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Estimation in nonlinear systems with transport delay.

The problem of estimation of state in nonlinear dynamical systems containing time delays is studied. The plant is specified by a set of nonlinear differential-difference equations. Observations are a nonlinear function of current and/or delayed states. Both contain additive disturbances. The criterion used for the optimal estimates is the integral of the weighted squared error. Using the theory of the calculus of variations, equations are developed for the estimation. They are first expressed in the form of a split boundary value problem, which is then converted to an initial value problem for on-line estimation. The result yields a sequential estimation scheme in which filtered and smoothed estimates are computed in a sequential manner. The applicability of the procedure is demonstrated by a practical example.

Stoller, R. L.

Dynamic mode decomposition for gyrokinetic eigenmode analysis

Dynamic mode decomposition (DMD) is a post-processing approach to decompose a complex time series into a set of modes via spectral analysis. DMD provides a new and powerful method to recover gyrokinetic drift-wave eigenfrequencies and eigenfunctions based only on the solution of the gyrokinetic-Maxwell initial value problem with almost no added cost to the initial value solver. In the present paper, DMD is applied to the CGYRO gyrokinetic code using a newly-developed CGYRO-DMD post-processor. CGYRO-DMD is numerically efficient, even on a single CPU. It does not set any restrictions on the plasma shape, beta (ratio of the plasma pressure to the magnetic field pressure), collisionality or number of species, and allows one to resolve numerous eigenmodes, even of comparable growth rates. In addition, DMD is not limited to unstable modes, but rather can capture stable and unstable branches simultaneously. In this work, we illustrate the accuracy of DMD through gyrokinetic analysis of mode transition for electromagnetic drift wave instabilities.

drift-wave eigenmodes

Thermal microwave emission from a random inhomogeneous layer over a homogeneous medium using the method of invariant imbedding

The paper studies thermal microwave emission from an inhomogeneous slab of a random medium, with possible nonuniform absorption, scattering, and temperature profiles, bounded by different dielectrics on both sides. The invariant imbedding method is used to cast the boundary value problem of the radiative transfer equations into an initial value problem at zero slab thickness. As a numerical example, the angular and polarization variations of brightness temperatures for ice over water are considered.

Tsang, L.

The theoretical accuracy of Runge-Kutta time discretizations for the initial boundary value problem: A careful study of the boundary error

The conventional method of imposing time dependent boundary conditions for Runge-Kutta (RK) time advancement reduces the formal accuracy of the space-time method to first order locally, and second order globally, independently of the spatial operator. This counter intuitive result is analyzed in this paper. Two methods of eliminating this problem are proposed for the linear constant coefficient case: (1) impose the exact boundary condition only at the end of the complete RK cycle, (2) impose consistent intermediate boundary conditions derived from the physical boundary condition and its derivatives. The first method, while retaining the RK accuracy in all cases, results in a scheme with much reduced CFL condition, rendering the RK scheme less attractive. The second method retains the same allowable time step as the periodic problem. However it is a general remedy only for the linear case. For non-linear hyperbolic equations the second method is effective only for for RK schemes of third order accuracy or less. Numerical studies are presented to verify the efficacy of each approach.

Carpenter, Mark H.

Numerical Simulation of Rotation-Driven Plasma Transport In the Jovian Magnetosphere

A Jupiter version of the Rice Convection Model (RCM-J) was developed with support of an earlier NASA SR&T grant. The conversion from Earth to Jupiter included adding currents driven by centrifugal force, reversing the planetary magnetic field, and rescaling various parameters. A series of informative runs was carried out, all of them solving initial value problems. The simulations followed an initial plasma torus configuration as it fell apart by interchange instability. Some conclusions from the simulations were the following: 1. We confirmed that, for conventional values of the torus density and ionospheric conductance, the torus disintegrates by interchange instability on a time scale of approx. one day, which is 1-2 orders of magnitude shorter than the best estimates of the average residence time of plasma in the torus. 2. In the model, the instability could be slowed to an arbitrary degree by the addition of sufficient impounding energetic particles, as suggested earlier by Siscoe et al (1981). However, the observed energetic particles do not seem sufficient to guarantee impoundment (e.g., Mauk et al., 1996). 3. Whether inhibited by impoundment or not, the interchange was found to proceed by the formation of long fingers, which get thinner as they get longer. This picture differed dramatically from the conventional radial-diffusion picture (e.g., Siscoe and Summers (1981)), more superficially with the outward-moving-blob picture (Pontius and Hill, 1989). The obvious limitation of the original RCM-J was that it could not represent a plasma source. We could represent the decay of a pre-existing torus, but we could not represent the way ionization of material from Io continually replenishes the plasma. We consequently were precluded from studying a whole set of fundamental issues of torus theory, including whether the system can come to a steady state.

Wolf, Richard A.

Numerical integration of kinetic equations.

Relaxation to Maxwellian of Balescu-Lenard equation solved numerically as initial value problem for isotropic distributions of electrons in background of positive charge

MAXWELL DISTRIBUTION

Applications of estimation theory to numerical weather prediction

Numerical weather prediction (NWP) is an initial value problem for a system of nonlinear partial differential equations in which the initial values are known only incompletely and inaccurately. Data at initial time can be supplemented, however, by observations of the system distributed over a time interval preceding it. Estimation theory was successful in approaching such problems for models governed by systems of ordinary differential equations and of linear PDEs. Estimation-theoretic methods for NWP are developed. A model exhibiting many features of large scale atmospheric flow important in NWP is the one governed by the shallow fluid equations. The estimation problem for a linearized formulation of these equations is studied. A finite difference version of the equations is used as a forecast model to simulate the numerical models used in NWP.

Cohn, S.