A very high-order finite volume method for the time-dependent convection-diffusion problem with Butcher tableau extension

The time discretization of a very high-order finite volume method may give rise to new numerical difficulties resulting into accuracy degradations. Indeed, for the simple one-dimensional unstationary convection-diffusion equation for instance, a conflicting situation between the source term time discretization and the boundary conditions may arise when using the standard Runge-Kutta method. We propose an alternative procedure by extending the Butcher Tableau to overcome this specific difficulty and achieve fourth-, sixth- or eighth-order of accuracy schemes in space and time. To this end, a new finite volume method is designed based on specific polynomial reconstructions for the space discretization, while we use the Extended Butcher Tableau to perform the time discretization. A large set of numerical tests has been carried out to validate the proposed method.Fundação para a Ciência e a Tecnologia (FCT

An a posteriori strategy for adaptive schemes in time for one-dimensional advection-diffusion transport equations

Stability condition is a more restrictive constraint that leads to unnecessary small-time steps with respect to the accuracy and results in computational time wastage. We propose a node by node adaptive time scheme to relax the stability constraint enabling a larger global time step for all the nodes. A nonlinear procedure for optimising both the schemes in time and space is proposed in view of increasing the numerical efficiency and reducing the computational time. The method is based on a four-parameter family of schemes we shall tune in function of the physical data (velocity, diffusion), the characteristic size in time and space, and the local regularly of the function leading to a nonlinear procedure. The a posteriori strategy we adopt consists in, given the solution at time t(n), computing a candidate solution with the highest accurate schemes in time and space for all the nodes. Then, for the nodes that present some instabilities, both the schemes in time and space are modified and adapted in order to preserve the stability with a large time step. The updated solution is computed with node-dependent schemes both in time and space. For the sake of simplicity, only convection-diffusion problems are addressed as a prototype with a two-parameters five-points finite difference method for the spatial discretisation together with an explicit time two-parameters four-stages Runge-Kutta method. We prove that we manage to obtain an optimal time-step algorithm that produces accurate numerical approximations exempt of non-physical oscillations.G.J. Machado and S. Clain acknowledge the financial support by FEDER -Fundo Europeu de Desenvolvimento Regional, through COMPETE 2020 -Programa Operacional Fatores de Competitividade, and the National Funds through FCT -Fundacao para a Ciencia e a Tecnologia, project no. UID/FIS/04650/2019.M.T. Malheiro acknowledge the financial support by Portuguese Funds through FCT (Fundacao para a Ciencia e a Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020 of CMAT-UM.M.T. Malheiro, G.J. Machado, and S. Clain acknowledge the financial support by FEDER - Fundo Europeu de Desenvolvimento Regional, through COMPETE 2020 -Programa Operacional Fatores de Competitividade, and the National Funds through FCT -Fundacao para a Ciencia e a Tecnologia, project no. POCI-01-0145-FEDER-028118

Upper gastro-intestinal fibre-optic endoscopy

A study of 3 000 upper gastro-intestinal fibre-optic endoscopies over a 3-year period is reported. The majority of patients were endoscoped after barium meals and in 69% of the cases the endoscopic findings correlated well with the radiological findings. In 28% of the cases the endoscopic findings differed from the radiological findings. Endoscopy is more accurate than radiology in the examination of the postgastrectomy stomach, in acute gastro-intestinal haemorrhage, in the radiologically distorted gastric antrum and in the follow-up of gastric ulcer healing. The localisation, healing rate and incidence of recurrence in gastric ulcers and the accuracy of endoscopic biopsy and cytology are reported. The value of endoscopic examination of the stomach and duodenum is shown, but it is stressed that endoscopy is complementary to and not exclusive of radiology.S. Afr. Med. J., 48, 857 (1974)

6th-order finite volume approximations for the stokes equations with a curved boundary

A new solver for the Stokes equations based on the finite volume method is proposed using very accurate polynomial reconstruction to provide a 6th-order scheme. We face two main difficulties: the gradient-divergence duality where the divergence free condition will impose the pressure gradient, and on the other hand, we assume that the domain has a regular curved boundary. The last point implies that a simple approximation of the boundary using piecewise segment lines dramatically reduces the scheme accuracy to at most a second-order one. We propose a new and simple technology which enables to restore the full scheme accuracy based on a specific polynomial reconstruction only using the Gauss points of the curved boundary and does not require any geometrical transformation.Fundação para a Ciência e a Tecnologia (FCT)This research was financed by FEDER Funds through Programa Operational Fatores de Competitividade — COMPETE and by Portuguese Funds FCT — Fundação para a Ciência e a Tecnologia, within the Projects PEst-C/MAT/UI0013/2014, PTDC/MAT/121185/2010 and FCT-ANR/MAT-NAN/0122/201

6th-order finite volume approximation for the steady-state burger and euler equations: the mood approach

We propose an innovative method based on the MOOD technology (Multi-dimensional Optimal Order Detection) to provide a 6th-order finite volume approximation for the one-dimensional steady-state Burger and Euler equations. The main ingredient consists in using an 'a posteriori' limiting strategy to eliminate non physical oscillations deriving from the Gibbs phenomenon while keeping a high accuracy for the smooth part. A short overview of the MOOD method will be presented and numerical tests with regular or discontinuous solutions will assess the method capacity to produce excellent approximations. In the latter situation, the numerical results enable to detect the zone where it is necessary to reduce the degree of the polynomial reconstructions to preserve the scheme robustness.Fundação para a Ciência e a Tecnologia (FCT

Compact schemes in time with applications to partial differential equations

We propose a new class of fourth-and sixth-order schemes in time for parabolic and hyperbolic equations. The method follows the compact scheme methodology by elaborating implicit relations between the approximations of the function and its derivatives. We produce a series of A-stable methods with low dispersion and high accuracy. Several benchmarks for linear and non-linear Ordinary Differential Equations demonstrate the effectiveness of the method. Then a second set of numerical benchmarks for Partial Differential Equations such as convection-diffusion, Schrodinger equation, wave equation, Burgers, and Euler system give the numerical evidences of the superior advantage of the method with respect to the traditional Runge-Kutta or multistep methods.S. Clain and G.J. Machado acknowledge the financial support by Portuguese Funds through Foundation for Science and Technology (FCT) in the framework of the Strategic Funding UIDB/04650/2020. M.T. Malheiro acknowledges the financial support by Portuguese Funds through Foundation for Science and Technology (FCT) in the framework of the Projects UIDB/00013/2020 and UIDP/00013/2020 of CMAT-UM. S. Clain, G.J. Machado, and M.T. Malheiro acknowledge the fi-nancial support by FEDER - Fundo Europeu de Desenvolvimento Regional, through COMPETE 2020 - Programa Operacional Fatores de Competitividade, POCI-01-0145-FEDER-028118 and PTDC/MAT-APL/28118/2017

New cell-vertex reconstruction for finite volume scheme : application to the convection-diffusion-reaction equation

The design of efficient, simple, and easy to code, second-order finite volume methods is an important challenge to solve practical problems in physics and in engineering where complex and very accurate techniques are not required. We propose an extension of the original Frink's approach based on a cell-to-vertex interpolation to compute vertex values with neighbor cell values. We also design a specific scheme which enables to use whatever collocation point we want in the cells to overcome the mass centre point restrictive choice. The method is proposed for two- and three-dimension geometries and a second-order extension time-discretization is given for time-dependent equation. A large number of numerical simulations are carried out to highlight the performance of the new method.Fundação para a Ciência e a Tecnologia (FCT

Soft tissue modelling for analysis of errors in breast reduction surgery

Breast reduction is one of the most common procedures in breast surgery. The aim of this work is to develop a computational model allowing one to forecast the final breast geometry according to the incision marking parameters. This model can be used in surgery simulators that provide preoperative planning and training, allowing the study of the errors origin in breast reduction

Isotopic signals in an agricultural watershed suggest denitrification is locally intensive in riparian areas but extensive in upland soils

© The Author(s), 2022. This article is distributed under the terms of the Creative Commons Attribution License. The definitive version was published in Sigler, W. A., Ewing, S. A., Wankel, S. D., Jones, C. A., Leuthold, S., Brookshire, E. N. J., & Payn, R. A. Isotopic signals in an agricultural watershed suggest denitrification is locally intensive in riparian areas but extensive in upland soils. Biogeochemistry, 158, (2022): 251–268, https://doi.org/10.1007/s10533-022-00898-9.Nitrogen loss from cultivated soils threatens the economic and environmental sustainability of agriculture. Nitrate (NO3−) derived from nitrification of nitrogen fertilizer and ammonified soil organic nitrogen may be lost from soils via denitrification, producing dinitrogen gas (N2) or the greenhouse gas nitrous oxide (N2O). Nitrate that accumulates in soils is also subject to leaching loss, which can degrade water quality and make NO3− available for downstream denitrification. Here we use patterns in the isotopic composition of NO3− observed from 2012 to 2017 to characterize N loss to denitrification within soils, groundwater, and stream riparian corridors of a non-irrigated agroecosystem in the northern Great Plains (Judith River Watershed, Montana, USA). We find evidence for denitrification across these domains, expressed as a positive linear relationship between δ15N and δ18O values of NO3−, as well as increasing δ15N values with decreasing NO3− concentration. In soils, isotopic evidence of denitrification was present during fallow periods (no crop growing), despite net accumulation of NO3− from the nitrification of ammonified soil organic nitrogen. We combine previous results for soil NO3− mass balance with δ15N mass balance to estimate denitrification rates in soil relative to groundwater and streams. Substantial denitrification from soils during fallow periods may be masked by nitrification of ammonified soil organic nitrogen, representing a hidden loss of soil organic nitrogen and an under-quantified flux of N to the atmosphere. Globally, cultivated land spends ca. 50% of time in a fallow condition; denitrification in fallow soils may be an overlooked but globally significant source of agricultural N2O emissions, which must be reduced along-side other emissions to meet Paris Agreement goals for slowing global temperature increase.National Institute of Food and Agriculture, 2011–51130-31121, S. A. Ewing, 2011, S. A. Ewing, 2016–67026-25067, S. A. Ewing, Montana State University Extension, Montana Fertilizer Advisory Committee, Montana Agricultural Experiment Station, Montana State University Vice President of Research, Montana State University College of Agriculture, Montana Institute on Ecosystems, NSF EPSCoR, OIA-1757351, S. A. Ewing, OIA-1443108, S. A. Ewing, EPS-110134, S. A. Ewing