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.

Optimization of multiclass queueing networks with changeover times via the achievable region method: Part II, the multi-station case

We address the problem of scheduling a multi-station multiclassqueueing network (MQNET) with server changeover times to minimizesteady-state mean job holding costs. We present new lower boundson the best achievable cost that emerge as the values ofmathematical programming problems (linear, semidefinite, andconvex) over relaxed formulations of the system's achievableperformance region. The constraints on achievable performancedefining these formulations are obtained by formulatingsystem's equilibrium relations. Our contributions include: (1) aflow conservation interpretation and closed formulae for theconstraints previously derived by the potential function method;(2) new work decomposition laws for MQNETs; (3) new constraints(linear, convex, and semidefinite) on the performance region offirst and second moments of queue lengths for MQNETs; (4) a fastbound for a MQNET with N customer classes computed in N steps; (5)two heuristic scheduling policies: a priority-index policy, anda policy extracted from the solution of a linear programmingrelaxation.

The development of a stable oral solution of captopril for paediatric patients

Study objectives: Many major drugs are not available in paediatric form. The aim of this study was to develop a stable liquid solution of captopril for oral paediatric use allowing individualised dosage and easy administration to newborn and young patients. Methods: A specific HPLC-UV method was developed. In a pilot study, a number of formulations described in the literature as affording one-month stability were examined. In the proper long-term study, the formulation that gave the best results was then prepared in large batches and its stability monitored for two years at 5°C and room temperature, and for one year at 40°C. Results: Most formulations described in the literature were found wanting in our pilot study. A simple solution of the drug (1 mg/mL) in purified water (European Pharmacopeia) containing 0.1% disodium edetate (EDTA-Na) as preservative proved chemically and microbiologically stable at 5°C and room temperature for two years. Conclusion: The proposed in-house formulation fulfils stringent criteria of purity and stability and is fully acceptable for oral administration to newborn and young patients.

A pseudo-spectral method for simulating of poro-elastic seismic wave propagation in complex borehole environments

We present a novel numerical approach for the comprehensive, flexible, and accurate simulation of poro-elastic wave propagation in cylindrical coordinates. An important application of this method is 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 consisting of three concentric domains representing the borehole fluid in the center, the borehole casing and the surrounding porous formation. The spatial discretization is based on a Chebyshev expansion in the radial direction, Fourier expansions in the other directions, and a Runge-Kutta integration scheme for the time evolution. A domain decomposition method based on the method of characteristics is used to match the boundary conditions at the fluid/porous-solid and porous-solid/porous-solid interfaces. The viability and accuracy of the proposed method has been tested and verified in 2D polar coordinates through comparisons with analytical solutions as well as with the results obtained with a corresponding, previously published, and independently benchmarked solution for 2D Cartesian coordinates. 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 handled adequately.

Nitrate and ammonium in soil solution in tobacco management systems

Tobacco farmers of southern Brazil use high levels of fertilizers, without considering soil and environmental attributes, posing great risk to water resources degradation. The objective of this study was to monitor nitrate and ammonium concentrations in the soil solution of an Entisol in and below the root zone of tobacco under conventional tillage (CT), minimum tillage (MT) and no-tillage (NT). The study was conducted in the small-watershed Arroio Lino, in Agudo, State of Rio Grande do Sul, Brazil. A base fertilization of 850 kg ha-1 of 10-18-24 and topdressing of 400 kg ha-1 of 14-0-14 NPK fertilizer were applied. The soil solution was sampled during the crop cycle with a tension lysimeter equipped with a porous ceramic cup. Ammonium and nitrate concentrations were analyzed by the distillation and titration method. Nitrate concentrations, ranging from 8 to 226 mg L-1, were highest after initial fertilization and decreased during the crop cycle. The average nitrate (N-NO3-) concentration in the root zone was 75 in NT, 95 in MT, and 49 mg L-1 in CT. Below the root zone, the average nitrate concentration was 58 under NT, 108 under MT and 36 mg L-1 under CT. The nitrate and ammonium concentrations did not differ significantly in the management systems. However, the nitrate concentrations measured represent a contamination risk to groundwater of the watershed. The ammonium concentration (N-NH4+) decreased over time in all management systems, possibly as a result of the nitrification process and root uptake of part of the ammonium by the growing plants.

Tree and Brush Control For County Road Right-of-Way, Ocotober 2002

This manual summarizes the roadside tree and brush control methods used by all of Iowa's 99 counties. It is based on interviews conducted in Spring 2002 with county engineers, roadside managers and others. The target audience of this manual is the novice county engineer or roadside manager. Iowa law is nearly silent on roadside tree and brush control, so individual counties have been left to decide on the level of control they want to achieve and maintain. Different solutions have been developed but the goal of every county remains the same: to provide safe roads for the traveling public. Counties in eastern and southern Iowa appear to face the greatest brush control challenge. Most control efforts can be divided into two categories: mechanical and chemical. Mechanical control includes cutting tools and supporting equipment. A chain saw is the most widely used cutting tool. Tractor mounted boom mowers and brush cutters are used to prune miles of brush but have significant safety and aesthetic limitations and boom mowers are easily broken by inexperienced operators. The advent of tree shears and hydraulic thumbs offer unprecedented versatility. Bulldozers are often considered a method of last resort since they reduce large areas to bare ground. Any chipper that violently grabs brush should not be used. Chemical control is the application of herbicide to different parts of a plant: foliar spray is applied to leaves; basal bark spray is applied to the tree trunk; a cut stump treatment is applied to the cambium ring of a cut surface. There is reluctance by many to apply herbicide into the air due to drift concerns. One-third of Iowa counties do not use foliar spray. By contrast, several accepted control methods are directed toward the ground. Freshly cut stumps should be treated to prevent resprouting. Basal bark spray is highly effective in sensitive areas such as near houses. Interest in chemical control is slowly increasing as herbicides and application methods are refined. Fall burning, a third, distinctly separate technique is underused as a brush control method and can be effective if timed correctly. In all, control methods tend to reflect agricultural patterns in a county. The use of chain saws and foliar sprays tends to increase in counties where row crops predominate, and boom mowing tends to increase in counties where grassland predominates. For counties with light to moderate roadside brush, rotational maintenance is the key to effective control. The most comprehensive approach to control is to implement an integrated roadside vegetation management (IRVM) program. An IRVM program is usually directed by a Roadside Manager whose duties may be shared with another position. Funding for control programs comes from the Rural Services Basic portion of a county's budget. The average annual county brush control budget is about $76,000. That figure is thought not to include shared expenses such as fuel and buildings. Start up costs for an IRVM program are less if an existing control program is converted. In addition, IRVM budgets from three different northeastern Iowa counties are offered for comparison in this manual. The manual also includes a chapter on temporary traffic control in rural work zones, a summary of the Iowa Code as it relates to brush control, and rules on avoiding seasonal disturbance of the endangered Indiana bat. Appendices summarize survey and forest cover data, an equipment inventory, sample forms for record keeping, a sample brush control policy, a few legal opinions, a literature search, and a glossary.

An improved method for measuring soil microbial activity by gas phase flow injection analysis

The rate of carbon dioxide production is commonly used as a measure of microbial activity in the soil. The traditional method of CO2 determination involves trapping CO2 in an alkali solution and then determining CO2 concentration indirectly by titration of the remaining alkali in the solution. This method is still commonly employed in laboratories throughout the world due to its relative simplicity and the fact that it does not require expensive, specific equipment. However, there are several drawbacks: the method is time-consuming, requires large amounts of chemicals and the consistency of results depends on the operator's skills. With this in mind, an improved method was developed to analyze CO2 captured in alkali traps, which is cheap and relatively simple, with a substantially shorter sample handling time and reproducibility equivalent to the traditional titration method. A comparison of the concentration values determined by gas phase flow injection analysis (GPFIA) and titration showed no significant difference (p > 0.05), but GPFIA has the advantage that only a tenth of the sample volume of the titration method is required. The GPFIA system does not require the purchase of new, costly equipment but the device was constructed from items commonly found in laboratories, with suggestions for alternative configurations for other detection units. Furthermore, GPFIA for CO2 analysis can be equally applied to samples obtained from either the headspace of microcosms or from a sampling chamber that allows CO2 to be released from alkali trapping solutions. The optimised GPFIA method was applied to analyse CO2 released from degrading hydrocarbons from a site contaminated by diesel spillage.

Soil CO2 flux: a method comparison of closed static chambers in a sugarcane field

A large variety of techniques have been used to measure soil CO2 released from the soil surface, and much of the variability observed between locations must be attributed to the different methods used by the investigators. Therefore, a minimum protocol of measurement procedures should be established. The objectives of this study were (a) to compare different absorption areas, concentrations and volumes of the alkali trapping solution used in closed static chambers (CSC), and (b) to compare both, the optimized alkali trapping solution and the soda-lime trapping using CSC to measure soil respiration in sugarcane areas. Three CO2 absorption areas were evaluated (7; 15 and 20 % of the soil emission area or chamber); two volumes of NaOH (40 and 80 mL) at three concentrations (0.1, 0.25 and 0.5 mol L-1). Three different types of alkaline traps were tested: (a), 80 mL of 0.5 mol L-1 NaOH in glass containers, absorption area 15 % (V0.5); (b) 40 mL of 2 mol L-1 NaOH retained in a sponge, absorption area 80 % (S2) and (c) 40 g soda lime, absorption area 15 % (SL). NaOH concentrations of 0.5 mol L-1 or lower underestimated the soil CO2-C flux or CO2 flux. The lower limit of the alkali trap absorption area should be a minimum of 20 % of the area covered by the chamber. The 2 mol L-1 NaOH solution trap (S2) was the most efficient (highest accuracy and highest CO2 fluxes) in measuring soil respiration.

Modelling tree population dynamics at the alpine and boreal tree-line ecotones in response to climate and land-use change

Summary Ecotones are sensitive to change because they contain high numbers of species living at the margin of their environmental tolerance. This is equally true of tree-lines, which are determined by attitudinal or latitudinal temperature gradients. In the current context of climate change, they are expected to undergo modifications in position, tree biomass and possibly species composition. Attitudinal and latitudinal tree-lines differ mainly in the steepness of the underlying temperature gradient: distances are larger at latitudinal tree-lines, which could have an impact on the ability of tree species to migrate in response to climate change. Aside from temperature, tree-lines are also affected on a more local level by pressure from human activities. These are also changing as a consequence of modifications in our societies and may interact with the effects of climate change. Forest dynamics models are often used for climate change simulations because of their mechanistic processes. The spatially-explicit model TreeMig was used as a base to develop a model specifically tuned for the northern European and Alpine tree-line ecotones. For the latter, a module for land-use change processes was also added. The temperature response parameters for the species in the model were first calibrated by means of tree-ring data from various species and sites at both tree-lines. This improved the growth response function in the model, but also lead to the conclusion that regeneration is probably more important than growth for controlling tree-line position and species' distributions. The second step was to implement the module for abandonment of agricultural land in the Alps, based on an existing spatial statistical model. The sensitivity of its most important variables was tested and the model's performance compared to other modelling approaches. The probability that agricultural land would be abandoned was strongly influenced by the distance from the nearest forest and the slope, bath of which are proxies for cultivation costs. When applied to a case study area, the resulting model, named TreeMig-LAb, gave the most realistic results. These were consistent with observed consequences of land-abandonment such as the expansion of the existing forest and closing up of gaps. This new model was then applied in two case study areas, one in the Swiss Alps and one in Finnish Lapland, under a variety of climate change scenarios. These were based on forecasts of temperature change over the next century by the IPCC and the HadCM3 climate model (ΔT: +1.3, +3.5 and +5.6 °C) and included a post-change stabilisation period of 300 years. The results showed radical disruptions at both tree-lines. With the most conservative climate change scenario, species' distributions simply shifted, but it took several centuries reach a new equilibrium. With the more extreme scenarios, some species disappeared from our study areas (e.g. Pinus cembra in the Alps) or dwindled to very low numbers, as they ran out of land into which they could migrate. The most striking result was the lag in the response of most species, independently from the climate change scenario or tree-line type considered. Finally, a statistical model of the effect of reindeer (Rangifer tarandus) browsing on the growth of Pinus sylvestris was developed, as a first step towards implementing human impacts at the boreal tree-line. The expected effect was an indirect one, as reindeer deplete the ground lichen cover, thought to protect the trees against adverse climate conditions. The model showed a small but significant effect of browsing, but as the link with the underlying climate variables was unclear and the model was not spatial, it was not usable as such. Developing the TreeMig-LAb model allowed to: a) establish a method for deriving species' parameters for the growth equation from tree-rings, b) highlight the importance of regeneration in determining tree-line position and species' distributions and c) improve the integration of social sciences into landscape modelling. Applying the model at the Alpine and northern European tree-lines under different climate change scenarios showed that with most forecasted levels of temperature increase, tree-lines would suffer major disruptions, with shifts in distributions and potential extinction of some tree-line species. However, these responses showed strong lags, so these effects would not become apparent before decades and could take centuries to stabilise. Résumé Les écotones son sensibles au changement en raison du nombre élevé d'espèces qui y vivent à la limite de leur tolérance environnementale. Ceci s'applique également aux limites des arbres définies par les gradients de température altitudinaux et latitudinaux. Dans le contexte actuel de changement climatique, on s'attend à ce qu'elles subissent des modifications de leur position, de la biomasse des arbres et éventuellement des essences qui les composent. Les limites altitudinales et latitudinales diffèrent essentiellement au niveau de la pente des gradients de température qui les sous-tendent les distance sont plus grandes pour les limites latitudinales, ce qui pourrait avoir un impact sur la capacité des espèces à migrer en réponse au changement climatique. En sus de la température, la limite des arbres est aussi influencée à un niveau plus local par les pressions dues aux activités humaines. Celles-ci sont aussi en mutation suite aux changements dans nos sociétés et peuvent interagir avec les effets du changement climatique. Les modèles de dynamique forestière sont souvent utilisés pour simuler les effets du changement climatique, car ils sont basés sur la modélisation de processus. Le modèle spatialement explicite TreeMig a été utilisé comme base pour développer un modèle spécialement adapté pour la limite des arbres en Europe du Nord et dans les Alpes. Pour cette dernière, un module servant à simuler des changements d'utilisation du sol a également été ajouté. Tout d'abord, les paramètres de la courbe de réponse à la température pour les espèces inclues dans le modèle ont été calibrées au moyen de données dendrochronologiques pour diverses espèces et divers sites des deux écotones. Ceci a permis d'améliorer la courbe de croissance du modèle, mais a également permis de conclure que la régénération est probablement plus déterminante que la croissance en ce qui concerne la position de la limite des arbres et la distribution des espèces. La seconde étape consistait à implémenter le module d'abandon du terrain agricole dans les Alpes, basé sur un modèle statistique spatial existant. La sensibilité des variables les plus importantes du modèle a été testée et la performance de ce dernier comparée à d'autres approches de modélisation. La probabilité qu'un terrain soit abandonné était fortement influencée par la distance à la forêt la plus proche et par la pente, qui sont tous deux des substituts pour les coûts liés à la mise en culture. Lors de l'application en situation réelle, le nouveau modèle, baptisé TreeMig-LAb, a donné les résultats les plus réalistes. Ceux-ci étaient comparables aux conséquences déjà observées de l'abandon de terrains agricoles, telles que l'expansion des forêts existantes et la fermeture des clairières. Ce nouveau modèle a ensuite été mis en application dans deux zones d'étude, l'une dans les Alpes suisses et l'autre en Laponie finlandaise, avec divers scénarios de changement climatique. Ces derniers étaient basés sur les prévisions de changement de température pour le siècle prochain établies par l'IPCC et le modèle climatique HadCM3 (ΔT: +1.3, +3.5 et +5.6 °C) et comprenaient une période de stabilisation post-changement climatique de 300 ans. Les résultats ont montré des perturbations majeures dans les deux types de limites de arbres. Avec le scénario de changement climatique le moins extrême, les distributions respectives des espèces ont subi un simple glissement, mais il a fallu plusieurs siècles pour qu'elles atteignent un nouvel équilibre. Avec les autres scénarios, certaines espèces ont disparu de la zone d'étude (p. ex. Pinus cembra dans les Alpes) ou ont vu leur population diminuer parce qu'il n'y avait plus assez de terrains disponibles dans lesquels elles puissent migrer. Le résultat le plus frappant a été le temps de latence dans la réponse de la plupart des espèces, indépendamment du scénario de changement climatique utilisé ou du type de limite des arbres. Finalement, un modèle statistique de l'effet de l'abroutissement par les rennes (Rangifer tarandus) sur la croissance de Pinus sylvestris a été développé, comme première étape en vue de l'implémentation des impacts humains sur la limite boréale des arbres. L'effet attendu était indirect, puisque les rennes réduisent la couverture de lichen sur le sol, dont on attend un effet protecteur contre les rigueurs climatiques. Le modèle a mis en évidence un effet modeste mais significatif, mais étant donné que le lien avec les variables climatiques sous jacentes était peu clair et que le modèle n'était pas appliqué dans l'espace, il n'était pas utilisable tel quel. Le développement du modèle TreeMig-LAb a permis : a) d'établir une méthode pour déduire les paramètres spécifiques de l'équation de croissance ä partir de données dendrochronologiques, b) de mettre en évidence l'importance de la régénération dans la position de la limite des arbres et la distribution des espèces et c) d'améliorer l'intégration des sciences sociales dans les modèles de paysage. L'application du modèle aux limites alpines et nord-européennes des arbres sous différents scénarios de changement climatique a montré qu'avec la plupart des niveaux d'augmentation de température prévus, la limite des arbres subirait des perturbations majeures, avec des glissements d'aires de répartition et l'extinction potentielle de certaines espèces. Cependant, ces réponses ont montré des temps de latence importants, si bien que ces effets ne seraient pas visibles avant des décennies et pourraient mettre plusieurs siècles à se stabiliser.

Application of alkaline waste from pulp industry to acid soil with pine

In Brazil extensive areas are covered with pine forests, planted for pulp and paper production. This industry generates solid alkaline waste, such as dregs. The application of this dregs to forest soils is an alternative for soil acidity correction and plant nutrient supply, as well as a solution for its proper disposal. The purpose of this study was to compare the residual effect of surface application of dregs and dolomitic lime on (a) changes in the physical and chemical properties of an acidic soil and (b) pine tree development. The experiment was carried out in 2004 in Bocaina do Sul, Santa Catarina, consisting of the application of increasing dreg and lime rates to a Pinus taeda L. production area, on a Humic Cambisol, in a randomized block design with four replications and 10 x 10 m plots. The treatments consisted of levels of soil acidity amendments corresponding to the recommendations by the SMP method to reach pH 5.5 in the 0-20 cm layer, as follows: no soil amendment; dregs at 5.08 (1/4 SMP), 10.15 (1/2 SMP) and 20.3 Mg ha-1 (1 SMP); and lime at 8.35 (1/2 SMP) and 16.7 Mg ha-1 (1 SMP). Soil layers were sampled in 2010 for analyses of soil chemical and physical properties. The diameter at breast height of the 6.5 year old pine trees was also evaluated. Surface application of dregs improved soil chemical fertility by reducing acidity and increasing base saturation, similar to liming, especially in surface layers. Dregs, comparable to lime, reduced the degree of clay flocculation, but did not affect the soil physical quality. There was no effect of the amendments on increase in pine tree diameter. Thus, the alternative to raise the pH in forest soils to 5.5 with dregs is promising for the forestry sector with a view to dispose of the waste and increase soil fertility.

Exact solution to the mean exit time problem for free inertial processes driven by Gaussian white noise

We obtain the exact analytical expression, up to a quadrature, for the mean exit time, T(x,v), of a free inertial process driven by Gaussian white noise from a region (0,L) in space. We obtain a completely explicit expression for T(x,0) and discuss the dependence of T(x,v) as a function of the size L of the region. We develop a new method that may be used to solve other exit time problems.

Exact solution to the exit-time problem for an undamped free particle driven by Gaussian white noise

In a recent paper [Phys. Rev. Lett. 75, 189 (1995)] we have presented the exact analytical expression for the mean exit time, T(x,v), of a free inertial process driven by Gaussian white noise out of a region (0,L) in space. In this paper we give a detailed account of the method employed and present results on asymptotic properties and averages of T(x,v).

An adaptive multiscale method for density-driven instabilities

Accurate modeling of flow instabilities requires computational tools able to deal with several interacting scales, from the scale at which fingers are triggered up to the scale at which their effects need to be described. The Multiscale Finite Volume (MsFV) method offers a framework to couple fine-and coarse-scale features by solving a set of localized problems which are used both to define a coarse-scale problem and to reconstruct the fine-scale details of the flow. The MsFV method can be seen as an upscaling-downscaling technique, which is computationally more efficient than standard discretization schemes and more accurate than traditional upscaling techniques. We show that, although the method has proven accurate in modeling density-driven flow under stable conditions, the accuracy of the MsFV method deteriorates in case of unstable flow and an iterative scheme is required to control the localization error. To avoid large computational overhead due to the iterative scheme, we suggest several adaptive strategies both for flow and transport. In particular, the concentration gradient is used to identify a front region where instabilities are triggered and an accurate (iteratively improved) solution is required. Outside the front region the problem is upscaled and both flow and transport are solved only at the coarse scale. This adaptive strategy leads to very accurate solutions at roughly the same computational cost as the non-iterative MsFV method. In many circumstances, however, an accurate description of flow instabilities requires a refinement of the computational grid rather than a coarsening. For these problems, we propose a modified iterative MsFV, which can be used as downscaling method (DMsFV). Compared to other grid refinement techniques the DMsFV clearly separates the computational domain into refined and non-refined regions, which can be treated separately and matched later. This gives great flexibility to employ different physical descriptions in different regions, where different equations could be solved, offering an excellent framework to construct hybrid methods.

Discriminant ECOC: a heuristic method for application dependent design of error correcting output codes

We present a heuristic method for learning error correcting output codes matrices based on a hierarchical partition of the class space that maximizes a discriminative criterion. To achieve this goal, the optimal codeword separation is sacrificed in favor of a maximum class discrimination in the partitions. The creation of the hierarchical partition set is performed using a binary tree. As a result, a compact matrix with high discrimination power is obtained. Our method is validated using the UCI database and applied to a real problem, the classification of traffic sign images.

A multilevel multiscale finite volume method

The Multiscale Finite Volume (MsFV) method has been developed to efficiently solve reservoir-scale problems while conserving fine-scale details. The method employs two grid levels: a fine grid and a coarse grid. The latter is used to calculate a coarse solution to the original problem, which is interpolated to the fine mesh. The coarse system is constructed from the fine-scale problem using restriction and prolongation operators that are obtained by introducing appropriate localization assumptions. Through a successive reconstruction step, the MsFV method is able to provide an approximate, but fully conservative fine-scale velocity field. For very large problems (e.g. one billion cell model), a two-level algorithm can remain computational expensive. Depending on the upscaling factor, the computational expense comes either from the costs associated with the solution of the coarse problem or from the construction of the local interpolators (basis functions). To ensure numerical efficiency in the former case, the MsFV concept can be reapplied to the coarse problem, leading to a new, coarser level of discretization. One challenge in the use of a multilevel MsFV technique is to find an efficient reconstruction step to obtain a conservative fine-scale velocity field. In this work, we introduce a three-level Multiscale Finite Volume method (MlMsFV) and give a detailed description of the reconstruction step. Complexity analyses of the original MsFV method and the new MlMsFV method are discussed, and their performances in terms of accuracy and efficiency are compared.