DataSheet1_An adaptive time stepping stiffness confinement method for solving reactor dynamics equations.docx

The stiffness confinement method (SCM) is frequently employed to solve the reactor dynamics equations because it confines the stiffness of the problem by frequency transformation. However, the balance between the error and efficiency of the SCM has not been well studied. This paper reports the error analysis of the SCM. The error by SCM is derived mathematically and written as integral error, driven by the integral of the frequency interpolation function. An easy-to-implement adaptive time-stepping (ATS) algorithm is proposed based on the error analysis by controlling the neutron flux amplitude error. First, a fine-step PKE is leveraged to estimate the second-order derivative of the flux amplitude-frequency, which is used to predict the error of the neutron flux amplitude. The low cost of solving the PKE incurs a negligible effect on the algorithm’s efficiency. Second, based on the error analysis, an error estimator proposed to determine an optimal time-step size for the neutron temporal-spatial equation. With a pre-set error tolerance, the ATS algorithm is exempted from the empirical selection of the time-step size in transient simulations. Numerical tests with TWIGL and modified 2D LMW benchmark problems show that the optimal time-step size effectively confines the local truncation error of the flux amplitude within the pre-set tolerance. The ATS algorithm yields a higher accuracy at a commensurate computational cost than calculations with fixed time-steps.</p

The most likely high risk clusters of HFMD cases detected using the purely spatial analysis.

<p>The most likely high risk clusters of HFMD cases detected using the purely spatial analysis.</p

Annual incidence rates of HFMD in 0–5 years old children (/100,000) per county in Shandong Province, 2007–2011.

<p>Annual incidence rates of HFMD in 0–5 years old children (/100,000) per county in Shandong Province, 2007–2011.</p

The detected spatial-temporal clusters of HFMD in Shandong Province during 2007 to 2011, and the distribution of HFMD pathogens within these clusters.

<p>Each pie shows the proportion of different HFMD pathogen for each spatial-temporal cluster, but no pathogen data in 2007.</p

Social-demographic characteristics of HFMD cases and the pathogen types of some cases in Shandong Province, 2007–2011.

<p>Social-demographic characteristics of HFMD cases and the pathogen types of some cases in Shandong Province, 2007–2011.</p

The results of the spatial autocorrelation test on HFMD cases in Shandong Province, 2007–2011.

<p>The results of the spatial autocorrelation test on HFMD cases in Shandong Province, 2007–2011.</p

The location of study area, Shandong Province in China.

<p>The location of study area, Shandong Province in China.</p

The spatial-temporal high risk clusters of HFMD cases detected using Space-Time Permutation model, 2007–2011.

#<p>:‘1’ represents ‘Most likely cluster’; ‘2–7’ represent six ‘Secondary clusters’.</p

Monthly distribution of HFMD cases (0–5 years), 2007–2011.

<p>Monthly distribution of HFMD cases (0–5 years), 2007–2011.</p