On spurious steady-state solutions of explicit Runge-Kutta schemes

The bifurcation diagram associated with the logistic equation v sup n+1 = av sup n (1-v sup n) is by now well known, as is its equivalence to solving the ordinary differential equation u prime = alpha u (1-u) by the explicit Euler difference scheme. It has also been noted by Iserles that other popular difference schemes may not only exhibit period doubling and chaotic phenomena but also possess spurious fixed points. Runge-Kutta schemes applied to both the equation u prime = alpha u (1-u) and the cubic equation u prime = alpha u (1-u)(b-u) were studied computationally and analytically and their behavior was contrasted with the explicit Euler scheme. Their spurious fixed points and periodic orbits were noted. In particular, it was observed that these may appear below the linearized stability limits of the scheme and, consequently, computation may lead to erroneous results

Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. Part 1: The ODE connection and its implications for algorithm development in computational fluid dynamics

Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit

Drifting diffusion on a circle as continuous limit of a multiurn Ehrenfest model

We study the continuous limit of a multibox Erhenfest urn model proposed before by the authors. The evolution of the resulting continuous system is governed by a differential equation, which describes a diffusion process on a circle with a nonzero drifting velocity. The short time behavior of this diffusion process is obtained directly by solving the equation, while the long time behavior is derived using the Poisson summation formula. They reproduce the previous results in the large $M$ (number of boxes) limit. We also discuss the connection between this diffusion equation and the Schr$\ddot{\rm o}$dinger equation of some quantum mechanical problems.Comment: 4 pages prevtex4 file, 1 eps figur

Spheromaks, solar prominences, and Alfvén instability of current sheets

Three related efforts underway at Caltech are discussed: experimental studies of spheromak formation, experimental simulation of solar prominences, and Alfvén wave instability of current sheets. Spheromak formation has been studied by using a coaxial magnetized plasma gun to inject helicity-bearing plasma into a very large vacuum chamber. The spheromak is formed without a flux conserver and internal λ profiles have been measured. Spheromak-based technology has been used to make laboratory plasmas having the topology and dynamics of solar prominences. The physics of these structures is closely related to spheromaks (low β, force-free, relaxed state equilibrium) but the boundary conditions and symmetry are different. Like spheromaks, the equilibrium involves a balance between hoop forces, pinch forces, and magnetic tension. It is shown theoretically that if a current sheet becomes sufficiently thin (of the order of the ion skin depth or smaller), it becomes kinetically unstable with respect to the emission of Alfvén waves and it is proposed that this wave emission is an important aspect of the dynamics of collisionless reconnection

Realistic Magnetohydrodynamical Simulation of Solar Local Supergranulation

Three-dimensional numerical simulations of solar surface magnetoconvection using realistic model physics are conducted. The thermal structure of convective motions into the upper radiative layers of the photosphere, the main scales of convective cells and the penetration depths of convection are investigated. We take part of the solar photosphere with size of 60x60 Mm in horizontal direction and by depth 20 Mm from level of the visible solar surface. We use a realistic initial model of the Sun and apply equation of state and opacities of stellar matter. The equations of fully compressible radiation magnetohydrodynamics with dynamical viscosity and gravity are solved. We apply: 1) conservative TVD difference scheme for the magnetohydrodynamics, 2) the diffusion approximation for the radiative transfer, 3) dynamical viscosity from subgrid scale modeling. In simulation we take uniform two-dimesional grid in gorizontal plane and nonuniform grid in vertical direction with number of cells 600x600x204. We use 512 processors with distributed memory multiprocessors on supercomputer MVS-100k in the Joint Computational Centre of the Russian Academy of Sciences.Comment: 6 pages, 5 figures, submitted to the proceedings of the GONG 2008 / SOHO XXI conferenc

On spurious asymptotic numerical solutions of explicit Runge Kutta methods

The bifurcation diagram associated with the logistic equation vn+1 = avn(1 - vn) is by now well known, as is its equivalence to solving the ordinary differential equation (ODE) u\u27 = αu(1 - u) by the explicit Euler difference scheme. It has also been noted by Iserles that other popular difference schemes may not only exhibit period doubling and chaotic phenomena but also possess spurious fixed points. We investigate, both analytically and computationally, Runge-Kutta schemes applied to the equation u\u27 = f(u), for f(u) = αu(1 - u) and f(u) = αu (1 - u)(b - u), contrasting their behavior with the explicit Euler scheme. We determine and provide a local analysis of bifurcations to spurious fixed points and periodic orbits. In particular we show that these may appear below the linearised stability limit of the scheme, and may consequently lead to erroneous computational results

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

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 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

Non-perturbative approach for the time-dependent symmetry breaking

We present a variational method which uses a quartic exponential function as a trial wave-function to describe time-dependent quantum mechanical systems. We introduce a new physical variable $y$ which is appropriate to describe the shape of wave-packet, and calculate the effective action as a function of both the dispersion $\sqrt{}$ and $y$. The effective potential successfully describes the transition of the system from the false vacuum to the true vacuum. The present method well describes the long time evolution of the wave-function of the system after the symmetry breaking, which is shown in comparison with the direct numerical computations of wave-function.Comment: 8 pages, 3 figure

Structure-based stabilization of insulin as a therapeutic protein assembly via enhanced aromatic-aromatic interactions

Key contributions to protein structure and stability are provided by weakly polar interactions, which arise from asymmetric electronic distributions within amino acids and peptide bonds. Of particular interest are aromatic side chains whose directional π-systems commonly stabilize protein interiors and interfaces. Here, we consider aromatic-aromatic interactions within a model protein assembly: the dimer interface of insulin. Semi-classical simulations of aromatic-aromatic interactions at this interface suggested that substitution of residue TyrB26 by Trp would preserve native structure while enhancing dimerization (and hence hexamer stability). The crystal structure of a [TrpB26]insulin analog (determined as a T3Rf3 zinc hexamer at a resolution of 2.25 Å) was observed to be essentially identical to that of WT insulin. Remarkably and yet in general accordance with theoretical expectations, spectroscopic studies demonstrated a 150-fold increase in the in vitro lifetime of the variant hexamer, a critical pharmacokinetic parameter influencing design of long-acting formulations. Functional studies in diabetic rats indeed revealed prolonged action following subcutaneous injection. The potency of the TrpB26-modified analog was equal to or greater than an unmodified control. Thus, exploiting a general quantum-chemical feature of protein structure and stability, our results exemplify a mechanism-based approach to the optimization of a therapeutic protein assembly