^{*}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The predictions from air quality models are subject to many sources of uncertainty; among them, grid resolution has been viewed as one that is limited by the availability of computational resources. A large grid size can lead to unacceptable errors for many pollutants formed via nonlinear chemical reactions. Further, insufficient grid resolution limits the ability to perform accurate exposure assessments. To address this issue in parallel to increasing computational power, modeling techniques that apply finer grids to areas of interest and coarser grids elsewhere have been developed. Techniques using multiple grid sizes are called nested grid or multiscale modeling techniques. These approaches are limited by uncertainty in the placement of finer grids since pertinent locations may not be known

Air quality modeling is the computational science of developing mathematical models that describe the behavior of pollutants in the atmosphere. Pollutants are emitted from various sources, including natural sources. However, it is the fate of emissions resulting from human activity that is typically the focus of air quality modeling. Once introduced into the atmosphere, pollutants are subject to dynamics and chemistry that transport and transform them continuously. Among the transport processes are advection by wind, turbulent diffusion, cloud convection, and deposition. Transformation processes include gas and aqueous phase chemistry, phase changes, and particle nucleation and growth. Heavy loads of pollutants, such as volcanic ash or smoke from wildland fires, can change the dynamics of the atmosphere, coupling the dynamics to the chemistry. Additionally, transformation processes can couple pollutants to each other and create secondary pollutants that are not emitted but formed in the atmosphere after emission of precursor species. Air quality models aim to represent all these processes in the most comprehensive manner.

Air quality models are either of Eulerian grid type or Lagrangian particle models with underlying grids. Cell size is the measure of the scales that can be explicitly resolved by a grid model. Smaller-scale phenomenon can only be represented through subgrid scale parameterizations. The dynamic and chemical processes mentioned above involve a wide range of scales. Complex interactions between processes occurring at different scales make it necessary to resolve the finest relevant scales. For example, it is well known that emissions from urban and industrial centers are responsible for regional or global air quality problems. On the other hand, if emission plumes are injected into relatively coarse grid cells in a regional or global scale model, they are immediately diluted with the contents of the cell and the details of their chemical interaction with the surrounding atmosphere are lost. In the early days of regional modeling 100 km resolution was typical; today, models are pushing the 1 km barrier and the quest for higher resolution continues. With higher grid resolution come improved resolution of complex terrain topography, land use, land cover, cloud cover, and other data used in subgrid scale parameterizations, enabling an improvement to the overall resolution of the models.

Historically, the motivation for air pollution modeling has been air quality management. Development of control strategies has been the initial driving force behind modeling research and advancement. With the confidence gained in describing historic pollution events, models were applied in air quality forecasting. Today, models are used for much broader purposes that connect air quality to other fields. The feedback of pollution on meteorology is receiving more attention with a focus on problems such as smoke from wildfires and volcanic ash, leading to the coupling of air quality models with numerical weather prediction models that were historically used to provide meteorological inputs only. Along with the study of climate and global scale air quality problems, air quality models are beginning to merge with global circulation and global chemistry models. These trends considerably expand the domains of the models from their original dimensions.

Modeling large geographic regions with high resolution is a challenging computational problem. The models' demand for computational resources escalates rapidly with increasing resolution. For example, consider the changes to the operation count in a model that uses an explicit numerical scheme. An explicit scheme advances a system from its current state

For models that use implicit schemes, the computational resource demand grows more rapidly with increasing resolution. In an implicit scheme, both the current and future states of the system appear in an equation of the following form:

This equation can be solved directly or iteratively. Direct solutions involve a matrix inversion and are typically more expensive than iterative solutions. Depending on the selected solution method, the operation count of an implicit scheme is generally between ^{2}

Multiscale modeling techniques emerged as a solution to this gargantuan computational challenge. The goal is to develop models capable of applying the appropriate scale or sufficient resolution where and when it is needed. The goal of multiscale models is to encompass different scales (e.g., local, urban, regional, global) in a unified modeling system to better capture the interactions among the processes relevant at each scale. There are various multiscale modeling techniques; two methods that gained popularity in air quality models will be described here. The first features grids that can be nested multiple levels deep for better resolution of finer scale processes. The second involves grid adaptation in response to the needs of a particular simulation, either by refining a structured mesh or by locally enriching an unstructured mesh with added cells. The paper will focus on the adaptive grid method and continue with a review of its use in air quality modeling. It will end with remarks on the prospect of growing use of adaptive grids in atmospheric modeling.

Nested-grid modeling techniques have been and still are very popular in atmospheric modeling (e.g., [

There are two types of grid nesting: one-way and two-way. In one-way nesting, the CG solution provides boundary conditions to the FG and there is no feedback from the FG back to the CG. Because of this, the CG simulation can be run and the solution stored before the FG simulation begins. Boundary conditions are generated by interpolating the CG solution. In two-way nesting, the FG solution is used to update the CG solution when the two solutions are synchronized; therefore, all grids must be simulated simultaneously. Typically, the CG solution is marched first by one time step. This solution yields the boundary conditions needed by the FG. When the FG solution synchronizes with the CG solution, its resolution is restricted (e.g., by averaging) to update the CG solution. Some two-way nested models use a weighted average of the CG and FG solutions, while others completely overwrite the CG solution in the region where the two solutions overlap, assuming that the finer resolution leads to a more accurate solution. A good restriction operator should transfer the maximum amount of information from the FG to the CG, but should not generate any noise due to aliasing of scales that are not resolvable by the CG. Averaging is the most common restriction approach where the weighted sum of FG values takes the place of the CG value [

In one-way nesting, one of the primary concerns is mass conservation at the CG-to-FG interface where boundary conditions are input to the FG using the CG solution. Mass conservation can be ensured by setting the flux at the FG inflow boundaries equal to the flux passing through this interface as determined by the CG solution. In two-way nesting, the mass in each CG cell is usually updated with the average mass of all overlapping FG cells. Special care is required to guarantee mass conservation. Since nonlinear reactions may have transformed the chemical species during the FG solution, the mass of basic chemical elements (e.g., sulfur, nitrogen, carbon), and not single species, should be conserved. The FG solution is used to compute the flux of each element at the FG-to-CG interface. When conservation principles are applied to the FG-to-CG interface, as described above, the CG concentrations near the interface must be corrected. This can be done by renormalizing the concentration of each species assuming that the ratio of species mass to chemical element mass remains constant before and after the correction.

Another concern in grid nesting is deformation of waves traveling through CG-FG boundaries. Grids with different resolutions act as different media to these waves, subjecting them to reflection and distortion at their interfaces. Ideally, waves leaving the CG must enter the FG without any distortion, while waves leaving the FG must pass the interface without reflections. Distortions at the CG-to-FG interface can be inhibited by using advection equivalent interpolation to interpolate the CG solution onto the FG boundary [

As for waves leaving the FG, it is more difficult to inhibit reflections at the FG-to-CG interface. In one-way nesting, phase speed differences of waves traveling through CG and FG grow, leading to considerable noise at the interface. On the other hand, in two-way nesting the interaction of CG and FG solutions with each other keeps them in phase; therefore, the reflections and refractions are kept to a minimum. Typically, diffusive filtering must be used to avoid reflections. The most successful filters for this purpose are those that differentiate the frequencies of traveling waves and selectively reduce the amplitudes of short-wave signals associated with noise at the interface [

While static grid nesting is a practical multiscale modeling technique, it has limitations. The domains and resolutions of nests are selected either arbitrarily or based on conceptual models that may not be very accurate. The quality and accuracy of the solution obtained greatly depends on the initial selections. Once the grids are set, it may not be simple to change them since input data must be reprocessed for the new grid settings. This is the case, for example, in forecasting operations. A common problem associated with grid nesting methods is that they often lead to spurious oscillations at grid interfaces. This is particularly problematic when an interface coincides with an area of large physical gradients because the filters used to remove the oscillations can also reduce the amplitudes of resolvable waves. Large refinement ratios between CG and FG amplify the oscillations and necessitate more vigorous filtering [

The objective of an adaptive grid method is to increase solution accuracy by providing dynamic refinement at regions and instances where accuracy is most dependent on resolution. This can be achieved by restructuring the grid on which solution fields are estimated to better fit the needs of the system being numerically described. Adaptive gridding techniques can be classified as h-refinement or r-refinement depending on the type of grid restructuring employed.

H-refinement relies on increasing the total number of grid elements (e.g., nodes or cells) within a base grid for which the original structure remains fixed. The technique, also known as mesh enrichment or local refinement, modifies the grid at regions tagged for increased resolution. Frequently, the method is carried out by subdividing grid elements into smaller self-similar components. In

H-refinement can be realized by applying two distinct methodologies. One approach is to include added grid elements directly into the original base grid; all elements are solved on a single grid throughout the simulation. This method requires data storage procedures and solvers capable of handling unstructured grids. Alternatively, h-refinement can be achieved by considering the additional elements at increased resolution levels as distinct grids that are dynamically created or removed. In this manner, each refined grid can remain structured and be individually solved using algorithms developed exclusively for uniform or rectangular grids. Here, communication among different levels of refinement takes the form of boundary and initial conditions setting for the different grids.

R-refinement techniques, commonly referred to as mesh moving or global refinement, relocate mesh nodes to regions warranting increased resolution and subsequently increase grid element concentration at the areas with the greatest inaccuracies. However, the total number of grid points is maintained constant. Unlike h-refinement, r-refinement around a region is necessarily accompanied by coarsening at another.

Both adaptive gridding techniques have advantages and drawbacks associated with them. An advantageous distinguishing feature of r-refinement is smoother transitions in grid resolution. In contrast, h-refinement ordinarily operates on grids with abrupt discontinuities among the different refinement levels allowed. Acute disruptions in resolution are undesirable and may lead to interface problems. The constant number of grid elements maintained throughout a simulation using r-refinement could prove beneficial; a fixed number of elements simplifies solution algorithms and is helpful in parallel computing implementations. It could appear that the achievable improvement to solution accuracy with r-refinement is limited by the total number of grid elements or nodes, and that such a restraint is not inherent to h-refinement. However, it is important to note that the available computational resources, and not the number nodes, are the true limitation to accuracy. At the limit of computational capacity, both h- and r-refinement can only increase local refinement at the expense of coarsening elsewhere. More accurately, r-refinement's disadvantage lies in its need to determine global resolution (

Grid structures applied in numerical modeling may be categorized as structured or unstructured. The distinguishing feature between them refers to the data structure associated with each, rather than visual traits. Data on a structured grid can be arranged into a rectangular matrix; cells and nodes can be identified through integer indices (e.g., i, j, k). The requirement brings forth regular grid patterns and most often quadrilateral or hexahedral elements. Additionally, the data arrangement in structured grids reduces memory requirements compared to unstructured ones.

In contrast to structured grids, data from unstructured grids cannot be arranged by applying simple integer indexing and full connectivity must be defined and stored for each node. Unstructured grid cells are defined as a group of nodes that encompass an element of any geometry. These grids can be built using triangles, quadrilaterals, tetrahedra, hexahedra, or any polygon or polyhedron, including combinations of elements with different geometries. The irregularity of unstructured grids allows greater versatility in their assembly compared to structured grids. This flexibility in domain discretization has made unstructured meshes popular in simulations dealing with complex geometries and a common choice to model fluid mechanics [

Typically, finite-difference methods are used with structured grids and finite-element (or finite-volume) methods are used with unstructured grids. Both methods are well developed for the solution of hyperbolic partial differential equations of the type encountered in atmospheric models. Higher-order spatial approximations are available for both types of methods; therefore, the desired level of accuracy can be achieved with either. In general, higher-order methods in atmospheric models must be used together with flux-correction or filtering algorithms to avoid the generation of spurious oscillations that may lead to undesirable effects such as non-monotonic and unphysical (e.g., negative concentrations in air quality models) solutions or even instabilities. Using higher-order difference approximations or higher-order elements is known as p-refinement, an alternative to the adaptive grid h- or r-refinement techniques described above for increasing solution accuracy. The decision of using an explicit or an implicit scheme must made for time integration; it does not depend on the spatial grid and whether it is structured or unstructured. Hence, either option is available both with finite-difference and finite-element methods.

Mesh enrichment (

Mesh movement (

The objective of increasing solution accuracy through adaptive gridding can only be met if adaptation is driven by an efficient indicator of the solution error in a spatial field. The concept of error equidistribution has been used to describe the adaptive grid process; grids are reconfigured to result in an equal amount of error for all grid elements [

Typically, adaptive grids use an error indicator in place of solution error to drive refinement. The error indicator may be a rudimentary calculation related to the error. For instance, estimates of the truncation or interpolation error may be applied. It is also common to rely on physical features that are known to efficiently signal locations where the error is most sensitive to grid resolution. In air quality modeling, for example, concentration gradients or curvatures can be used to trigger refinement and effectively increase solution accuracy. The selection of an error indicator to drive refinement is more critical to r-refinement than h-refinement. For mesh moving refinement, adaptation acts as an optimization processes by which error is minimized with a fixed amount of available resolution. Since refinement at one region necessitates coarsening at another, effective adaptation criteria truly representative of solution error are crucial. In mesh enrichment, base-level solution accuracy is generally guaranteed. Refinement augments resolution at selected locations and increases total grid resolution. An optimal grid configuration is not necessarily achieved nor pursued. Instead, mesh enrichment may simply allocate additional resolution in an attempt to sufficiently increase solution accuracy.

Research efforts investigating adaptive gridding in meteorological modeling precede those exploring the technique as an option for air quality simulation. The motivations and methodologies for adaptive grid refinement in both atmospheric realms are very much alike. The earliest attempt to simulate atmospheric flows using an adaptive grid was reported by Jones

Subsequent applications of adaptive gridding in atmospheric modeling were described by Skamarock

An alternative strategy to adaptive gridding in meteorological modeling was reported by Dietachmayer and Droegemeier [

Application of adaptive grids in air quality modeling has been explored for over 10 years. Three distinct efforts to implement adaptive gridding techniques into Eulerian air pollution models can be identified. These undertakings can each be traced back to Tomlin

An adaptive grid method applicable to air pollution modeling using unstructured grids is described by Tomlin _{x} plume within background atmospheres with varying degrees of pollution. The NO concentration field was selected to estimate spatial error and drive refinement limited to a maximum of two levels. In the simulation, refinement winds up concentrated around pronounced spatial gradients. The adaptive grid captures features in the ozone and NO_{2} concentration structures unseen in a simulation without refinement, near the point source and further downwind. Additionally, a significant difference in total NO_{2} mass is observed between adapted and unadapted simulations. The authors attribute the discrepancy to nonlinear chemistry, which renders ozone and NO_{x} concentrations grid resolution dependent.

The adaptive gridding method of Tomlin

The use of mesh movement (r-refinement) as an adaptive grid strategy in air quality modeling was first reported by Srivastava

Several tests were performed using this adaptive grid algorithm to evaluate its performance including simulations of rotating conical concentration distributions [

To demonstrate the computational expense and accuracy gained, CPU times with static and adaptive grids were compared for the reactive pollutants puff test [

Implementation of r-refinement into a comprehensive regional chemical-transport model was first reported by Odman

AG-CMAQ performance was evaluated by applying the model to simulate the air pollution impacts from an actual biomass burning event affecting air quality over a large urban area.

The adaptive grid method described by Constantinescu

The adaptive grid methodology of Constantinescu _{2}, and HCHO) produced better results. An increase in accuracy was observed when applying an adaptive grid relative to coarse resolution results. Two interesting observations are also discussed: (1) the increase in solution accuracy is highly dependent on the user defined refinement tolerances, and (2) the total number of grid cells decreases with time until becoming fairly stable. Special attention was given to implementation of the adaptive grid method into parallel computing systems, facilitated by the domain's division into self-similar blocks. A computational cost is acknowledged for adaptive gridding. However, the computational requirements were significantly lower than those necessary for simulations performed at the finest grid resolution allowed in the adaptive grid application. For the application described, wall-clock times of STEM runs with static and adaptive grids were compared. An adaptive grid run of comparable accuracy required only a quarter of the time spent in a static fine grid run.

The methodologies described in Section 4.1 follow different approaches to adaptive grid integration into air pollution modeling. However, all attempts have encountered common challenges. These common obstacles and the strategies applied to overcome them merit attention and are discussed in this subsection.

We have previously mentioned that an essential component of any grid refinement technique is an error estimation that serves as a refinement criterion. All applications of adaptive gridding to air quality modeling previously reported have relied on atmospheric concentration spatial fields. An error estimate calculated from the difference in first-order and second-order approximations of local concentrations was described in Tomlin

A need to constrain refinement intensity has also been identified in all previous investigations. Uncontrolled adaptation may lead to excessive refinement. Control parameters are implemented into the refinement criteria estimates previously described to normalize and smooth spatial error calculations. Mesh enrichment methods additionally require that user-defined upper and lower tolerances be set to trigger adaptation, directly controlling the degree to which grids are reconfigured. Furthermore, mesh enrichment techniques constrain adaptation to a maximum number of refinement levels. In the iterative mesh moving method previously described, a maximum number of relocation operations and minimum node movement relative to initial grid spacing are defined to stop adaptation.

Some concern about abrupt transitions in grid resolution has been expressed in past adaptive gridding exercises. The existence of neighboring grid elements refined to very different degrees may lead to errors and the loss of features resolved under the fine resolution if advection transports the features into coarse resolution regions prior to adaptation. A safety layer encompassing high resolution regions is proposed as a solution by Tomlin et

As adaptability is integrated into more complex atmospheric systems, the grid refinement process may become increasingly taxing, especially when applied to comprehensive chemical-transport models. Therefore, it may be necessary or desirable to apply grid adaptation at intervals greater than solution frequency. Ghorai

Preadaptation has also been identified as a process that might be important in effective grid refinement. The strategy is of particular relevance to point sources which might immediately dilute into coarse cells and lose plume features prior to any adaptation. The plume simulation carried out by Tomlin

Interpolation of coarse or uniformly distributed data onto adapted grids is an important component of all adaptive air quality modeling methods reported. The operation is essential during three processes of an adaptive grid air quality model: emissions, meteorology, and solution redistribution. The procedure for processing emissions must be applied after each grid adaptation operation. Different interpolation schemes have been used to interpolate area sources onto adapted non-uniform grids. Additionally, point sources must be relocated to appropriate cells after grid reconfiguration. Emissions processing can become taxing, especially for mesh moving methods that use cells with irregular geometries [

One characteristic of adaptive grid air quality models that can severely hinder performance is excessively small process time steps applied uniformly throughout the domain. It is common practice in air pollution models to apply the domain-minimum time step, typically fixed by the characteristic time for advection, at every point [

Finally, an especially significant concern about adaptive grid air quality modeling is the applicability of subgrid parameterizations to grids with non-uniform resolution and highly refined regions. The issue has not yet been addressed in reported research efforts. Parameterizations for physical processes (e.g., turbulence, cloud processes) must be valid for the refined grid. The dependence of these parameterizations on grid resolution may bring into question their validity across the entire domain if considerable differences in grid resolution are allowed. Furthermore, if resolution is sufficiently increased, some parameterizations might not be required and their inclusion could adversely affect solution accuracy. The concern is also relevant to adaptive grid modeling in other atmospheric realms. For instance, one adaptive grid weather prediction model assigns an adjustment factor, determined by cell area, to each cell and uses this factor to regulate the degree to which the convective parameterization is applied at individual cells [

All adaptive grid air quality modeling applications previously described find that dynamic mesh refinement significantly increases the accuracy of results. Observed differences in simulated concentration fields using adaptive and static grids demonstrate a legitimate need for increased resolution in air quality modeling. Adaptive gridding has consistently proven to be an adequate and highly attractive option to meet increased resolution requirement. Additionally, the different adaptive grid methods reported outperform grid nesting and increase computational efficiency compared to high resolution simulations capable of providing the same level of accuracy. Adaptive grids significantly decrease numerical diffusion while revealing detailed pollutant concentration structures and features that cannot be resolved with static grids, uniform or nested, using comparable computational resources. Decreased errors associated with pollutant levels modeled using adaptive grids can be perceived near emissions sources as well as at considerable downwind distances.

In the adaptive grid applications described, the largest spatial error, and therefore the highest degree of refinement needed to reduce this error, is typically observed at regions with pronounced concentration gradients. Large gradients can be usually found near pollutant sources and along plume edges further downwind. The nonlinear nature of atmospheric chemical transformations further strengthens the case for adaptive gridding. Studies have repeatedly determined that pollutant concentrations subjected to nonlinear chemistry (e.g., ozone, nitrogen oxides) are mesh dependent. Under these circumstances, integrated pollutant concentrations can change significantly when applying grid refinement techniques that can better resolve the chemical processes. As chemical transport models incorporate detailed secondary organic aerosol processes, nonlinear as well, the advantages of adaptive grids may become even more apparent.

Vertical grid adaptation has only been incorporated into a single adaptive grid model. Tomlin

Finally, all developers agree that the power of adaptive gridding will become more apparent as resolution of model inputs is increased to better match refined grids. This includes emissions, meteorological fields, and boundary conditions. Accuracy has been seen to improve when applying high resolution emissions [

As adaptive grid modeling is being explored in air quality modeling, concurrent efforts to integrate the technique into weather prediction and global circulation models have been reported. Attempts to apply adaptive grids in comprehensive meteorological models have been limited with few recent developments. A noteworthy application is the Operational Multiscale Environmental model with Grid Adaptivity (OMEGA) [

Much of the progress recently reported pertaining to adaptive grid modeling in atmospheric science involves incorporation of the technique into general circulation models. The motivation to integrate adaptive gridding into global models is largely analogous to that driving incorporation into air quality and weather models: models simulate atmospheric processes covering a wide range of scales and available computational resources are unable to explicitly resolve all processes involved. Therefore, it is not surprising to find that adaptive gridding techniques explored in climate models are frequently equivalent to those applied to air pollution modeling. However, the spatial and temporal scales of interest are very different to those in air quality simulations. In addition, the spherical nature of the global modeling domain and differences in simulated processes result in very different grid refinement requirements.

Nonetheless, future applications of adaptive gridding in air quality modeling may benefit from previous and ongoing efforts to integrate the technique into global models. Hubbard and Nikiforakis [

Adaptive grid methods have not been fully explored in atmospheric modeling. Early attempts in weather prediction did not flourish and this field remained dominated by static nested grid methods. There have been a few recent attempts to integrate adaptive grids into air quality modeling but once again these attempts did not enjoy wide acceptance. Future undertakings should consider several key factors if greater interest in the technique is sought. The prevalence of community-driven models in atmospheric sciences today makes compatibility with existing model frameworks an indispensible requirement for increased use of adaptive gridding methodologies. Additional applications that go beyond typical power plant plume and ozone chemistry simulations are necessary to further demonstrate the worth of dynamic grid refinement. Finally, adaptive grid air quality models will have to be accompanied by equivalent increases to the resolution of emissions and meteorological inputs to truly reach their full potential.

What looks promising for adaptive grids is the presence of a vibrant support by the global climate community who acknowledge reaching a plateau after years of improvement in accuracy, mainly driven by progress in high performance computing. Adaptive grid methods are viewed as a means to reignite model advancement and a long term solution for dealing with the multiscale nature of the climate system [

The potential returns of adaptive grids are enormous while the risks are relatively small. The methods have matured through advancements in computational fluid dynamics and wide use, for example, in aerospace engineering applications. The difficulties will be faced in developing parameterizations for processes that cannot be resolved by adaptive gridding. In our view, the problem will eventually die out as adaptive grids continue to provide increased grid resolution. Nonetheless, the issue must be dealt with in the interim. The wide range of scales provided by adaptive grids will necessitate resolution dependent parameterizations and this may appear as a daunting task. However, what is faced today by increasing grid resolution uniformly or through grid nesting is the exact same problem: parameterizations that made sense for coarse grid resolution must be rethought as grid resolution increases. We must simply develop a culture of designing resolution dependent parameterizations upfront instead of revising the parameterizations every time model resolutions change. For these reasons, we believe that the time is right to invest in the development of adaptive grid models.

Uniform grid (left) and examples of h-refinement (center) and r-refinement (right).

Refined unstructured triangular grid covering the United Kingdom. Reprinted with permission from [

Adapted grid during a biomass burning plume simulation with AG-CMAQ.

Block-structured dynamically refined grid over East Asia. Reprinted with permission from [

OMEGA grid dynamically adapted to a pollutant plume. Reprinted with permission from [

Block-structured adaptive grid on a global domain. Reprinted with permission from [

Adaptive Grid Methods in Air Quality Modeling.

Tomlin |
Unstructured triangular or tetrahedral | Single grid h-refinement | 1st and 2nd order solution difference; concentration gradient | Eulerian grid model | Yes |

Srivastava |
Structured quadrilateral | R-refinement | Interpolation error; concentration curvature | CMAQ | No |

Constantinescu |
Uniform quadrilateral with nested refinements | Multigrid h-refinement | Concentration curvature | STEM | No |

This research was supported in part by the U.S. Department of Defense, through the Strategic Environmental Research and Development Program (SERDP) under SERDP project number RC-1647 and by the U.S. Department of Agriculture Forest Service through the Joint Fire Science Program (JFSP) under JFSP project number 08-1-6-04.