The Semi Implicit Gradient Augmented Level Set Method

Here a semi-implicit formulation of the gradient augmented level set method is presented. By tracking both the level set and it's gradient accurate subgrid information is provided,leading to highly accurate descriptions of a moving interface. The result is a hybrid Lagrangian-Eulerian method that may be easily applied in two or three dimensions. The new approach allows for the investigation of interfaces evolving by mean curvature and by the intrinsic Laplacian of the curvature. In this work the algorithm, convergence and accuracy results are presented. Several numerical experiments in both two and three dimensions demonstrate the stability of the scheme.Comment: 19 Pages, 14 Figure

A comparative study of the efficiency of jet schemes

We present two versions of third order accurate jet schemes, which achieve high order accuracy by tracking derivative information of the solution along characteristic curves. For a benchmark linear advection problem, the efficiency of jet schemes is compared with WENO and Discontinuous Galerkin methods of the same order. Moreover, the performance of various schemes in tracking solution contours is investigated. It is demonstrated that jet schemes possess the simplicity and speed of WENO schemes, while showing several of the advantages as well as the accuracy of DG methods.Comment: 12 pages, 6 figures, presented at the conference Mathematical Modeling and Applications to Industrial Problems 201

Coupling surface flow with porous media flow

This thesis proposes a model for the interaction between ground flow and surface flow using a coupled system of the Navier-Stokes and Darcy equations. The coupling of surface flow with porous media flow has important applications in science and engineering. This work is motivated by applications to geo-sciences. This work couples the two flows using interface conditions that incorporate the continuity of the normal component, the balance of forces and the Beaver-Joseph-Saffman Law. The balance of forces condition can be written with or without inertial forces from the free fluid region. This thesis provides both theoretical and numerical analysis of the effect of the inertial forces on the model. Flow in porous media is often simulated over large domains in which the actual permeability is heterogeneous with discontinuities across the domain. The discontinuous Galerkin method is well suited to handle this problem. On the other hand, the continuous finite element is adequate for the free flow problems considered in this work. As a result this thesis proposes coupling the continuous finite element method in the free flow region with the discontinuous Galerkin method in the porous medium. Existence and uniqueness results of a weak solution and numerical scheme are proved. This work also provides derivations of optimal a priori error estimates for the numerical scheme. A two-grid approach to solving the coupled problem is analyzed. This method will decouple the problem naturally into two problems, one in the free flow domain and other in the porous medium. In applications for this model, it is often the case that the areas of interest (faults, kinks) in the porous medium are small compared to the rest of the domain. In view of this fact, the rest of the thesis is dedicated to a coupling of the Discontinuous Galerkin method in the problem areas with a cheaper method on the rest of the domain. The finite volume method will be coupled with the Discontinuous Galerkin method on parts of the domain on which the permeability field varies gradually to decrease the problem sizes and thus make the scheme more efficient

Meshfree Finite Differences for Vector Poisson and Pressure Poisson Equations with Electric Boundary Conditions

We demonstrate how meshfree finite difference methods can be applied to solve vector Poisson problems with electric boundary conditions. In these, the tangential velocity and the incompressibility of the vector field are prescribed at the boundary. Even on irregular domains with only convex corners, canonical nodalbased finite elements may converge to the wrong solution due to a version of the Babuška paradox. In turn, straightforward meshfree finite differences converge to the true solution, and even high-order accuracy can be achieved in a simple fashion. The methodology is then extended to a specific pressure Poisson equation reformulation of the Navier-Stokes equations that possesses the same type of boundary conditions. The resulting numerical approach is second order accurate and allows for a simple switching between an explicit and implicit treatment of the viscosity terms. Keywords: Meshfree Finite-differences; Navier-Stokes; Incompressible; Vector Poisson equation; Pressure Poisson equation; Reformulation; Manufactured solution; High-orderNational Science Foundation (U.S.) (Grant DMS–1318942

Unstructured, curved elements for the two-dimensional high order discontinuous control-volume/finite-element method

Quadrilateral and triangular elements with curved edges are developed in the framework of spectral, discontinuous, hybrid control-volume/finite-element method for elliptic problems. In order to accommodate hybrid meshes, encompassing both triangular and quadrilateral elements, one single mapping is used. The scheme is applied to two-dimensional problems with discontinuous, anisotropic diffusion coefficients, and the exponential convergence of the method is verified in the presence of curved geometries

Independent and combined effects of improved water, sanitation, and hygiene, and improved complementary feeding, on child stunting and anaemia in rural Zimbabwe: a cluster-randomised trial.

BACKGROUND: Child stunting reduces survival and impairs neurodevelopment. We tested the independent and combined effects of improved water, sanitation, and hygiene (WASH), and improved infant and young child feeding (IYCF) on stunting and anaemia in in Zimbabwe. METHODS: We did a cluster-randomised, community-based, 2 × 2 factorial trial in two rural districts in Zimbabwe. Clusters were defined as the catchment area of between one and four village health workers employed by the Zimbabwe Ministry of Health and Child Care. Women were eligible for inclusion if they permanently lived in clusters and were confirmed pregnant. Clusters were randomly assigned (1:1:1:1) to standard of care (52 clusters), IYCF (20 g of a small-quantity lipid-based nutrient supplement per day from age 6 to 18 months plus complementary feeding counselling; 53 clusters), WASH (construction of a ventilated improved pit latrine, provision of two handwashing stations, liquid soap, chlorine, and play space plus hygiene counselling; 53 clusters), or IYCF plus WASH (53 clusters). A constrained randomisation technique was used to achieve balance across the groups for 14 variables related to geography, demography, water access, and community-level sanitation coverage. Masking of participants and fieldworkers was not possible. The primary outcomes were infant length-for-age Z score and haemoglobin concentrations at 18 months of age among children born to mothers who were HIV negative during pregnancy. These outcomes were analysed in the intention-to-treat population. We estimated the effects of the interventions by comparing the two IYCF groups with the two non-IYCF groups and the two WASH groups with the two non-WASH groups, except for outcomes that had an important statistical interaction between the interventions. This trial is registered with ClinicalTrials.gov, number NCT01824940. FINDINGS: Between Nov 22, 2012, and March 27, 2015, 5280 pregnant women were enrolled from 211 clusters. 3686 children born to HIV-negative mothers were assessed at age 18 months (884 in the standard of care group from 52 clusters, 893 in the IYCF group from 53 clusters, 918 in the WASH group from 53 clusters, and 991 in the IYCF plus WASH group from 51 clusters). In the IYCF intervention groups, the mean length-for-age Z score was 0·16 (95% CI 0·08-0·23) higher and the mean haemoglobin concentration was 2·03 g/L (1·28-2·79) higher than those in the non-IYCF intervention groups. The IYCF intervention reduced the number of stunted children from 620 (35%) of 1792 to 514 (27%) of 1879, and the number of children with anaemia from 245 (13·9%) of 1759 to 193 (10·5%) of 1845. The WASH intervention had no effect on either primary outcome. Neither intervention reduced the prevalence of diarrhoea at 12 or 18 months. No trial-related serious adverse events, and only three trial-related adverse events, were reported. INTERPRETATION: Household-level elementary WASH interventions implemented in rural areas in low-income countries are unlikely to reduce stunting or anaemia and might not reduce diarrhoea. Implementation of these WASH interventions in combination with IYCF interventions is unlikely to reduce stunting or anaemia more than implementation of IYCF alone. FUNDING: Bill & Melinda Gates Foundation, UK Department for International Development, Wellcome Trust, Swiss Development Cooperation, UNICEF, and US National Institutes of Health.The SHINE trial is funded by the Bill & Melinda Gates Foundation (OPP1021542 and OPP113707); UK Department for International Development; Wellcome Trust, UK (093768/Z/10/Z, 108065/Z/15/Z and 203905/Z/16/Z); Swiss Agency for Development and Cooperation; US National Institutes of Health (2R01HD060338-06); and UNICEF (PCA-2017-0002)