Non-linear CFL Conditions Issued from the von Neumann Stability Analysis for the Transport Equation
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Non-linear CFL Conditions Issued from the von Neumann Stability Analysis for the Transport Equation Erwan Deriaz1
· Pierre Haldenwang2
Received: 26 November 2019 / Revised: 24 July 2020 / Accepted: 2 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper presents a theory of the possible non-linear stability conditions encountered in the simulation of convection dominated problems. Its main objective is to study and justify original CFL-like stability conditions thanks to the von Neumann stability analysis. In particular, we exhibit a wide variety of stability conditions of the type t ≤ Cx α with t the time step, x the space step, and α a rational number within the interval [1, 2]. Numerical experiments corroborate these theoretical results. Keywords CFL condition · von Neumann stability · Transport equation · Runge–Kutta schemes · Finite differences · Turbulence Mathematics Subject Classification 65M02 · 34D02 · 41A02
Introduction This paper prospects the stability conditions coming from the von Neumann stability analysis of the transport equation with various finite difference discretizations. It provides a large variety of CFL-like stability conditions. These conditions observed in finite difference methods [9] also apply to other numerical methods such as Discontinuous Galerkin [25], Continuous Finite Elements [7,13] as well as Cartesian Level Set methods [10]. In the following we denote the time step by t, the space step by x and the velocity by a. When a is omitted then x stands for x/a. The CFL condition, a physical criterion from the founding paper [4] based on the influence areas of the points, asserts that the numerical
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Erwan Deriaz [email protected] Pierre Haldenwang [email protected]
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Institut Jean Lamour, Campus Artem, 2 allée André Guinier, BP 50840, 54011 Nancy Cedex, France
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Laboratoire de Mécanique, Modélisation et Procédés Propres, 38 rue Frédéric Joliot-Curie, 13451 Marseille Cedex 20, France 0123456789().: V,-vol
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Page 2 of 17
Journal of Scientific Computing
(2020) 85:5
simulation of a transport phenomenon at speed a has to satisfy t ≤ C x/a with C a constant close to 1. It is a necessary condition for the numerical stability of explicit schemes. Nevertheless, the linear CFL condition may not be sufficient to ensure the numerical stability. Thanks to the von Neumann stability analysis [1], and allowing a controlled exponential growth of the error, it is possible to establish stability conditions of the type t ≤ C x α with p(2q−1) α ∈ [1, 2] a rational number. This exponent α is given by α = q(2 p−1) with p and q integers such that q ≥ p > 0. This result was announced but not justified in [6]. Here we detail the mathematical premise. Then we also prove and test these non-linear CFL conditions. The organization of the paper is the following: in the first part we briefly review the stability domain of the time schemes, mainly studied for the ordinary differential equation theory and comprehen
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