Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. I. The dynamics of time discretization and its implications for algorithm development in computational fluid dynamics

Abstract

The goal of this paper is to utilize the theory of nonlinear dynamics approach to investigate the possible sources of errors and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic and parabolic partial differential equations terms. This interdisciplinary research belongs to a subset of a new field of study in numerical analysis sometimes referred to as "the dynamics of numerics and the numerics of dynamics." At the present time, this new interdisciplinary topic is still the property of an isolated discipline with all too little effort spent in pointing out an underlying generality that could make it adaptable to diverse fields of applications. This is the first of a series of research papers under the same topic. Our hope is to reach researchers in the fields of computational fluid dynamics (CFD) and, in particular, hypersonic and combustion related CFD. By simple examples (in which the exact solutions of the governing equations are known), the application of the apparently straightforward numerical technique to genuinely nonlinear problems can be shown to lead to incorrect or misleading results. One striking phenomenon is that with the same initial data, the continuum and its discretized counterpart can asymptotically approach different stable solutions. This behavior is especially important for employing a time-dependent approach to the steady state since the initial data are usually not known and a freest ream condition or an intelligent guess for the initial conditions is often used. With the unique property of the different dependence of the solution on initial data for the partial differential equation and the discretized counterpart, it is not easy to delineate the * An abbreviated version appeared in the "All rights of reproduction in any form reserved. 250YEE, SWEBY, AND GRIFFITHS true physics from numerical artifacts when numerical methods are the sole source of solution procedure for the continuum. Part I concentrates on the dynamical behavior of time discretization for scalar nonlinear ordinary differential equations in order to motivate this new yet unconventional approach to algorithm development in CFD and to serve as an introduction for parts II and III of the same series of research papers. .: [)

Numerical Experiments with a Turbulent Single-Mode Rayleigh-Taylor Instability

“…These are valid solutions of the difference equations, but they are a qualitatively different class of solution than the approximate numerical solutions to the underlying differential equations found in the limit of infinite spatial and temporal resolution. Such chaos has been observed in COYOTE [16] as well as other codes [17]- [20]. It occurs when the code is run near its stability limit.…”

Construction of low dissipative high-order well-balanced filter schemes for non-equilibrium flows

“…where δ 1 is the entropy fix parameter (see [36] for a discussion).Q l j+1/2 is an unbiased limiter function which can bê…”

Building Blocks for Reliable Complex Nonlinear Numerical Simulations

“…This includes adaptive temporal and spatial schemes, grid adaptation as an integral part of the numerical solution process, and, most of all, adaptive numerical dissipation controls. Using tools from dynamical systems, Yee et al (1991Yee et al ( -1997, Yee & Sweby (1993-1997, Griffiths et al (1992a,b) and Lafon & Yee (1991 showed that adaptive temporal and adaptive spatial schemes are important in minimizing numerically induced chaos, numerically induced chaotic transients and the false prediction of flow instability by direct numerical simulation (DNS). Their studies further indicate the need in the development of practical adaptive temporal schemes based on error controls to minimize spurious numerics due to the full diseretizations.…”

“…It was shown in Yee et al (1991) and Griffiths et al (1992a,b) that spurious discrete traveling waves can exist, depending on the method of discretizing the source term. When physical diffusion is added, it is not known what type of numerical difficulties will surface.…”

“…The following isolates some of the key elements and issues of numerical uncertainties in time-marching to the steady state. Studies in Yee et al (1991Yee et al ( -1997 indicate that each of the following can affect not only the convergence rate but also spurious numerics other than standard stability and accuracy linearized numerical analysis problems.…”