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

Author

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• J. Frederic Bonnans

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

• Xiaolu Tan

(CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique - Polytechnique - X - 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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File URL: https://hal.inria.fr/inria-00634417/document

References listed on IDEAS

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1. Ioannis Karatzas & Fridrik M. Baldursson, 1996. "Irreversible investment and industry equilibrium (*)," Finance and Stochastics, Springer, vol. 1(1), pages 69-89.
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