Method of lines transpose: High order L-stable O(N) schemes for parabolic equations using successive convolution

We present a new solver for nonlinear parabolic problems that is L-stable and achieves high order accuracy in space and time. The solver is built by first constructing a single-dimensional heat equation solver that uses fast O(N) convolution. This fundamental solver has arbitrary order of accuracy in space, and is based on the use of the Green's function to invert a modified Helmholtz equation. Higher orders of accuracy in time are then constructed through a novel technique known as successive convolution (or resolvent expansions). These resolvent expansions facilitate our proofs of stability and convergence, and permit us to construct schemes that have provable stiff decay. The multi-dimensional solver is built by repeated application of dimensionally split independent fundamental solvers. Finally, we solve nonlinear parabolic problems by using the integrating factor method, where we apply the basic scheme to invert linear terms (that look like a heat equation), and make use of Hermite-Birkhoff interpolants to integrate the remaining nonlinear terms. Our solver is applied to several linear and nonlinear equations including heat, Allen-Cahn, and the Fitzhugh-Nagumo system of equations in one and two dimensions

An explicit high-order single-stage single-step positivity-preserving finite difference WENO method for the compressible Euler equations

In this work we construct a high-order, single-stage, single-step positivity-preserving method for the compressible Euler equations. Space is discretized with the finite difference weighted essentially non-oscillatory (WENO) method. Time is discretized through a Lax-Wendroff procedure that is constructed from the Picard integral formulation (PIF) of the partial differential equation. The method can be viewed as a modified flux approach, where a linear combination of a low- and high-order flux defines the numerical flux used for a single-step update. The coefficients of the linear combination are constructed by solving a simple optimization problem at each time step. The high-order flux itself is constructed through the use of Taylor series and the Cauchy-Kowalewski procedure that incorporates higher-order terms. Numerical results in one- and two-dimensions are presented

Method of Lines Transpose: High Order L-Stable {O}(N) Schemes for Parabolic Equations Using Successive Convolution

We present a new solver for nonlinear parabolic problems that is L-stable and achieves high order accuracy in space and time. The solver is built by first constructing a one-dimensional heat equation solver that uses fast $\mathcal O(N)$ convolution. This fundamental solver has arbitrary order of accuracy in space and is based on the use of the Green\u27s function to invert a modified Helmholtz equation. Higher orders of accuracy in time are then constructed through a novel technique known as successive convolution (or resolvent expansions). These resolvent expansions facilitate our proofs of stability and convergence, and permit us to construct schemes that have provable stiff decay. The multidimensional solver is built by repeated application of dimensionally split independent fundamental solvers. Finally, we solve nonlinear parabolic problems by using the integrating factor method, where we apply the basic scheme to invert linear terms (that look like a heat equation), and make use of Hermite--Birkhoff interpolants to integrate the remaining nonlinear terms. Our solver is applied to several linear and nonlinear equations including heat, Allen--Cahn, and the FitzHugh--Nagumo system of equations in one and two dimensions

Excess dietary iodine intake in long-term African refugees

Abstract Objective To assess the iodine status of long-term refugees dependent on international food aid and humanitarian assistance. Design A series of cross-sectional two-stage cluster or systematic random sample surveys which assessed urinary iodine excretion and the prevalence of visible goitre. Salt samples were also collected and tested for iodine content by titration. Setting Six refugee camps in East, North and Southern Africa. Subjects Male and female adolescents aged 10-19 years. Main results The median urinary iodine concentration (UIC) ranged from 254 to 1200 μg l−1 and in five of the camps exceeded the recommended maximum limit of 300 μg l−1, indicating excessive iodine intake. Visible goitre was assessed in four surveys where it ranged from 0.0 to 7.1%. The camp with the highest UIC also had the highest prevalence of visible goitre. The iodine concentrations in 11 salt samples from three camps were measured by titration and six of these exceeded the production-level concentration of 20 to 40 ppm recommended by the World Health Organization (WHO), but were all less than 100 ppm. Conclusions Excessive consumption of iodine is occurring in most of the surveyed populations. Urgent revision of the level of salt iodisation is required to meet current WHO recommendations. However, the full cause of excessive iodine excretion remains unknown and further investigation is required urgently to identify the cause, assess any health impact and identify remedial actio

Prevalence of wasting among under 6-month-old infants in developing countries and implications of new case definitions using WHO growth standards: a secondary data analysis

Objectives To determine wasting prevalence among infants aged under 6 months and describe the effects of new case definitions based on WHO growth standards.Design Secondary data analysis of demographic and health survey datasets.Setting 21 developing countries.Population 15 534 infants under 6 months and 147 694 children aged 6 to under 60 months (median 5072 individuals/country, range 1710-45 398). Wasting was defined as weight-for-height z-score < -2, moderate wasting as -3 to <-2 z-scores, severe wasting as z-score <-3.Results Using National Center for Health Statistics (NCHS) growth references, the nationwide prevalence of wasting in infant under-6-month ranges from 1.1% to 15% (median 3.7%, IQR 1.8-6.5%; similar to 3 million wasted infants < 6 months worldwide). Prevalence is more than doubled using WHO standards: 2.0-34% (median 15%, IQR 6.2-17%; similar to 8.5 million wasted infants < 6 months worldwide). Prevalence differences using WHO standards are more marked for infants under 6 months than children, with the greatest increase being for severe wasting (indicated by a regression line slope of 3.5 for infants < 6 months vs 1.7 for children). Moderate infant-6-month wasting is also greater using WHO, whereas moderate child wasting is 0.9 times the NCHS prevalence.Conclusions Whether defined by NCHS references or WHO standards, wasting among infants under 6 months is prevalent in many of the developing countries examined in this study. Use of WHO standards to defi ne wasting results in a greater disease burden, particularly for severe wasting. Policy makers, programme managers and clinicians in child health and nutrition programmes should consider resource and risk/benefit implications of changing case defi nitions

Admission profile and discharge outcomes for infants aged less than 6 months admitted to inpatient therapeutic care in 10 countries. A secondary data analysis.

Evidence on the management of acute malnutrition in infants aged less than 6 months (infants <6mo) is scarce. To understand outcomes using current protocols, we analysed a sample of 24 045 children aged 0-60 months from 21 datasets of inpatient therapeutic care programmes in 10 countries. We compared the proportion of admissions, the anthropometric profile at admission and the discharge outcomes between infants <6mo and children aged 6-60 months (older children). Infants <6mo accounted for 12% of admissions. The quality of anthropometric data at admission was more problematic in infants <6mo than in older children with a greater proportion of missing data (a 6.9 percentage point difference for length values, 95% CI: 6.0; 7.9, P < 0.01), anthropometric measures that could not be converted to indices (a 15.6 percentage point difference for weight-for-length z-score values, 95% CI: 14.3; 16.9, P < 0.01) and anthropometric indices that were flagged as outliers (a 2.7 percentage point difference for any anthropometric index being flagged as an outlier, 95% CI: 1.7; 3.8, P < 0.01). A high proportion of both infants <6mo and older children were discharged as recovered. Infants <6mo showed a greater risk of death during treatment (risk ratio 1.30, 95% CI: 1.09; 1.56, P < 0.01). Infants <6mo represent an important proportion of admissions to therapeutic feeding programmes, and there are crucial challenges associated with their care. Systematic compilation and analysis of routine data for infants <6mo is necessary for monitoring programme performance and should be promoted as a tool to monitor the impact of new guidelines on care