Abstract. In this paper we develop a novel two-stage fourth order time-accurate discretization for time-dependent flow problems, particularly for hyperbolic conservation laws. Different from the classical Runge-Kutta (R-K) temporal discretization for first order Riemann solvers as building blocks, the current approach is solely associated with Lax-Wendroff (L-W) type schemes as the building blocks. As a result, a two-stage procedure can be constructed to achieve a fourth order temporal accuracy, rather than using well-developed four stages for R-K methods. The generalized Riemann problem (GRP) solver is taken as a representative of L-W type schemes for the construction of a two-stage fourth order scheme.

BackgroundChildhood obesity has been increasing rapidly worldwide. There is limited evidence for effective lifestyle interventions to prevent childhood obesity worldwide, especially in developing countries like China. The objective of this study was to assess the effectiveness of a school-based multi-component lifestyle childhood obesity prevention program (the CLICK-Obesity study) in Mainland China.MethodsA cluster randomized controlled trial was developed among grade 4 students from 8 urban primary schools (638 students in intervention, 544 as control) in Nanjing City, China. Students were randomly allocated to the control or intervention group at school-level. A one-year multi-component intervention program (classroom curriculum, school environment support, family involvement and fun programs/events) together with routine health education was provided to the intervention group, while the control group received routine health education only. The main outcome variables assessed were changes in body mass index, obesity occurrence, obesity-related lifestyle behaviors and knowledge.ResultsOverall, 1108 (93.7%) of the 1182 enrolled students completed the intervention study. The intervention group had a larger marginal reduction than did the control group in overall mean BMI value (-0.32±1.36 vs. -0.29±1.40, p = 0.09), although this was not significant. Compared with the control group, the intervention group was more likely to decrease their BMI (OR = 1.44, 95%CI = 1.10, 1.87) by 0.5 kg/m2 or above, increase the frequency of jogging/running (OR = 1.55, 95%CI = 1.18, 2.02), decrease the frequency of TV/computer use (OR = 1.41, 95%CI = 1.09, 1.84) and of red meat consumption (OR = 1.50, 95%CI = 1.15, 1.95), change commuting mode to/from school from sedentary to active mode (OR = 2.24, 95%CI = 1.47, 3.40), and be aware of the harm of selected obesity risk factors.ConclusionsThe school-based lifestyle intervention program was practical and effective in improving health behaviors and obesity-related knowledge for children in China. This study provides important policy implications on school-based intervention programs for modifications of obesity-related lifestyles.Trial RegistrationChinese Clinical Trial Registry ChiCTR-ERC-11001819

Abstract. A direct Eulerian generalized Riemann problem (GRP) scheme is derived for compressible fluid flows. Riemann invariants are introduced as the main ingredient to resolve the generalized Riemann problem (GRP) directly for the Eulerian formulation. The crucial auxiliary Lagrangian scheme in the original GRP scheme is not necessary in the present framework. The delicate sonic cases can be easily treated and the extension to multidimensional cases is straightforward.

For computational fluid dynamics (CFD), the generalized Riemann problem (GRP) solver and the gas-kinetic kinetic scheme (GKS) provide a time-accurate flux function starting from a discontinuous piecewise linear flow distributions around each cell interface. With the use of time derivative of the flux function, a two-stage Lax-Wendroff-type (L-W for short) time stepping method has been recently proposed in the design of a fourth-order time accurate method [18]. In this paper, based on the same time-stepping method and the second-order GKS flux function [34], a fourth-order gas-kinetic scheme is constructed for the Euler and Navier-Stokes equations. In comparison with the formal one-stage time-stepping third-order gas-kinetic solver [21], the current fourth-order method not only reduces the complexity of the flux function, but also improves the accuracy of the scheme, even though the third-and fourth-order schemes have similar computation cost. Most importantly, the robustness of the fourth-order GKS is as good as the second-order one. Perfect numerical solutions can be obtained from the high Reynolds number boundary layer solutions to the hypersonic viscous heat conducting flow computations. Many numerical tests, including many difficult ones for the Navier-Stokes solvers, have been used to validate the current fourth-order method. Following the two-stage time-stepping framework, the one-stage third-order GKS can be easily extended to a fifth-order method with the usage of both first-order and second-order time derivatives of the flux function. The use of time-accurate flux function may have great impact on the development of higher-order CFD methods.

The Generalized Riemann Problem (GRP) for a nonlinear hyperbolic system of m balance laws (or alternatively "quasi-conservative" laws) in one space dimension is now well-known and can be formulated as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution at the discontinuity. In particular, the GRP numerical scheme (second-order high resolution) is based on an analytical evaluation of the first time derivative. It turns out that this derivative depends only on the first-order spatial derivatives, hence the initial data can be taken as piecewise linear. The analytical solution is readily obtained for a single equation (m = 1) and, more generally, if the system is endowed with a complete (coordinate) set of Riemann invariants. In this case it can be "diagonalized" and reduced to the scalar case. However, most systems with m > 2 do not admit such a set of Riemann invariants. This paper introduces a generalization of this concept: weakly coupled systems (WCS). Such systems have only "partial set" of Riemann invariants, but these sets are weakly coupled in a way which enables a "diagonalized" treatment of the GRP. An important example of a WCS is the Euler system of compressible, nonisentropic fluid flow (m = 3). The solution of the GRP discussed here is based on a careful analysis of rarefaction waves. A "propagation of singularities" argument is applied to appropriate Riemann invariants across the rarefaction fan. It serves to "rotate" initial spatial slopes into "time derivative". In particular, the case of a "sonic point" is incorporated easily into the general treatment. A GRP scheme based on this solution is derived, and several numerical examples are presented. Special attention is given to the "acoustic approximation" of the analytical solution. It can be viewed as a proper linearization (different from the approach of Roe) of the nonlinear system. The resulting numerical scheme is the simplest (second-order, high-resolution) generalization of the Godunov scheme.