counts for
23 papers
Stochastic perturbation and zero noise limit for scalar conservation laws
Ulrik S. Fjordholm, Magnus C. Ørke
Scalar conservation laws sit at the intersection between being simple enough to study analytically, while being complex enough to exhibit a wide range of nonlinear phenomena. We in…
Semi-discrete heat equations with variable coefficients and the parametrix method
Ulrik S. Fjordholm, Kenneth H. Karlsen, Peter H. C. Pang
We develop a parametrix approach for constructing solutions and establishing grid-size independent estimates for semi-discrete heat equations with variable coefficients. While the…
Transport equations for Osgood velocity fields
Ulrik Skre Fjordholm, Ola Isaac Høgåsen Mæhlen
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…
Convergent finite difference schemes for stochastic transport equations
Ulrik S. Fjordholm, Kenneth H. Karlsen, Peter H. C. Pang
We present difference schemes for stochastic transport equations with low-regularity velocity fields. We establish stability and convergence of the difference approximations…
The particle paths of hyperbolic conservation laws
Ulrik S. Fjordholm, Ola H. Mæhlen, Magnus C. Ørke
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 v…
The zero-noise limit of SDEs with drift
Ulrik Skre Fjordholm, Markus Musch, Andrey Pilipenko
We study the zero-noise limit for autonomous, one-dimensional ordinary differential equations with discontinuous right-hand sides. Although the deterministic equation might have in…
On the vanishing viscosity limit of statistical solutions of the incompressible Navier-Stokes equations
Ulrik Skre Fjordholm, Siddhartha Mishra, Franziska Weber
We study statistical solutions of the incompressible Navier-Stokes equation and their vanishing viscosity limit. We show that a formulation using correlation measures, which are pr…
Well-posedness theory for nonlinear scalar conservation laws on networks
Ulrik Skre Fjordholm, Markus Musch, Nils Henrik Risebro
We consider nonlinear scalar conservation laws posed on a network. We establish stability, and thus uniqueness, for weak solutions satisfying the entropy condition. We apply…
Convergence rates of monotone schemes for conservation laws for data with unbounded total variation
Ulrik Skre Fjordholm, Kjetil Olsen Lye
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 ini…
Second-order accurate TVD numerical methods for nonlocal nonlinear conservation laws
Ulrik Skre Fjordholm, Adrian Montgomery Ruf
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 b…
Convergence of second-order, entropy stable methods for multi-dimensional conservation laws
Neelabja Chatterjee, Ulrik Skre Fjordholm
High-order accurate, numerical methods for hyperbolic conservation laws have attracted much interest over the last decade, but only a few rigorous converg…
Statistical solutions of hyperbolic systems of conservation laws: numerical approximation
Ulrik Skre Fjordholm, Kjetil Lye, Siddhartha Mishra +1
Statistical solutions are time-parameterized probability measures on spaces of integrable functions, that have been proposed recently as a framework for global solutions and uncert…
A convergent finite volume method for the Kuramoto equation and related non-local conservation laws
Neelabja Chatterjee, Ulrik Skre Fjordholm
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…
Sharp uniqueness conditions for one-dimensional, autonomous ordinary differential equations
Ulrik Skre Fjordholm
We give two conditions that are necessary and sufficient for the uniqueness of Filippov solutions of scalar, autonomous ordinary differential equations with discontinuous velocity…
A second-order numerical method for the aggregation equations
José A. Carrillo, Ulrik Skre Fjordholm, Susanne Solem
Inspired by so-called TVD limiter-based second-order schemes for hyperbolic conservation laws, we develop a second-order accurate numerical method for multi-dimensional aggregation…
Numerical approximation of statistical solutions of scalar conservation laws
Ulrik Skre Fjordholm, Kjetil Lye, Siddhartha Mishra
We propose efficient numerical algorithms for approximating statistical solutions of scalar conservation laws. The proposed algorithms combine finite volume spatio-temporal approxi…
Statistical solutions and Onsager's conjecture
Ulrik Skre Fjordholm, Emil Wiedemann
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 F…
Stability properties of the ENO method
Ulrik Skre Fjordholm
We review the currently available stability properties of the ENO reconstruction procedure, such as its monotonicity and non-oscillatory properties, the sign property, upper bounds…
Statistical solutions of hyperbolic conservation laws I: Foundations
Ulrik Skre Fjordholm, Samuel Lanthaler, Siddhartha Mishra
We seek to define statistical solutions of hyperbolic systems of conservation laws as time-parametrized probability measures on -integrable functions. To do so, we prove the equ…
Second-order convergence of monotone schemes for conservation laws
Ulrik S. Fjordholm, Susanne Solem
We prove that a class of monotone, \emph{-contractive} schemes for scalar conservation laws converge at a rate of in the Wasserstein distance (-distance), whenever…
A sign preserving WENO reconstruction method
Ulrik S. Fjordholm, Deep Ray
We propose a third-order WENO reconstruction which satisfies the sign property, required for constructing high resolution entropy stable finite difference scheme for conservation l…
Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws
Ulrik S. Fjordholm, Roger Käppeli, Siddhartha Mishra +1
Entropy solutions have been widely accepted as the suitable solution framework for systems of conservation laws in several space dimensions. However, recent results in \cite{CDL1,C…
ENO reconstruction and ENO interpolation are stable
Ulrik S. Fjordholm, Siddhartha Mishra, Eitan Tadmor
We prove stability estimates for the ENO reconstruction and ENO interpolation procedures. In particular, we show that the jump of the reconstructed ENO pointvalues at each cell int…