adrianru

• 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.

• Badwaik, Jayeesh; Klingenberg, Christian; Risebro, Nils Henrik & Ruf, Adrian Montgomery (2021). Multilevel Monte Carlo finite volume methods for random conservation laws with discontinuous flux. Modélisation mathématique et analyse numérique. ISSN 0764-583X. 55(3), p. 1039–1065. doi: 10.1051/m2an/2021011. Full text in Research Archive Show summary

  We consider conservation laws with discontinuous flux where the initial datum, the flux function, and the discontinuous spatial dependency coefficient are subject to randomness. We establish a notion of random adapted entropy solutions to these equations and prove well-posedness provided that the spatial dependency coefficient is piecewise constant with finitely many discontinuities. In particular, the setting under consideration allows the flux to change across finitely many points in space whose positions are uncertain. We propose a single- and multilevel Monte Carlo method based on a finite volume approximation for each sample. Our analysis includes convergence rate estimates of the resulting Monte Carlo and multilevel Monte Carlo finite volume methods as well as error versus work rates showing that the multilevel variant outperforms the single-level method in terms of efficiency. We present numerical experiments motivated by two-phase reservoir simulations for reservoirs with varying geological properties.

• Ruf, Adrian Montgomery; Sande, Espen & Solem, Susanne (2019). The Optimal Convergence Rate of Monotone Schemes for Conservation Laws in the Wasserstein Distance. Journal of Scientific Computing. ISSN 0885-7474. 80(3), p. 1764–1776. doi: 10.1007/s10915-019-00996-1. Full text in Research Archive Show summary

  In 1994, Nessyahu, Tadmor and Tassa studied convergence rates of monotone finite volume approximations of conservation laws. For compactly supported, Lip+ -bounded initial data they showed a first-order convergence rate in the Wasserstein distance. Our main result is to prove that this rate is optimal. We further provide numerical evidence indicating that the rate in the case of Lip+ -unbounded initial data is worse than first-orde

• Ridder, Johanna & Ruf, Adrian Montgomery (2019). A convergent finite difference scheme for the Ostrovsky–Hunter equation with Dirichlet boundary conditions. BIT Numerical Mathematics. ISSN 0006-3835. 59, p. 775–796. doi: 10.1007/s10543-019-00746-7. Full text in Research Archive
• Risebro, Nils Henrik & Ruf, Adrian Montgomery (2019). Numerical investigations into a model of partially incompressible two-phase flow in pipes. SeMA Journal - Boletin de la Sociedad Española de Matemática Aplicada. ISSN 2254-3902. p. 1–17. doi: 10.1007/s40324-019-00207-9. Full text in Research Archive Show summary

  We consider a model for flow of liquid and gas in a pipe. We assume that the gas is ideal and that the liquid is incompressible. Under this assumption the resulting system of equations, expressing conservation of mass and momentum, splits into two subsystems such that the gas flow is independent of the liquid flow, and the liquid flow is described by a conservation law parameterized by the mass fraction of gas. When solving these equations numerically, we propose to stagger the gas and liquid variables with respect to each other. The advantage of this is that in finite volume methods one can use numerical flux functions designed for 2×2 systems of hyperbolic conservation laws to solve both the gas flow and the liquid flow, rather than a much more complicated numerical flux for the whole 4×4 system. We test this using the Roe numerical flux for both subsystems, and compare the results with results produced by using the second-order Nessyahu–Tadmor scheme for the second subsystem.

• Ruf, Adrian Montgomery (2017). Convergence of a full discretization for a second-order nonlinear elastodynamic equation in isotropic and anisotropic Orlicz spaces. Zeitschrift für Angewandte Mathematik und Physik. ISSN 0044-2275. 68:118(5), p. 1–24. doi: 10.1007/s00033-017-0863-z. Full text in Research Archive Show summary

  In this paper, we study a second-order, nonlinear evolution equation with damping arising in elastodynamics. The nonlinear term is monotone and possesses a convex potential but exhibits anisotropic and nonpolynomial growth. The appropriate setting for such equations is that of monotone operators in Orlicz spaces. Global existence of solutions in the sense of distributions is shown via convergence of the backward Euler scheme combined with an internal approximation. Moreover, we show uniqueness in a class of sufficiently smooth solutions and provide an a priori error estimate for the temporal semidiscretization.

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• Ruf, Adrian Montgomery (2019). A second-order scheme for a class of nonlocal conservation laws. Full text in Research Archive
• Ruf, Adrian Montgomery (2019). The optimal convergence rate of monotone schemes for conservation laws in the Wasserstein distance. Full text in Research Archive
• Ruf, Adrian Montgomery (2019). Second-order numerical methods for nonlocal conservation laws. Full text in Research Archive
• Ruf, Adrian Montgomery (2018). The Ostrovsky-Hunter equation with Dirichlet boundary conditions. Full text in Research Archive
• Ruf, Adrian Montgomery (2018). A second-order method for nonlocal conservation laws. Full text in Research Archive
• Ruf, Adrian Montgomery (2018). The Ostrovsky-Hunter equation with Dirichlet boundary conditions. Full text in Research Archive
• Ruf, Adrian Montgomery (2017). What is a Researcher? Full text in Research Archive
• Ruf, Adrian Montgomery & Liu, Qinghua (2017). The complete guide to a PhD for new PhD students by slightly less new PhD students. Full text in Research Archive
• Ruf, Adrian Montgomery (2017). Multiphase Flow in Pipelines. Full text in Research Archive
• Ruf, Adrian Montgomery (2016). A class of nonlinear evolution equations of second order. Full text in Research Archive

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