The Stability of Linear Diffusion Acceleration Relative to CMFD

Coarse Mesh Finite Difference (CMFD) is a widely-used iterative acceleration method for neutron transport problems in which nonlinear terms are introduced in the derivation of the low-order CMFD diffusion equation. These terms, including the homogenized diffusion coefficient, the current coupling coefficients, and the multiplicative prolongation constant, are subject to numerical instability when a scalar flux estimate becomes sufficiently small or negative. In this paper, we use a suite of contrived problems to demonstrate the susceptibility of CMFD to failure for each of the vulnerable quantities of interest. Our results show that if a scalar flux estimate becomes negative in any portion of phase space, for any iterate, numerical instability can occur. Specifically, the number of outer iterations required for convergence of the CMFD-accelerated transport problem can increase dramatically, or worse, the iteration scheme can diverge. An alternative Linear Diffusion Acceleration (LDA) scheme addresses these issues by explicitly avoiding local nonlinearities. Our numerical results show that the rapid convergence of LDA is unaffected by the very small or negative scalar flux estimates that can adversely affect the performance of CMFD. Therefore, our results demonstrate that LDA is a robust alternative to CMFD for certain sensitive problems in which CMFD can exhibit reduced effectiveness or failure.

Error Analysis and Sampling Strategy Design for Using Fixed or Mobile Platforms to Estimate Ocean Flux

For estimating lateral flux in the ocean using fixed or mobile platforms, the authors present a method of analyzing the estimation error and designing the sampling strategy. When an array of moorings is used, spatial aliasing leads to an error in flux estimation. When an autonomous underwater vehicle (AUV) is run, measurements along its course are made at different times. Such nonsynopticity in the measurements leads to an error in flux estimation. It is assumed that the temporal–spatial autocovariance function of the flux variable can be estimated from historical data or ocean models (as in this paper). Using the temporal–spatial autocovariance function of the flux variable, the mean-square error of the flux estimate by fixed or mobile platforms is derived. The method is used to understand the relative strengths of moorings and AUVs (assumed here to be able to maintain constant speed) under different scenarios of temporal and spatial variabilities. The flux estimate by moorings through trapezoidal approximation generally carries a bias that drops quadratically with the number of moorings. The authors also show that a larger number of slower AUVs may achieve a more accurate flux estimate than a smaller number of faster AUVs under the same cumulative speed, but the performance margin shrinks with the increase of the cumulative speed. Using the error analysis results, one can choose the type of platforms and optimize the sampling strategy under resource constraints. To verify the theoretical analysis, the authors run simulated surveys in synthesized ocean fields. The flux estimation errors agree very well with the analytical predictions. Using an ocean model dataset, the authors estimate the lateral heat flux across a section in Monterey Bay, California, and also compare the flux estimation errors with the analytical predictions.