An integrated method of advanced anisotropic hr-adaptive mesh and discretization numerical techniques has been, for first time, applied to modelling of multiscale advection–diffusion problems, which is based on a discontinuous Galerkin/control volume discretization on unstructured meshes. Over existing air quality models typically based on static-structured grids using a locally nesting technique, the advantage of the anisotropic hr-adaptive model has the ability to adapt the mesh according to the evolving pollutant distribution and flow features. That is, the mesh resolution can be adjusted dynamically to simulate the pollutant transport process accurately and effectively. To illustrate the capability of the anisotropic adaptive unstructured mesh model, three benchmark numerical experiments have been set up for two-dimensional (2-D) advection phenomena. Comparisons have been made between the results obtained using uniform resolution meshes and anisotropic adaptive resolution meshes. Performance achieved in 3-D simulation of power plant plumes indicates that this new adaptive multiscale model has the potential to provide accurate air quality modelling solutions effectively.

It is well-known that the interaction of multiscale physical processes in
atmospheric phenomena poses a formidable challenge for numerical modelling

So far, the accurate numerical modelling of advection (or transport) remains
a central problem for many applications such as air pollution, atmospheric
chemistry, meteorology and other physical sciences. There have been many
studies on the numerical advection schemes (e.g. PPM, Bott and Walcek
etc.) that have been used in many air quality models (e.g. CMAQ,
CMAx, NAQPMS etc.)

In contrast to locally nested mesh techniques, adaptive mesh techniques can not
only resolve multiscale processes in a consistent way, but also
enable to follow and capture the features of flows as time evolves. Dynamic
mesh adaptation can be achieved, either by re-locating mesh nodes or by
locally increasing (and decreasing) the number of nodes in time and space.
The former, known as mesh movement (i.e. r-adaptive technique), can be used to
improve the accuracy of solutions by optimally re-locating mesh nodes to
resolve the small-scale features of interest

This article applies a new anisotropic hr-adaptive mesh technique into air quality transport (advection) modelling. This adaptive unstructured mesh technique provides the dynamic spatial and temporal resolution to capture moving features, e.g. moving fronts or power plant plumes. Using the hr-adaptive technique, existing elements can be split (h-adaptive) or element vertices can be moved (r-adaptive), to periodically modify the mesh geometry. Hence, the purpose of this article is to demonstrate, through example problems, the capability of anisotropic mesh adaptivity for modelling of multiscale transport phenomena.

The remaining structure of this article is as follows:
Sect.

As a model problem, we consider the generic transport equation for a scalar quantity,

Integrating Eq. (

Basic configuration for FEM_Adapt_L and FEM_Fix_L schemes (where FEM represents CV or DG; the maximum mesh size is set to be 0.2).

Initial distribution/exact solution at

As a locally conservative, stable and high-order accurate method, the
discontinuous Galerkin methods can easily construct discontinuous
approximations on unstructured meshes to capture highly complex solutions and
are well-suited for hr-adaptivity and parallelization

Integrating Eq. (

Case one – solid body revolution: the errors in the

Case one – solid body revolution: the evolution of number of nodes for

In the upwind flux formulation, the value of

In local Lax–Friedrichs flux formulation, the tracer advection is given by

To ensure nonlinear stability and effectively suppress spurious oscillations,
the slope limiting techniques are used here

Case one – solid body revolution: the results from the fixed and adaptive
mesh schemes using almost the same node number

The control volume discretization uses a dual mesh constructed around the
nodes of the parent finite element mesh. Once the dual control volume mesh
has been defined, it is possible to discretize the transport
Eq. (

For diffusion term

Case two – swirling flow: the errors in the

Case two – swirling flow: the evolution of number of nodes for

Case two – swirling flow: the results from the fixed and adaptive mesh
schemes using almost the same node number

Case two – swirling flow: the results from the fixed and adaptive mesh
schemes using almost the same node number

Case two – swirling flow: the evolution of the adaptive mesh coloured with tracer value

Case three – swirling deformation: initial distribution and velocity field.

Case three – swirling deformation: comparison of the analytical solution
with the results from different schemes using almost the same number of nodes

Case three – swirling deformation: comparison of the analytical solution
with the results from different schemes using almost the same number of nodes

The semi-discrete matrix form of Eq. (

For discontinuous Galerkin discretization, the explicit Euler scheme (

For control volume discretization, an explicit scheme is simple but strictly
limited by the CFL number, which can be restrictive on adaptive meshes as the
minimum mesh size can be very small. Here, we adopt a new time stepping

The optimization-based adaptivity technique, developed by the Applied
Modelling and Computation Group (AMCG) at Imperial College London

To represent small-scale dynamics, a relative error metric formulation is suggested:

Case three – swirling deformation: the evolution of the adaptive mesh coloured with tracer value

Case three – swirling deformation: the evolution of the adaptive mesh
coloured with tracer value

Case three – swirling deformation: the evolution of

Case three – swirling deformation: the distribution of CFLNumber for CV_Adapt_128 scheme at

Case four – power plant plumes:

Case four – power plant plumes: simulated

Case four – power plant plumes: the evolution of the adaptive mesh coloured with

Case four – power plant plumes: the evolution of 3-D plumes visualization, surface

To guide refinement/coarsening of the mesh, the maximum and minimum mesh
sizes are set to allow one to impose different limits in different directions
(for details, see

Another key issue of mesh adaptivity is to interpolate any necessary
data from the previous mesh to the adapted one. The consistent interpolation is often adopted in mesh adaptivity.
However, the consistent interpolation can introduce a suboptimal interpolation error, unsuitability for discontinuous fields, and lack of conservation.
An alternative conservative interpolation approach, the Galerkin projection is proposed for discontinuous fields.
A supermeshing algorithm

The new multiscale air quality transport model has been developed with a 3-D unstructured and adaptive mesh model (Fluidity; developed by the AMCG at Imperial College London). Fluidity, an open-source LGPL model, numerically solves the 2-D/3-D Navier–Stokes equation (being non-hydrostatic, to model dense water formation and flows over steep topography) and field equations with a range of control volume and finite element discretization methods. It includes a number of novel, advanced methods based upon adapting and moving anisotropic unstructured meshes, advanced finite element and control volume discretization, and a range of numerical stabilization and large-eddy simulation (LES) turbulence models. Among existing unstructured mesh models, Fluidity is the only model that can simultaneously resolve both small- and large-scale fluid flows while smoothly varying resolution and conforming to complex topography. The model employs 3-D anisotropic mesh adaptivity to resolve and reveal fine-scale features as they develop while reducing resolution elsewhere. A number of interpolation methods (e.g. non-conservative pointwise and conservative methods) are available for mesh-to-mesh interpolations between adaptations.

Fluidity is parallelized using MPI and is capable of scaling to many
thousands of processors. It has a user-friendly GUI and a python interface
that can be used to calculate diagnostic fields, set prescribed fields or
set user-defined boundary conditions (for details see

To illustrate the efficiency and accuracy of anisotropic adaptive schemes, four benchmark
problems have been adopted, which are representative and challenging enough to predict
how the new adaptive multiscale model would behave in future real-life applications

In the following comparative study, we consider FEM_Fix and FEM_Adapt
schemes (FEM represents CV or DG) based on the control volume and
discontinuous Galerkin discretization. The CV_Fix_L and DG_Fix_L schemes
use fixed uniform triangular meshes, while the CV_Adapt_L and DG_Adapt_L
schemes use adaptive meshes (where L represents the different mesh schemes,
as shown in Table

To assess the difference between the analytical solution

All computations were performed on a workstation using the Gfortran Compiler
for Linux. The simulation workstation has 8 processors and a 4

A standard test problem applied to the advection Eq. (

Figure

Figure

The capability of the adaptive mesh model has been further demonstrated in
modelling swirling flow phenomena. The set-up of the simulation in this case
is similar with case one; however the velocity field is provided by the
formula

The initial mass distribution will be deformed by the time-dependent velocity
field, which gradually slows down to zero and reverses its direction at

A comparison of results using fixed and adaptive meshes is illustrated in
Figs.

The numerical solutions in Figs.

Figure

A comparison of the anisotropic adaptive mesh schemes with the Walcek (or
Bott) scheme

It can be observed the initial

The sequence of triangulations presented in
Fig.

Figure

In this case, the anisotropic adaptive mesh model is applied to an
advection–diffusion problem (Eq.

We started with a numerical investigation of a simplified 2-D test. The mixing
layer height is 600

Figure

To further demonstrate the adaptive mesh model's ability in 3-D modelling, we
extended the above 2-D case to 3-D dispersion of plumes. According to the
terrain data of the modelling domain, the initial 2-D adaptive mesh (see
Fig.

In this paper, a new anisotropic adaptive mesh technique has been introduced and applied to modelling of multiscale transport phenomena, which is a central component in air quality modelling systems. The first two benchmark test cases using the fixed mesh and adapted mesh schemes have been set up to illustrate the efficiency and accuracy of anisotropic adaptive mesh technique, which is an important means to improve the competitiveness of unstructured mesh air quality models. The third case presents the irreplaceable advantage of this new adaptive mesh method to reveal detailed small-scale plume structure (large gradients) that cannot be resolved with static grids, using comparable computational resources. Dispersion of power plant plumes, as a real model problem, has been simulated in the last case to illustrate that the adaptive algorithm is able to capture the detailed small-scale plume structures near each point source as well as the regional high concentrations at large downwind distances.

It is demonstrated that the dynamic anisotropic adaptive mesh technique can
be used to automatically adapt the mesh resolution to follow the evolving
pollutant and transient flow features in time and space, thus reducing the
CPU time and memory requirement significantly. In combination with the
time stepping

The third test case serves as a proof-of-concept to further illustrate the capability of anisotropic mesh adaptivity techniques. In this case, the swirling deformation flow exhibits very high aspect ratios (1000, for example), which means that the pollutant distribution can possess very strong anisotropies as time evolves. Hence, the anisotropic mesh adaptation provides a very useful and effective way to simulate and represent this special atmospheric phenomena.

In summary, the results obtained in this work show the capability and potential of adaptive mesh methods to simulate multiscale air pollutant transport problems (spanning a range of scales) with higher numerical accuracy. The mesh adaptation can be used to improve the mesh resolution when and where it is needed without performing successive global refinement, which is prohibitively expensive, and therefore, not feasible for realistic applications. Future work will consider chemical reactions to further demonstrate the capability of dynamic adaptive mesh techniques.

Fluidity code developed by the Applied Modelling and Computation Group (AMCG)
at Imperial College London is available under the GNU General Public License
(

This work was carried out under funding from the Chinese Academy of Sciences (CAS) Strategic Priority Research Program (grant no. XDB05030200), the UK's Natural Environment Research Council (projects NER/A/S/2003/00595,NE/C52101X/1 and NE/C51829X/1), the Engineering and Physical Sciences Research Council (GR/R60898, EP/I00405X/1 and EP/J002011/1) and the Imperial College High Performance Computing Service. The authors would like to thanks to J. Percival for many helpful discussions and support from others in AMCG. The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007–2013) under grant agreement no. 603663 for the research project PEARL (Preparing for Extreme And Rare events in coastaL regions). Pain is grateful for the support of the EPSRC MEMPHIS multi-phase flow programme grant. The authors acknowledge the reviewers and Editor for their in depth perspicacious comments that contributed to improving the presentation of this paper. Edited by: P. Jöckel