Is the CFL Condition Sufficient? Some Remarks
We present some remarks about the CFL condition for explicit time discretization methods of Adams–Bashforth and Runge–Kutta type and show that for convection-dominated problems stability conditions of the type Δt≤CΔx α are found for high order space discr
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Abstract We present some remarks about the CFL condition for explicit time discretization methods of Adams–Bashforth and Runge–Kutta type and show that for convection-dominated problems stability conditions of the type Δt ≤ CΔx α are found for high order space discretizations, where the exponent α depends on the order of the time scheme. For example, for second order Adams–Bashforth and Runge–Kutta schemes we find α = 4/3. Keywords Explicit time discretization · Stability · CFL condition · Runge–Kutta · Adams–Bashforts · Computational fluid dynamics · Convection dominated problems
1 Introduction This discussion paper presents some reflections about the stability of time discretization schemes for convection-dominated problems, presented by the first author at the conference “CFL-condition, 80 years gone by”, held in Rio de Janeiro in May 2010. In Computational Fluid Dynamics, explicit schemes are typically used for the nonlinear convection term. Thus for stability reasons, the celebrated Courant– Friedrichs–Lewy (CFL) condition [3] has to be satisfied, which states that the time step should be proportional to the space step, with a constant depending on the magnitude of the velocity. The aim of the paper is to revisit the time-stability issue for some higher order time schemes. We present several numerical experiments using either onestep methods of Runge–Kutta type or multi-step methods of Adams–Bashforth type applied to the one-dimensional Burgers equations and to the two-dimensional Euler/Navier–Stokes equations. The numerical results, using a spectral discretization in space, illustrate that for stability the classical CFL condition is not sufficient and that the time step is limited by non-integer powers (larger than one) of the spatial grid size. K. Schneider () · D. Kolomenskiy · E. Deriaz M2P2–CNRS, Aix-Marseille Université, 38 rue Joliot-Curie, 13451 Marseille cedex 20, France e-mail: [email protected] C.A. de Moura, C.S. Kubrusly (eds.), The Courant–Friedrichs–Lewy (CFL) Condition, DOI 10.1007/978-0-8176-8394-8_9, © Springer Science+Business Media New York 2013
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The remainder of the manuscript is organized as follows. In Sect. 2, explicit onestep and multi-step time schemes are recalled, together with their stability domains. Section 3 presents some numerical examples for the inviscid Burgers equation in one space dimension and for the two-dimensional incompressible Navier–Stokes equation. Finally, some conclusions are drawn.
2 Stability of Time Schemes We consider the general form of an evolutionary partial differential equation ∂t u = H (u)
(1)
where H (u) contains all the spatial derivatives. The above equation is completed with suitable initial and boundary conditions. In fluid mechanics, one typically encounters equations where H is the sum of a nonlinear term with first order derivatives and a linear term with second order derivatives. For simplicity, we consider here a convection–diffusion equation, i.e., H (u) = −a∂x u + ν∂xx u, where a is a constant con
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