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Monotonicity condition for the $\theta$-scheme for diffusion equations

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  • J. Frederic Bonnans

    () (Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems - CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique - X - École polytechnique - CNRS - Centre National de la Recherche Scientifique - Inria Saclay - Ile de France - Inria - Institut National de Recherche en Informatique et en Automatique - UMA - Unité de Mathématiques Appliquées - Univ. Paris-Saclay, ENSTA ParisTech - École Nationale Supérieure de Techniques Avancées - X - École polytechnique - CNRS - Centre National de la Recherche Scientifique, CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique - X - École polytechnique - CNRS - Centre National de la Recherche Scientifique)

  • Xiaolu Tan

    (CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique - X - École polytechnique - CNRS - Centre National de la Recherche Scientifique)

Abstract

We derive the necessary and sufficient condition for the $L^{\infty}-$monotonicity of finite difference $\theta$-scheme for a diffusion equation. We confirm that the discretization ratio $\Delta t = O(\Delta x^2)$ is necessary for the monotonicity except for the implicit scheme. In case of the heat equation, we get an explicit formula, which is weaker than the classical CFL condition.

Suggested Citation

  • J. Frederic Bonnans & Xiaolu Tan, 2011. "Monotonicity condition for the $\theta$-scheme for diffusion equations," Working Papers inria-00634417, HAL.
  • Handle: RePEc:hal:wpaper:inria-00634417
    Note: View the original document on HAL open archive server: https://hal.inria.fr/inria-00634417
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