An inverse Lax-Wendroff method for boundary conditions applied to Boltzmann type models

International audienceIn this paper we present a new algorithm based on Cartesian meshes for the numerical approximation of kinetic models set in an arbitrary geometry. Due to the high dimensional property of kinetic models, numerical algorithms based on unstructured meshes are not really appropriate since most of numerical methods (semi-Lagrangian, spectral methods) are particularly efficient on structured grids. Here we propose to adapt the inverse Lax-Wendroff procedure, which has been recently introduced for conservation laws [21], for kinetic equations. Numerical simulations in 1D x 3D and 2D x 3D based on this approach are proposed for Boltzmann type operators (BGK, ES-BGK models)

Mixed semi-Lagrangian/finite difference methods for plasma simulations

In this paper, we present an efficient algorithm for the long time behavior of plasma simulations. We will focus on 4D drift-kinetic model, where the plasma's motion occurs in the plane perpendicular to the magnetic field and can be governed by the 2D guiding-center model. Hermite WENO reconstructions, already proposed in \cite{YF15}, are applied for solving the Vlasov equation. Here we consider an arbitrary computational domain with an appropriate numerical method for the treatment of boundary conditions. Then we apply this algorithm for plasma turbulence simulations. We first solve the 2D guiding-center model in a D-shape domain and investigate the numerical stability of the steady state. Then, the 4D drift-kinetic model is studied with a mixed method, i.e. the semi-Lagrangian method in linear phase and finite difference method during the nonlinear phase. Numerical results show that the mixed method is efficient and accurate in linear phase and it is much stable during the nonlinear phase. Moreover, in practice it has better conservation properties.Comment: arXiv admin note: text overlap with arXiv:1312.448

Numerical study of bio-fluids and mass transfer processes through membranes

La reologia de la sang (hemoreología) exerceix un ppel important en la perforació de teixits i l'alteració de les seves condicions fisiològiques és gairebé sempre la principal causa de patologies cardiovasculars. Per tant, l'estudi dels perfils de velocitat i l'esforç de tall de la sang al llarg de micro-venes és important en la investigació de malalties cardiovasculars. Recents avenços en organs-on-a-chip remarquen la possibilitat d'usar lung-on-a-xips el qual ha estat desenvolupat per reemplaçar les funcions respiratòries de l'home en assajos farmaceuticos. S'ha avançat en l'estudi de processos de micro-separació a través de membranes micro-poroses desenvolupant una eina numèrica per modelar el comportaments dels micro-dispositius usant la geometria dels lung-on-a-chip en dues i tres dimensions, on una membrana artificial separa dos canals amb dos diferents fluids i així dos diferents règims de flux. A causa de que és un problema de múltiples escales, el nou codi consisteix d'un model híbrid LBM-FD (Lattice Boltzmann - diferències finites) sobre una malla no uniforme que modela processos de transferència de massa en fluxos no newtonians. El model híbrid LBM-FD va ser usat per estudiar el transport de massa a través d'una membrana hidrofòbica i microporosa ocalizada entre un flux co-corrent passant a través de canals rectangulars, similar als microdispositius usats en el projecte lung-on-a- xip. Estecódigo ha estat usat per fer un estudi paramètric en la recerca de la correlació entre el nombre de Peclet al canal permeat i els processos de transferència de massa a través de la membrana (Sherwood number mitjana ). Les correlacions al microdispositiu bidimensional reprodueix correctament la relació lineal de amb el nombre de porus. Les correlacions mostren un valor de l'exponencial en la llei de potències de 1/3 (el que caracteritza el problema de Graetz-Leveque) de pel que fa a Pe. S'han trobat les correlacions en dues i tres dimensions. En el cas tridimensional es comparen els resultats obtinguts usant el flux del model de la llei de pontencias truncat amb un grau de pseudo-plasticitat de n = 0.7 pel que fa als resultats obtinguts per flux newtonià n = 1. El cas no-newtonià mostra un increment del 5% en la transferència de massa () respecte al cas newtonià.La reología de la sangre (hemoreología) desempeña un papel importante en la perforación de tejidos y la alteración de sus condiciones fisiológicas es casi siempre la principal causa de patologías cardiovasculares. Por lo tanto, el estudio de los perfiles de velocidad y el esfuerzo de corte de la sangre a lo largo de micro-venas es importante en la investigación de enfermedades cardiovasculares. Recientes avances remarcan la posibilidad de usar micro-dispositivos como lung-on-a-chip, el cual ha sido desarrollado para reemplazar las funciones respiratorias del hombre en ensayos farmaceuticos. Se ha avanzado en el estudio de procesos de micro-separación a través de membranas micro-porosas desarrollando una herramienta numérica para modelar el comportamientos de los micro-dispositivos usando la geometría de los lung-on-a-chip en dos y tres dimensiones. Debido a que es un problema de múltiples escalas, el nuevo código consiste de un modelo híbrido LBM-FD (Lattice Boltzmann - Diferencias finitas) sobre una malla no uniforme que modela procesos de transferencia de masa en flujos no Newtonianos. El modelo híbrido LBM-FD fue usado para estudiar el transporte de masa a través de una membrana hidrofóbica y microporosa localizada entre un flujo co-corriente pasando a través de canales rectangulares, similar a los microdispositivos usados en el proyecto lung-on-a-chip. Con este código, un estudio paramétrico en la busqueda de la correlación entre el Peclet en el canal permeado y el Sherwood promedio ha sido realizado. Las correlaciones en el microdispositivo bi-dimensional reproduce correctamente la relación lineal de con el número de poros. Las correlaciones muestran un valor del exponencial en la ley de potencias de 1/3 ( lo que caracteriza el problema de Graetz-Leveque) de con respecto a Pe. Se han hallado las correlaciones en dos y tres dimensiones. En el caso tri-dimensional se comparan los resultados obtenidos usando el flujo del modelo del modelo de la ley de pontencias truncado con un grado de pseudo-plasticidad de n=0.7 con respecto a los resultados obtenidos para flujo Newtoniano n=1. El caso no-Newtoniano muestra un incremento del 5% en la transferencia de masa () con respecto al caso Newtoniano.Blood rheology (haemorheology) plays a key role in tissue perfusion and its alteration from physiological conditions is often the main cause of cardiovascular pathologies. Therefore, the study of blood velocity profiles and wall shear stress distribution along micro-vessels is important in the field of cardiovascular diseases research. Recent advances in organ-on-a-chip highlighted the possibility of using artificial lung-on-chips which have been developed to replace the respiratory functions of the human lungs in pharmaceutical tests. We have expanded the study of micro-separation processes through micro-porous membranes by developing a numerical tool able to model the behavior of lung-on-a-chip micro-devices in both two and three dimensional geometries. As this is a multiscale problem, the new code consists in a hybrid LBM-FD (Lattice Boltzmann - finite difference) model on a non-uniform material grid, that models mass transfer processes in non-Newtonian flows. A part from the validation of the code, results obtained include the correlations of the non-dimensional numbers involved in mass transfer processes and the dependence on porosity, and the study of concentration profiles under steady (pipe flow) and the beginning of the study in non-steady (Womersley flow) conditions. The LBM-FD hybrid model was used to study the mass transport through a hydrophobic micro-porous membrane located in-between a co-current flow passing through rectangular channels, which is similar to the micro-device used in Lung-On-a-Chip research. This code has been used to perform a parametric study to find the empirical correlation between Peclet number in the permeate channel and the mass transfer processes across the membrane which is quantified by mean of Sherwood number. The correlations in the two-dimensional micro-device reproduce correctly the linear scaling law of with the number of pores. The correlations give a power value equal to 1/3 (which characteristic of the Graetz-Leveque problem) for the scaling exponent of the average Sherwood number with Pe. This has been done in 2D and 3D models. In the three-dimensional case, we compared the results obtained using the power-law flow with a shear-thinning degree of n=0.7 against the results obtained using the Newtonian hypothesis (n=1). The non-Newtonian case

High order numerical schemes for transport equations on bounded domains

This article is an account of the NABUCO project achieved during the summer camp CEMRACS 2019 devoted to geophysical fluids and gravity flows. The goal is to construct finite difference approximations of the transport equation with nonzero incoming boundary data that achieve the best possible convergence rate in the maximum norm. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at the incoming boundary. Optimal convergence rates are obtained by combining sharp stability estimates for extrapolation boundary conditions with numerical boundary layer expansions. We illustrate the results with the Lax-Wendroff and O3 schemes

Lattice Boltzmann methods for direct numerical simulation of turbulent fluid flows

We study the use of lattice Boltzmann (LB) methods for simulation of turbulent fluid flows motivated by their high computational throughput and amenability to highly parallel platforms such as graphics processing units (GPUs). Several algorithmic improvements are unearthed including work on non-unit Courant numbers, the force operator, use of alternative topologies based on face and body centered cubic lattices and a new formulation using a generalized eigendecomposition that allows a new freedom in tuning the eigenvectors of the linearised collision operator. Applications include a variable bulk viscosity and the use of a stretched grid, our implementation of which reduces errors present in previous efforts. We present details for numerous lattices including all required matrices, their moments the procedures and programs used to generate these and perform linear stability analysis. Small Mach number flows where density variations are negligible except in the buoyancy force term allow the use of a highly accurate finite volume solver to simulate the evolution of the buoyancy field which is coupled to the LB simulation as an external force. We use a multidimensional flux limited third order flux integral based advection scheme. The simplified algorithm we have devised is easier to implement, has higher performance and does not sacrifice any accuracy compared to the leading alternative. Our algorithm is particularly suited to an outflow based implementation which furthers the stated benefits. We present numerical experiments confirming the third order accuracy of our scheme when applied to multidimensional advection. The coupled solver is implemented in a new code that runs in parallel across multiple machines using GPUs. Our code achieves high computational throughput and accuracy and is used to simulate a range of turbulent flows. Details regarding turbulent channel flow and sheared convective boundary layer simulations are presented including some new insight into the scaling properties of the latter flow

Finite element solution of the Boltzmann equation for rarefied macroscopic gas flows.

This thesis presents research carried out at The Civil and Computational Engineering Centre at Swansea University between September 2004 and December 2007. The focus of the research was the application of modern finite element solution techniques to the governing equations of molecular gas dynamics in order to solve macroscopic gas flow problems. The journey of research began by considering and comparing various finite difference and finite element formulations in the solution of a simple scalar convection equation. This formed the basis for developing a solver for a variety of forms of the Boltzmann equation of molecular gas dynamics, and application of these solvers to a range of subsonic, transonic and supersonic gas flow problems. The merits and drawbacks of the molecular approach, particularly when compared with more traditional continuum CFD solvers, are identified along with possible extensions to the work presented here

High order numerical schemes for transport equations on bounded domains*

This article is an account of the NABUCO project achieved during the summer camp CEMRACS 2019 devoted to geophysical fluids and gravity flows. The goal is to construct finite difference approximations of the transport equation with nonzero incoming boundary data that achieve the best possible convergence rate in the maximum norm. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at the incoming boundary. Optimal convergence rates are obtained by combining sharp stability estimates for extrapolation boundary conditions with numerical boundary layer expansions. We illustrate the results with the Lax-Wendroff and O3 schemes