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Monotonization of a Family of Implicit Schemes for the Burgers Equation

In: Modeling, Simulation and Optimization of Complex Processes HPSC 2018

Author

Listed:
  • Alexander Kurganov

    (Southern University of Science and Technology, Department of Mathematics)

  • Petr N. Vabishchevich

    (Nuclear Safety Institute of the Russian Academy of Sciences
    North-Eastern Federal University)

Abstract

We study numerical methods for convection-dominated fluid dynamics problems. In particular, we consider initial-boundary value problems for the Burgers equation with small diffusion coefficients. Our goal is to investigate several strategies, which can be used to monotonize numerical methods and to ensure non-oscillatory and positivity-preserving properties of the computed solutions. We focus on fully implicit finite-element methods constructed using the backward Euler time discretization combined with high-order spatial approximations. We experimentally study the following three monotonization approaches: mesh refinement, increasing the time-step size and utilizing higher-order finite-element approximations. Feasibility of these three strategies is demonstrated on a number of numerical examples for both one- and two-dimensional Burgers equations.

Suggested Citation

  • Alexander Kurganov & Petr N. Vabishchevich, 2021. "Monotonization of a Family of Implicit Schemes for the Burgers Equation," Springer Books, in: Hans Georg Bock & Willi Jäger & Ekaterina Kostina & Hoang Xuan Phu (ed.), Modeling, Simulation and Optimization of Complex Processes HPSC 2018, pages 247-256, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-55240-4_12
    DOI: 10.1007/978-3-030-55240-4_12
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