Numerical Approximation of Oscillatory Solutions of Hyperbolic-Elliptic Systems of Conservation Laws by Multiresolution Schemes

Numerical Approximation of Oscillatory Solutions of Hyperbolic-Elliptic Systems of Conservation Laws by Multiresolution Schemes

Year:    2009

Author:    Stefan Berres, Raimund Bürger, Alice Kozakevicius

Advances in Applied Mathematics and Mechanics, Vol. 1 (2009), Iss. 5 : pp. 581–614

Abstract

The generic structure of solutions of initial value problems of hyperbolic-elliptic systems, also called mixed systems, of conservation laws is not yet fully understood. One reason for the absence of a core well-posedness theory for these equations is the sensitivity of their solutions to the structure of a parabolic regularization when attempting to single out an admissible solution by the vanishing viscosity approach. There is, however, theoretical and numerical evidence for the appearance of solutions that exhibit persistent oscillations, so-called oscillatory waves, which are (in general, measure-valued) solutions that emerge from Riemann data or slightly perturbed constant data chosen from the interior of the elliptic region. To capture these solutions, usually a fine computational grid is required. In this work, a version of the multiresolution method applied to a WENO scheme for systems of conservation laws is proposed as a simulation tool for the efficient computation of solutions of oscillatory wave type. The hyperbolic-elliptic $2 \times 2$ systems of conservation laws considered are a prototype system for three-phase flow in porous media and a system modeling the separation of a heavy-buoyant bidisperse suspension. In the latter case, varying one scalar parameter produces elliptic regions of different shapes and numbers of points of tangency with the borders of the phase space, giving rise to different kinds of oscillation waves.

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.09-m0935

Advances in Applied Mathematics and Mechanics, Vol. 1 (2009), Iss. 5 : pp. 581–614

Published online:    2009-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    34

Keywords:    Hyperbolic-elliptic system conservation law oscillation wave numerical simulation multiresolution method sedimentation model.

Author Details

Stefan Berres

Raimund Bürger

Alice Kozakevicius

  1. A parallel splitting wavelet method for 2D conservation laws

    Schmidt, Alex A. | Kozakevicius, Alice J. | Jakobsson, Stefan

    (2016) P.030018

    https://doi.org/10.1063/1.4951774 [Citations: 2]
  2. A Diffusively Corrected Multiclass Lighthill-Whitham-Richards Traffic Model with Anticipation Lengths and Reaction Times

    Bürger, Raimund | Mulet, Pep | Villada, Luis M.

    Advances in Applied Mathematics and Mechanics, Vol. 5 (2013), Iss. 05 P.728

    https://doi.org/10.4208/aamm.2013.m135 [Citations: 6]
  3. An adaptive finite-volume method for a model of two-phase pedestrian flow

    Berres, Stefan | Ruiz-Baier, Ricardo | Schwandt, Hartmut | M. Tory, Elmer

    Networks & Heterogeneous Media, Vol. 6 (2011), Iss. 3 P.401

    https://doi.org/10.3934/nhm.2011.6.401 [Citations: 16]
  4. Multiscale RBF-based central high resolution schemes for simulation of generalized thermoelasticity problems

    Yousefi, Hassan | Taghavi Kani, Alireza | Mahmoudzadeh Kani, Iradj

    Frontiers of Structural and Civil Engineering, Vol. 13 (2019), Iss. 2 P.429

    https://doi.org/10.1007/s11709-018-0483-5 [Citations: 7]
  5. New impressive performances for the analytical solutions to the (1 + 1)-dimensional van der-waals gas system against its numerical solutions

    Zahran, Emad H.M. | Ahmad, Hijaz | Askar, Sameh | Uzun Ozsahin, Dilber

    Results in Physics, Vol. 51 (2023), Iss. P.106667

    https://doi.org/10.1016/j.rinp.2023.106667 [Citations: 4]
  6. Vanishing viscosity limits of mixed hyperbolic–elliptic systems arising in multilayer channel flows

    Papaefthymiou, E S | Papageorgiou, D T

    Nonlinearity, Vol. 28 (2015), Iss. 6 P.1607

    https://doi.org/10.1088/0951-7715/28/6/1607 [Citations: 3]
  7. New kink solutions for the van der Waals p‐system

    Az‐Zo'bi, Emad A.

    Mathematical Methods in the Applied Sciences, Vol. 42 (2019), Iss. 18 P.6216

    https://doi.org/10.1002/mma.5717 [Citations: 20]
  8. A parallel wavelet adaptive WENO scheme for 2D conservation laws

    Schmidt, Alex A. | Kozakevicius, Alice de Jesus | Jakobsson, Stefan

    International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 27 (2017), Iss. 7 P.1467

    https://doi.org/10.1108/HFF-08-2016-0295 [Citations: 5]
  9. Multiresolution scheme for two-phase volcanic flows

    Schmidt, Alex A. | Kozakevicius, Alice J. | Zeidan, Dia | Jakobsson, Stefan

    CENTRAL EUROPEAN SYMPOSIUM ON THERMOPHYSICS 2019 (CEST), (2019), P.030024

    https://doi.org/10.1063/1.5114008 [Citations: 0]
  10. Semi-analytic treatment of mixed hyperbolic–elliptic Cauchy problem modeling three-phase flow in porous media

    Az-Zo’bi, Emad | Yildirim, Ahmet | Akinyemi, Lanre

    International Journal of Modern Physics B, Vol. 35 (2021), Iss. 29

    https://doi.org/10.1142/S0217979221502933 [Citations: 4]
  11. The residual power series method for the one-dimensional unsteady flow of a van der Waals gas

    Az-Zo’bi, Emad A. | Yıldırım, Ahmet | AlZoubi, Wael A.

    Physica A: Statistical Mechanics and its Applications, Vol. 517 (2019), Iss. P.188

    https://doi.org/10.1016/j.physa.2018.11.030 [Citations: 27]
  12. Solving a mixture model of two-phase flow with velocity non-equilibrium using WENO wavelet methods

    Kozakevicius, Alice de Jesus | Zeidan, Dia | Schmidt, Alex A. | Jakobsson, Stefan

    International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 28 (2018), Iss. 9 P.2052

    https://doi.org/10.1108/HFF-05-2017-0215 [Citations: 26]
  13. WENO wavelet method for a hyperbolic model of two-phase flow in conservative form

    Zeidan, Dia | Kozakevicius, Alice J. | Schmidt, Alex A. | Jakobsson, Stefan

    (2016) P.030022

    https://doi.org/10.1063/1.4951778 [Citations: 1]