Comparison of different numerical methods for the solution of the time-fractional reaction-diffusion equation with variable diffusion coefficient

In this work we perform a comparison of two different numerical schemes for the solution of the time-fractional diffusion equation with variable diffusion coefficient and a nonlinear source term. The two methods are the implicit numerical scheme presented in [M.L. Morgado, M. Rebelo, Numerical approximation of distributed order reaction- diffusion equations, Journal of Computational and Applied Mathematics 275 (2015) 216-227] that is adapted to our type of equation, and a colocation method where Chebyshev polynomials are used to reduce the fractional differential equation to a system of ordinary differential equations

Development of an integral viscoelastic flow solver in OpenFOAM®

The usual high cost of commercial codes, and some technical limitations, clearly limits the employment of numerical modelling tools in both industry and academia. Consequently, the number of companies that use numerical code is limited and there a lot of effort put on the development and maintenance of in-house academic based codes. Having in mind the potential of using numerical modelling tools as a design aid, of both products and processes, different research teams have been contributing to the development of open source codes/libraries. In this framework, any individual can take advantage of the available code capabilities and/or implement additional features based on his specific needs. These type of codes are usually developed by large communities, which provide improvements and new features in their specific fields of research, thus increasing significantly the code development process. Among others, OpenFOAM® multi-physics computational library, developed by a very large and dynamic community, nowadays comprises several features usually only available in their commercial counterparts; e.g. dynamic meshes, large diversity of complex physical models, parallelization, multiphase models, to name just a few. This computational library is developed in C++ and makes use of most of all language capabilities to facilitate the implementation of new functionalities. Concerning the field of computational rheology, OpenFOAM® solvers were recently developed to deal with the most relevant differential viscoelastic rheological models, and stabilization techniques are currently being verified. This work describes the implementation of a new solver in OpenFOAM® library, able to cope with integral viscoelastic models based on the deformation field method. The implemented solver is verified through the comparison of the predicted results with analytical solutions, results published in the literature and by using the Method of Manufactured Solutions.

A new integral viscoelastic flow solver in OpenFOAM®

The usual high cost of commercial codes, and some technical limitations, clearly limits the employment of numerical modelling tools in both industry and academia. Consequently, the number of companies that use numerical code is limited and there a lot of effort put on the development and maintenance of in-house academic based codes . Having in mind the potential of using numerical modelling tools as a design aid, of both products and processes, different research teams have been contributing to the development of open source codes/libraries. In this framework, any individual can take advantage of the available code capabilities and/or implement additional features based on his specific needs. These type of codes are usually developed by large communities, which provide improvements and new features in their specific fields of research, thus increasing significantly the code development process. Among others, OpenFOAM® multi-physics computational library, developed by a very large and dynamic community, nowadays comprises several features usually only available in their commercial counterparts; e.g. dynamic meshes, large diversity of complex physical models, parallelization, multiphase models, to name just a few. This computational library is developed in C++ and makes use of most of all language capabilities to facilitate the implementation of new functionalities. Concerning the field of computational rheology, OpenFOAM® solvers were recently developed to deal with the most relevant differential viscoelastic rheological models, and stabilization techniques are currently being verified. This work describes the implementation of a new solver in OpenFOAM® library, able to cope with integral viscoelastic models based on the deformation field method. The implemented solver is verified through the comparison of the predicted results with analytical solutions, results published in the literature and by using the Method of Manufactured Solutions

An Equation for the Prediction of Oxygen Consumption in a Brazilian Population

Background: The equations predicting maximal oxygen uptake (VO2max or peak) presently in use in cardiopulmonary exercise testing (CPET) softwares in Brazil have not been adequately validated. These equations are very important for the diagnostic capacity of this method. Objective: Build and validate a Brazilian Equation (BE) for prediction of VO2peak in comparison to the equation cited by Jones (JE) and the Wasserman algorithm (WA). Methods: Treadmill evaluation was performed on 3119 individuals with CPET (breath by breath). The construction group (CG) of the equation consisted of 2495 healthy participants. The other 624 individuals were allocated to the external validation group (EVG). At the BE (derived from a multivariate regression model), age, gender, body mass index (BMI) and physical activity level were considered. The same equation was also tested in the EVG. Dispersion graphs and Bland-Altman analyses were built. Results: In the CG, the mean age was 42.6 years, 51.5% were male, the average BMI was 27.2, and the physical activity distribution level was: 51.3% sedentary, 44.4% active and 4.3% athletes. An optimal correlation between the BE and the CPET measured VO2peak was observed (0.807). On the other hand, difference came up between the average VO2peak expected by the JE and WA and the CPET measured VO2peak, as well as the one gotten from the BE (p = 0.001). Conclusion: BE presents VO2peak values close to those directly measured by CPET, while Jones and Wasserman differ significantly from the real VO2peak.

Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions

In this paper, a new class of generalized backward doubly stochastic differential equations is investigated. This class involves an integral with respect to an adapted continuous increasing process. A probabilistic representation for viscosity solutions of semi-linear stochastic partial differential equations with a Neumann boundary condition is given.

Reflected Backward Stochastic Differential Equation with Jumps and RCLL Obstacle

In this paper we study one-dimensional reflected backward stochastic differential equation when the noise is driven by a Brownian motion and an independent Poisson point process when the solution is forced to stay above a right continuous left-hand limited obstacle. We prove existence and uniqueness of the solution by using a penalization method combined with a monotonic limit theorem.

Quantifying rates of landscape evolution and tectonic processes by thermochronology and numerical modeling of crustal heat transport using PECUBE

PECUBE is a three-dimensional thermal-kinematic code capable of solving the heat production-diffusion-advection equation under a temporally varying surface boundary condition. It was initially developed to assess the effects of time-varying surface topography (relief) on low-temperature thermochronological datasets. Thermochronometric ages are predicted by tracking the time-temperature histories of rock-particles ending up at the surface and by combining these with various age-prediction models. In the decade since its inception, the PECUBE code has been under continuous development as its use became wider and addressed different tectonic-geomorphic problems. This paper describes several major recent improvements in the code, including its integration with an inverse-modeling package based on the Neighborhood Algorithm, the incorporation of fault-controlled kinematics, several different ways to address topographic and drainage change through time, the ability to predict subsurface (tunnel or borehole) data, prediction of detrital thermochronology data and a method to compare these with observations, and the coupling with landscape-evolution (or surface-process) models. Each new development is described together with one or several applications, so that the reader and potential user can clearly assess and make use of the capabilities of PECUBE. We end with describing some developments that are currently underway or should take place in the foreseeable future. (C) 2012 Elsevier B.V. All rights reserved.

Rigorous derivation of a nonlinear diffusion equation as fast-reaction limit of a continuous coagulation-fragmentation model with diffusion

Weak solutions of the spatially inhomogeneous (diffusive) Aizenmann-Bak model of coagulation-breakup within a bounded domain with homogeneous Neumann boundary conditions are shown to converge, in the fast reaction limit, towards local equilibria determined by their mass. Moreover, this mass is the solution of a nonlinear diffusion equation whose nonlinearity depends on the (size-dependent) diffusion coefficient. Initial data are assumed to have integrable zero order moment and square integrable first order moment in size, and finite entropy. In contrast to our previous result [CDF2], we are able to show the convergence without assuming uniform bounds from above and below on the number density of clusters.

A mixed finite element method for nonlinear diffusion equations

We propose a mixed finite element method for a class of nonlinear diffusion equations, which is based on their interpretation as gradient flows in optimal transportation metrics. We introduce an appropriate linearization of the optimal transport problem, which leads to a mixed symmetric formulation. This formulation preserves the maximum principle in case of the semi-discrete scheme as well as the fully discrete scheme for a certain class of problems. In addition solutions of the mixed formulation maintain exponential convergence in the relative entropy towards the steady state in case of a nonlinear Fokker-Planck equation with uniformly convex potential. We demonstrate the behavior of the proposed scheme with 2D simulations of the porous medium equations and blow-up questions in the Patlak-Keller-Segel model.

Gaussian estimates for the density of the non-linear stochastic heat equation in any space dimension

In this paper, we establish lower and upper Gaussian bounds for the probability density of the mild solution to the stochastic heat equation with multiplicative noise and in any space dimension. The driving perturbation is a Gaussian noise which is white in time with some spatially homogeneous covariance. These estimates are obtained using tools of the Malliavin calculus. The most challenging part is the lower bound, which is obtained by adapting a general method developed by Kohatsu-Higa to the underlying spatially homogeneous Gaussian setting. Both lower and upper estimates have the same form: a Gaussian density with a variance which is equal to that of the mild solution of the corresponding linear equation with additive noise.

Error estimate and control in the MSFV method for multiphase flow in porous media

n this paper the iterative MSFV method is extended to include the sequential implicit simulation of time dependent problems involving the solution of a system of pressure-saturation equations. To control numerical errors in simulation results, an error estimate, based on the residual of the MSFV approximate pressure field, is introduced. In the initial time steps in simulation iterations are employed until a specified accuracy in pressure is achieved. This initial solution is then used to improve the localization assumption at later time steps. Additional iterations in pressure solution are employed only when the pressure residual becomes larger than a specified threshold value. Efficiency of the strategy and the error control criteria are numerically investigated. This paper also shows that it is possible to derive an a-priori estimate and control based on the allowed pressure-equation residual to guarantee the desired accuracy in saturation calculation.

A new automated method versus continuous positive airway pressure method for measuring pressure-volume curves in patients with acute lung injury

Objective: To compare pressure–volume (P–V) curves obtained with the Galileo ventilator with those obtained with the CPAP method in patients with ALI or ARDS receiving mechanical ventilation. P–V curves were fitted to a sigmoidal equation with a mean R2 of 0.994 ± 0.003. Lower (LIP) and upper inflection (UIP), and deflation maximum curvature (PMC) points calculated from the fitted variables showed a good correlation between methods with high intraclass correlation coefficients. Bias and limits of agreement for LIP, UIP and PMC obtained with the two methods in the same patient were clinically acceptable.

Standardisation of 18F by a coincidence method using full solid angle detectors.

A solution of (18)F was standardised with a 4pibeta-4pigamma coincidence counting system in which the beta detector is a one-inch diameter cylindrical UPS89 plastic scintillator, positioned at the bottom of a well-type 5''x5'' NaI(Tl) gamma-ray detector. Almost full detection efficiency-which was varied downwards electronically-was achieved in the beta-channel. Aliquots of this (18)F solution were also measured using 4pigamma NaI(Tl) integral counting and Monte Carlo calculated efficiencies as well as the CIEMAT-NIST method. Secondary measurements of the same solution were also performed with an IG11 ionisation chamber whose equivalent activity is traceable to the Système International de Référence through the contribution IRA-METAS made to it in 2001; IRA's degree of equivalence was found to be close to the key comparison reference value (KCRV). The (18)F activity predicted by this coincidence system agrees closely with the ionisation chamber measurement and is compatible within one standard deviation of the other primary measurements. This work demonstrates that our new coincidence system can standardise short-lived radionuclides used in nuclear medicine.

A new efficient approach to the direct restricted active space self-consistent field method

An implicitly parallel method for integral-block driven restricted active space self-consistent field (RASSCF) algorithms is presented. The approach is based on a model space representation of the RAS active orbitals with an efficient expansion of the model subspaces. The applicability of the method is demonstrated with a RASSCF investigation of the first two excited states of indole

A pseudo-spectral method for the simulation of poro-elastic seismic wave propagation in 2D polar coordinates using domain decomposition

We present a novel numerical approach for the comprehensive, flexible, and accurate simulation of poro-elastic wave propagation in 2D polar coordinates. An important application of this method and its extensions will be the modeling of complex seismic wave phenomena in fluid-filled boreholes, which represents a major, and as of yet largely unresolved, computational problem in exploration geophysics. In view of this, we consider a numerical mesh, which can be arbitrarily heterogeneous, consisting of two or more concentric rings representing the fluid in the center and the surrounding porous medium. The spatial discretization is based on a Chebyshev expansion in the radial direction and a Fourier expansion in the azimuthal direction and a Runge-Kutta integration scheme for the time evolution. A domain decomposition method is used to match the fluid-solid boundary conditions based on the method of characteristics. This multi-domain approach allows for significant reductions of the number of grid points in the azimuthal direction for the inner grid domain and thus for corresponding increases of the time step and enhancements of computational efficiency. The viability and accuracy of the proposed method has been rigorously tested and verified through comparisons with analytical solutions as well as with the results obtained with a corresponding, previously published, and independently bench-marked solution for 2D Cartesian coordinates. Finally, the proposed numerical solution also satisfies the reciprocity theorem, which indicates that the inherent singularity associated with the origin of the polar coordinate system is adequately handled.