Universidade Federal Fluminense
Technical Report 3/06
Setembro/2006
An adaptive triangular mesh refinement using
graph with autonomous nodes based on Finite
Volume method applied to evolutionary PDE’s
Sanderson L. Gonzaga de Oliveira and Mauricio Kischinhevski
Abstract: This work proposes an adaptive mesh refinement using any form of triangles based on FVM with cellcentered control volumes in order to numerically solve PDE’s. The mesh is represented by a graph, which nodes
are autonomous and parent nodes in the adaptive mesh refinement are not stored. This is less computationally
expensive and admits more flexibility to link the nodes among neighbors in different levels of refinement, when
comparing with a local refinement scheme that uses trees. A Sierpinski curve is used to link the graph nodes.
Triangular control volumes are refined by bisection, what permits straightforward update of the list that links the
graph nodes without mesh distortion. A Green-Gauss reconstruction is used to determine the gradients of diffusive
and viscous terms in the control volumes faces. This work permits using several schemes of linear interpolation to
determine the PDE dependent variable in the control volume face. Additionally, linear-system solvers based on the
minimization of functionals can be easily employed. Several geometric shapes in the initial domain are allowed.
The development of a computational program to numerically solve classical PDE’s will demonstrate its efficiency
and advantages in relation to other adaptive mesh refinement schemes.
Keywords: adaptive mesh refinement, space-filling curve, numerical method, numerical simulation of PDE’s,
Finite Volume method, Sierpinski curve, Green-Gauss reconstruction, numerical integration, linear interpolation.
1. Introduction
Many initial value and boundary problems for unsteady PDE’s use structures of small scale, which develop,
propagate, decline or disappear when the solution evolutes. Examples include boundary layers in viscous fluids,
shock wave discontinuities, and reaction zones in combustion processes, among others. The numerical solution of
those problems can be very difficult due to the location, time and nature of such structures are not ordinarily known
in the beginning of the process. Using a structured mesh (when all internal points/volumes/elements have the same
number of neighbors) to cover the differential problem is not appropriate in such situations because those meshes
do not evaluate the differential scales of the phenomena that is being studied. Usually, those meshes are
computationally expensive because they should have large number of points to furnish a quality solution.
Techniques that use an adaptive mesh refinement are less computationally expensive in such problems. Those
techniques are robust, reliable, and efficient.
Numerically solving PDE’s in a efficient time requires a mesh that designed points are more refined in regions
where the solution or its derivatives quickly change during time evolution. Some adaptive mesh refinement
strategy is required, especially in unsteady problems. The idea is to automatically build a coarse mesh where the
numerical solution furnishes an appropriate approximation among piecewise cells and to construct a fine mesh
where the numerical solution do not supplies an appropriate approximation among piecewise cells, such as
singularities, boundary layers, among others. Furthermore, it should have a smooth transition among neighbor
piecewise cells, which have different levels of refinement. Frequently, an adaptive mesh refinement enormously
reduces the required number of points to obtain an accurate numerical solution for problems that are almost
smooth, and can reach a numerical solution with the desired quality in non-smooth problems.
1
This work uses a graph data structure similar to one proposed by Burgarelli et al. (2006), namely Autonomous
Leave Graph (ALG). ALG numerically reconstructs the solution using a mesh composed by square shape cells and
the initial discrete domain is a unit square.
The novel proposed here technique uses a discrete adaptive mesh composed by control volumes with any triangular
shape. This technique operates all the required elements for adaptive mesh refinement in Finite Volume method
(FVM), which is applied to numerically solve PDE’s. An unstructured mesh composed of triangular control
volumes can be more appropriate near physical boundary regions and features of problem with complex
geometries.
After this brief introduction, section 2 compares some numerical methods to solve PDE’s. Section 3 deals with
discretization methods used in FVM with unstructured meshes. Section 4 cites some works that use FVM with
unstructured meshes. Section 5 presents the cell-centered Green-Gauss linear integration. Section 6 treats the
triangular adaptive refinement. Section 7 presents the graph in detail. Section 8 describes the Sierpinski spacefilling curve used to link the graph nodes. Section 9 shows the ordering mesh used. Finally, section 10 draws some
conclusions.
2. Numerical methods for solving PDE’s
It should be studied which method is more appropriate for each problem, with its features and singularities. Even
after a method is chosen, each method has its own characteristics, for example, which variational formulation
should be employed, the kind of mesh to be generated, which location the values of the PDE dependent variable
will be determined for each piecewise cell/volume/element. There is a certain quantity of methods for numerically
solving PDE’s. Some of them are for specific problems. However, the more employed and well-known are:
Finite Difference method (FDM): it approximates solutions to partial differential equations at nodal points, and
thus it is rather simple to write FDM codes for problems with simple geometries, regularly in the case of
rectangular domains, as long as the coefficients of the PDE are sufficiently smooth (Jeon and Sheen, 2005). In
most cases, the computational code for being simpler, it is less expensive and faster than other methods. In general,
when an unstructured mesh should be build, researchers choose other method. MDF do not supply a simple form of
stability and convergence analyses. It is used in problems that require large refinement of the whole mesh, such as
problems in Computational Fluid Dynamic (CFD);
Finite Element method (FEM): it is an approximation to the PDE solution. It is used in order to solve PDE’s in
more complicated domains with non-smooth coefficients. It has been well-developed in its theory and
implementation. FEM has an advanced base theory for developing the stability and convergence analyses. It
permits easy unstructured meshes use. Its computational implementation is more complicated than the other
methods;
Finite Volume method (FVM): it lies between the FDM and the FEM. It is a little bit more complicated than FDM
and can be friendlier than FEM. FVM is based in the mass conservation law and its application is simple and
convenient, mainly when boundary conditions are prescribed. Classically, it is applied to a PDE in its conservative
form, that is, with conservation laws. Such equations characteristically employ a divergent operator. Each equation
is located in each control volume and the Gauss Divergence theorem converts each equation to the integral of the
flux along the boundary of the control volume. It is required to consider carefully the mesh generation with a
proper generation of control volumes in mind in order to obtain an optimal order of convergence. The average flux
along each face of a control volume is then approximated by a special finite difference scheme by using data in the
neighboring cells. From the implementation point of view, it is easier to code FVM than FEM; for some simple
geometry it is as easy to implement the FVM as the FDM. Resolution of PDE’s in unstructured meshes is also
relatively easy. Basically, convergence analyses of the FVM can be obtained in the framework of the FEM. Indeed,
convergence can be analyzed directly in discrete norms or it can be done function space norms by constructing an
equivalent or asymptotically equivalent FEM. Some authors obtained stability and convergence analyses from
other forms, which do not use a FEM basis (Jeon and Sheen, 2005);
Boundary Element method (BEM): because it requires calculating only boundary values rather than values
throughout the space defined by a PDE, it is significantly more efficient in terms of computational resources for
problems where there is a small surface/volume ratio. Nevertheless, BEM is considerably less efficient than
2
volume-discretization methods for many problems because BEM formulations typically give rise to fully populated
matrices. This means that the storage requirements and computational time will tend to grow according to the
square of the problem size. By contrast, matrices of other methods are normally banded (components are only
locally connected) and storage requirements for the system matrices naturally grow quite linearly with the problem
size. Compression techniques can be used to ameliorate these problems, though at the cost of the added complexity
and with a success-rate that depends heavily on the nature of the problem being solved and the geometry involved.
BEM is applicable to problems for which Green’s functions can be calculated. These usually involve fields in
linear homogeneous media. This places considerable restrictions on the range and generality of problems to which
boundary elements can usefully be applied. Nonlinearities can be included in the formulation, although they will
generally introduce volume integrals which then require the volume to be discretized before solution can be
attempted, removing one of the most often cited advantages of BEM.
Table I depicts a brief summary comparing those four methods. It can be verified that each method presents
advantages and disadvantages and should be applied depending the problem features.
Table I: Advantages e disadvantages among four methods for numerically solving PDE’s
Method
FDM
FEM
FVM
BEM
Computational
High
Small
Relative
Relative
implementation ease
Computationally
efficiency comparing
to others for the
Yes
No
Yes
Yes
problems that is better
applicable
Applicability in
solving problems with
Depends to the
complex geometry and
Small
High
Small
problem
non-smooth
coefficients
Stability and
Depends the
Difficult
Easy
Difficult
convergence analyses
problem
Applicability with
Small
High
On average
Small
unstructured mesh
Usage in the last
Very high
High
On average
Small
decades
Applicability in PDE’s
in the conservation
On average
On average
High
On average
form
Requirements of
careful mesh
Small
High
High
High
generation
Usage in problems that
requires large
High
Small
High
Small
discretization, such
CFD’s problems
Besides those four cited methods, there are other specific methods for classes of problems or launched recently.
For example, Aftomis (1997) presented the Cartesian method in an aerodynamics problem. Koh and Tsai (2003)
presented this method for Euler solution. For a simple geometry, such as airfoils, it is not economical to employ a
Cartesian method since number of grid points required to ensure sufficient resolution is higher than body-fitted grid
method.
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Other example is the Cell Boundary Element method (CBEM) that is a FEM version with FVM concepts. It is also
based in the flux continuity. Firstly using a fundamental solution, the governing differential equation is converted
into the Laplace equation on each cell. Then the average flux on each face of a cell is evaluated by using the
Dirichlet-Neumann map. Using the continuity of fluxes along each interface, it is obtained the CBEM (Jeon and
Sheen, 2005). Currently, CBEM is being developed, for example, for evolutionary problems.
This present work uses an adaptive triangular mesh refinement developed for FVM, which has been applied to
approximate complex problems in several areas of science and engineering, especially for environmental draining,
scattering prediction of pollutants in the atmosphere, water and earth, in aerodynamic problems, draining study
over several surfaces, in petroleum reservoirs, in simulations to increase efficiency of petroleum recovering
available in a reservoir, among others.
Integral conservation law states that the total size rate of a substance with density  in a specific finite volume is
equal to the total flux of the substance through the finite volume boundary. In other words, the key concept used
during the FVM formulation is the principle of conservation of a specific physical quantity expressed by governing
equations over any finite volume.
The flux domain is discretized in a set of control volumes non-overlapped, which can be irregular in size and
shape. In other words, similarly to other methods that obtain approximated equations from balance equations, FVM
consists in the integration of the differential equation in the conservative form of the control volume.
3. FVM discretization schemes used with unstructured meshes
Commonly, there are two schemes for FVM mesh discretization: cell-centered and vertex-centered. Both
discretizations differ in the location of the control volumes in the mesh and the flux variable:
(i) In the cell-centered scheme, flux quantities are stored in the proper finite volume centroids and the mesh has
simple geometry;
(ii) In the vertex-centered scheme, flux variables are stored in the mesh vertices. Control volumes are composed of
sub-finite volumes, that is, parts of finite volumes, which the vertex belongs.
Convective and diffusive terms should be carefully considered. In FVM, it is required to evaluate the dependent
variable value and its gradient in the control volume face for applying Gauss Divergence theorem. Therefore,
methods pursue appropriate forms for those requirements, for example, to adequately establish a parallel segment
between evaluation points and the outward normal vector, which is used in the Gauss Divergence theorem.
In general, methods follow the formulation namely as reconstruction by Green-Gauss linear integration, which use
both theorems to evaluate the gradients in the control volume faces. It is the strict application of FVM basic
formulation for unstructured meshes. Convective and diffusive terms are evaluated in all control volume faces.
Green-Gauss reconstruction disadvantage is the angle evaluation between the segment between each control
volume centroids and faces. Some correction should be done and numerical oscillations can occur in turbulent
flows and discontinuities. Besides, special calculations, through physical and mathematical knowledge, in the
studied problem should be accomplished for higher order accuracy.
Habitually, works that reconstruct the solution through a cell-centered scheme are Green-Gauss reconstruction and
simplified least square gradient reconstruction, proposed by Barth in 1991 (Barth and Olberger, 2004), and can be
used also for vertex-centered schemes. Although simplified least square gradient reconstruction is computationally
more expensive than Green-Gauss reconstruction, for distorted meshes the simplified least square gradient
reconstruction gives better results than the Green-Gauss reconstruction.
In general, works use the following techniques for the vertex-centered schemes: Median dual for general simplex
meshes and Voronoi diagrams, which uses Delaunay triangulation. Both approaches can, coarsely, be said as
Green-Gauss reconstruction adapted to evaluation in the mesh vertices. Both create a dual mesh for determining
the required quantities. Voronoi diagrams take advantage that the faces that compose a control volume are
orthogonal to the segment between control volume centroids (centroids of simplices). Voronoi diagrams are
attractive for its simplicity in evaluation of dependent variable and gradients in the control volume faces, though, it
requires a strict dual mesh generation named Delaunay triangulation and hybrid meshes are not possible. There is
also the possibility of determination points out of the cells. It can reconstruct the entire mesh in local refinements.
4
Median dual can be used with general meshes. It follows a formulation such as FEM test functions. The Median
dual plays a special role because one can show that using a specific numerical quadrature, the FEM method with
linear elements and FVM on median duals are equivalent (Barth and Olberger, 2004). This enables stability and
convergence analyses directly from FEM. On the other hand, Median dual can be more expensive for the dual
mesh generation and the resolution of small linear systems required for each control volume, in the same form that
MEF does.
Cell-vertex schemes are first-order accurate on distorted grids. On Cartesian or on smooth grids the cell-vertex
schemes are second or higher order accurate depending on flux evaluation scheme. In the opposite, the
discretization error of cell-centered scheme depends strongly on the smoothness of the grid. In general, a cellcentered scheme on triangular/tetrahedral grid leads to about two/six times more control volumes. Hence, cellcentered schemes have more degrees (more unknowns) of freedom than Median dual scheme. In addition, control
volumes in cell-centered scheme are usually smaller than those in cell-vertex scheme. This suggests that cellcentered schemes are more accurate. However, residual of cell-centered-scheme results from much smaller number
of fluxes compared to Median dual scheme. Boundary condition implementation in cell-vertex schemes requires
additional logic in order to assure a consistent solution at boundary points, contrary to cell-centered schemes,
which it is simple. Thus, there is no clear evidence about which scheme is better (Yousuf, 2005). Nevertheless, the
data structure should be adapted for a vertex-centered scheme, whereas it is straightforward in a cell-centered
scheme.
Taking in account the above characteristics, this work determines the PDE dependent variable at the centroids
(barycentric is the triangle gravity center) of the proper triangular control volume. FVM discretization and
formulation using cell-centered control volumes are easily found in the open literature. Figure 1 depicts a triangular
mesh with centroids in the proper finite volume.
Figure 1: Example mesh with control volumes in the proper finite volume (Barth e Ohlberger, 2004).
Other reason for determining the PDE dependent variable in the triangular control volume centroids is to follow the
storage scheme of the data structure used in the present work. Those values should be stored uniquely in the graph
nodes and it is required using a list created by a modified Sierpinski curve for mesh ordering. If the values were
determined in the refined triangular control volumes vertices, the PDE dependent variable values would be doubled
in the graph nodes. In order to avoid this, it would be required a uniform refinement of the mesh to appear vertices
in the interior of the domain. Those vertices would be associated to three or more triangular control volume, what
do not agree with an adaptive mesh refinement. It would be possible the generation of ghost triangular control
volumes, what in practice, would create a dual mesh, hence, more computational effort. Creating local Delaunay
triangulation would require an adaptation of the whole mesh. Producing ghost triangular control volumes for
determination through Median dual scheme would require PDE dependent variable evaluation also in the vertices
that early existed, what would increase the linear system in n unknowns, where n is the number of the triangular
control volumes in the mesh.
Determining the PDE dependent variable in the triangular control volume centroids has a similar advantage of
Median dual, that is, using finite volume centroids for generating control volumes, it is assured that it will not have
exterior points in the mesh. In addition, it can be used general meshes. Opposite to control volumes created, for
example, from Voronoi diagrams, whose centroids are circumcenters and thus the centroids can be exterior to the
triangular control volumes because it requires Delaunay triangulation.
5
4. FVM with unstructured meshes
4.1. Vertex-centered control volume schemes
The majority of the works that use vertex-centered schemes uses either Voronoi diagrams for its simplicity in the
formulation or Median dual, which permits generic mesh usage. As examples, it can be cited the following works
that use Voronoi diagrams: Jameson et al. (1986) in aerodynamics problems; Miller and Wang (1994) in numerical
solution of two-dimensional unsteady incompressible flow problems; Turner and Ferguson (1995) applied a
hexagonal mesh in the numerical simulation of mass and heat transfer in porous media; Chou (1997) in the Stokes
problem; Perot (2000) in conservation properties of unstructured staggered mesh; Nallapati and Perot (2000) in
numerical simulation of free-surface flows using a moving unstructured mesh; Maliska and Vasconcellos (2000)
used the scheme to simulate flows with moving fronts using an unstructured mesh; Jayantha and Turner (2003a) to
simulate transport in orthotropic porous media; Jayantha and Turner (2003b) simulating transport in anisotropic
porous media; Ju (2004) shows an algorithm for mesh generation; among many others.
As examples of works that used Median dual scheme, it can be cited: Dwyer and Grcar (1998) employed the
method in the solution of the Navier-Stokes equation; Schulz and Kallinderis (1998) in unsteady flow structure
interaction for incompressible flows; Schneider and Maliska (2002) show an approach for numerical simulation of
two-dimensional unsteady convective-diffusive problems; Bastian and Lang (2004) in parallel procedures; Hainke
(2004) in a convective problem; among many others.
4.2. Cell-centered control volume schemes
In general, cell-centered control volume schemes use Green-Gauss reconstruction and simplified least-square
gradient scheme. Examples of Green-Gauss reconstruction are: Kim and Choi (2000) presented a second-order
time-accurate numerical solution for unsteady incompressible flow problems; Schneider and Maliska (2002)
showed an approach for numerical simulation of unsteady two-dimensional convective-diffusive problems; among
many other works. Examples for simplified least-square gradient reconstruction are: Kobayashi et al. (1999)
applied a numerical simulation in steady two-dimensional incompressible viscous recirculating flows; OllivierGooch and Altena (2002) applied the scheme in the numerical simulation of the advection-diffusion equation,
among other works.
4.3. Higher order accurate schemes
First-order accurate schemes are strongly diffusive. Higher order accurate schemes pursue to avoid numerical
oscillations. Some of those methods are: quadratic reconstruction, MUSCL (Monotone Upstream centered Scheme
for Conservation Laws), Frink’s reconstruction (Frink et al., 1991), ENO/WENO (Weighted Essentially NonOscillatory). Quadratic reconstruction was proposed by Barth in 1990 to vertex-centered control volumes (Barth e
Olberger, 2004). Delanaye and Essers, in 1997, developed a particular form of quadratic reconstruction for the cellcentered scheme which is computationally more efficient than the Barth method (Yousuf, 2005). MUSCL was
developed by van Leer in 1979. It is a second-order accurate extension of the scheme proposed by Godunov in
1959 (Barth e Ohlberger, 2004). ENO and WENO are also higher order accuracy that follows the Godunov
scheme. Frink’s reconstruction is an extension of the scheme proposed by Barth and Jaspersen in 1989 for threedimensional problems (Frink et al., 1991).
There are works that pursue second-order accuracy for specific classes of problems. Other works use both
approaches for vertex/cell-centered control volumes, for example, McManus et al. (2000) show a scalable strategy
for the parallelization of multiphysics unstructured mesh-iterative codes on distributed-memory systems, Barth and
Ohlberger (2004) show a complete FVM foundation and analysis of the earlier cited schemes. Besides, there are
several other works that use FVM with unstructured meshes either applied to specific problems or proposing also
specific solutions.
6
5. Green-Gauss reconstruction
Higher order accurate schemes pursue to avoid numerical oscillation; however, they are more computationally
expensive, such as Frick’s reconstruction and ENO/WENO schemes, as well as in the generation of a dual mesh of
the quadratic reconstruction. MUSCL obligates to include some non-linearity, namely limiters. The design and
implementation of limiters and especially multidimensional limiters is an active research field. Simplified leastsquare gradient reconstruction requires solving small linear systems to each cell average cell.
In general, those techniques are more expensive than Green-Gauss reconstruction. Green-Gauss reconstruction is
not appropriate to distorted meshes, however, this present work warrants the mesh quality because the refinement
adopted.
Pursuing a simple implementation and low computational cost, this work uses Green-Gauss reconstruction. It
pursues to determinate gradients (diffusive and viscous terms) in the control volume faces as easy as a vertexcentered dual scheme, however, without generating a dual mesh because the quality mesh is warranted.
It is presented an example of formulation applied to a two-dimensional unsteady convective-diffusive problem,
where velocity field is known and Cartesian coordinate system (x,y) is adopted:
 

 u       2  S 
(1)
t

where  is the diffusive coefficient, S represents source term, ρ is the specific fluid mass, u is the velocity vector
and t is the time. Figure 2 depicts two control volumes, where control volume centroids are represented by P and 1
points. The face between two neighbor control volumes is limited
by a and b points. Integration point is

represented by ip, and the normal outward vector is represented by n .
Figure 2: Two neighbor control volumes, whose centroids are P and 1, and ip is the integration point.
Points a and b of Figure 2 determine the effective area where element P1 changes flow. The segment between a

and b points is also the addition of area vectors of aip and ipb segments, represented by vector n over ip

integration point. Vector n is not, necessarily, parallel to P1 . Integration point, where flows are evaluated, is
located in the average point of P1 segment. Mass quantities of each sub-control volume are the half of Pa1b,
namely a diamond cell.
Governing equation is rewritten in divergent form and it follows the FMV basic formulation for unstructured
meshes. After Gauss Divergence theorem application, source term lineariazed, numerical integration in time and
space, yield to:
3
 

V
(2)
M PnPn  M Pn 1Pn 1  t [  (u  n )ipip   (  n )ip  ( S Pip  SC ) Pa1b ]
2
ip 1
where MP=ρ∆VP is the mass of the control volume and ∆VP is the P control volume area. When the quantity parcels
of all elements are added and boundary conditions applied, there is a conservative algebric equation of control
volume P, connected to its neighbors, that yields:
3
Aii   Bipip Sci
(3)
ip 1
7
where i indicates the number of control volumes, A is a diagonal matrix of the control volume mass, and B
represents the evaluated coefficients in the control volume.
5.1. Linear interpolation in the convective term
The used interpolation function evaluates the value of a generic property  in the control volume interface.
Differencing schemes apply to the linearization of advetive terms, i.e., the discretization of convected quantities.
Early attempts to solve advection-diffusion problems applied the Central Differencing scheme (CDS), but are
predominantly diffusive problems. For problems with predominant advection, solutions exhibited non-physical
behavior. Those issues initiated the development of a multitude of differencing schemes. Some of these are:
i) CDS is the most straightforward discretization of the convected variable, since it simply follows the linear
interpolation idea. In terms of a Taylor-series expansion, CDS is second-order accurate, but it is rarely used
nowadays owing to its conditional stability (Madsen, 1998). Oscillations in numerical solutions enable upstream
propagation of any disturbance even for purely advective cases. In the Figure 2 example, a Taylor-series expansion
around ip integration point, 1 and  2 values can be calculate as (Schneider and Maliska, 2002):

 2 ( L / 2) 2
1   ip   | ip ( L / 2)   2 | ip
 ...  ...
n
2
n

 2 ( L / 2) 2
 2   ip   | ip ( L / 2)   2 | ip
 ...  ...
n
2
n
In CDS,  ip value is given by adding equation (4) and (5), assuming second-order accuracy:
ip 
(4)
(5)
1  2
(6)
2
ii) Upwind Differencing Scheme (UDS) is a well-known remedy for the difficulties encountered in CDS. It was
first put forward by Courant, Isaacson and Rees in 1952 and subsequently reinvented by Gentry, Martin and Daly
in 1966, Brakat and Clark in 1966, and Runchal and Wolfshtein in 1969 (Patankar, 1980). It consists of setting the
cell-face value equal to the nearest cell-center value in the upstream direction (Madsen, 1998). UDS is only firstorder accurate but still an improvement over CDS, as it gets rid upstream propagation of disturbances. The low
accuracy of the relatively crude UDS is often interpreted as causing excessive numerical diffusion, and makes it
quite common to apply higher order upwind schemes including more upstream points for the interpolation of ip .
Value of
 ip is given by adding equations (4) and (5), assuming first-order accuracy:
ip  1 for cos β > 0
ip  2 for cos β < 0
(7)

u
(8)
where β is the angle of velocity vector with segment P1 in Figure 2.
iii) Weighted Upwind Differencing Scheme (WUDS) is a combination of CDS and UDS using weights;
iv) Hybrid Differencing Scheme (HDS) is also a combination of CDS and UDS, which was suggested by Spalding
in 1972 (Madsen, 1998);
v) Quick scheme was propose by Leonard in 1979 (Versteeg e Malalasekera, 1995). It performs the interpolation
by fitting a parabola through the two upstream points and the one downstream point nearest the face.
v) Higher order accurate schemes of interpolation use information from two or more points in the upstream
direction. For example, Skew Upwind Differencing Scheme (SUDS or Skew UDS), which interpolates values in
the faces using two points of the flow. Schneider and Maliska (2002) proposed:
1  cos 
1  cos 
ip 
1 
2
(9)
2
2
8
Schneider and Maliska (2002) still suggested two other schemes. Patankar (1980) also presents different schemes.
There are other schemes and for each one, it should be studied which one would be better applied.
5.2. Diffusive term
Several approaches are proposed in the open literature to deal with the diffusive term. The dot product between the
gradient vector of the PDE dependent variable and the normal outward vector can be given by (Schneider and
Maliska, 2002):
 A(1   P )
  n 
(10)
L cos 
where L and A are the size of P1 and ab segments, respectively, and α is the angle between normal outward vector
and P1 segment.
5.3. Higher order accurate approaches
To avoid numerical oscillations, several approaches were proposed in the open literature to reconstruction with
higher order accuracy. It can be cited the second-order accurate scheme propose by Kim and Choi (2000). Figure 3
depicts a similar scheme of Figure 2. It is defined a generalized coordinate system with covariant bases (e1, e2)
locally on each control volume face, where e1 and e2 are unit vectors from Pc (the cross-sectional point of P1 P2 and
Pa Pb ) to P2 and from Pc to Pb, respectively. Note that it is not the mid-point on the cell face. The gradient of  at Pc
can be expressed as:
 1  2
(11)
 
e 
e


where ξ and η represent the directions along e1 e e2, respectively, and e1 and e2 are corresponding contravariant
bases (Kim e Choi, 2000). Then, the normal component  in Pf can be written as:
 2  1 b  a


tan 
(12)
 |Pf    n 
n
1   2

where δ1 and δ2 are the normal distances to the control volume face, respectively, from P1 and P2, θ is the angle
between n and e1, and ∆η is the distance from Pa to Pb. It is interesting to note that equation (12) is composed of
two terms: the first one corresponds to the principal diffusion and the second one corresponds to the cross diffusion

as in the curvilinear coordinate system. When n is parallel to P1 P2 , the second term vanishes (Kim and Choi, 2000).
The C value at Pc is obtained with second-order accuracy (linear interpolation):
   
c  1 2 2 1
(14)
1   2
The  f value is evaluated by adding a correction term to C :
12   21
    21 a  b
     1 2

| |
1   2
1   2

where ε is the vector from Pc to Pf , |ε| is the magnitude of ε and:
f 
(15)
Ncs
 
 / A
i 1
Ncs
i
i
1/ A
i 1
(16)
i
where α=a or b, Ncs is the number of the control volumes sharing the vertex, i is the ith control volume value, and
Ai is the ith control volume area.
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Figure 3: Interpolation of flow variables at the mid-point on the control volume face (Kim e Choi, 2000).
5.4. Considerations
In the computational tests, it will be used prescribed flows for boundary condition. FVM for advection problems
are subjected to Courant-Friendrichs-Lewy (CFL) condition (Kim and Choi, 2000), which imposes that the
maximum stable time step for the entire mesh depends on the minimum control volume are. Avoiding CFL
condition is especially important in an adaptive mesh refinement because there are few control volumes very small,
which force a reduced time step for the whole mesh.
6. Adaptive mesh refinement technique with triangular cell-centered control volumes
This work uses terms ‘vertex’ and ‘face’ in relation to triangles and ‘nodes’ and ‘edges’ to deal with graphs. Figure
4a depicts an initial discretization scheme. Figure 4b depicts a graph of this initial scheme. The barycentric of the
triangle is represented as a black point in Figure 4a. Graph in Figure 4b presents two types of linked nodes: cell
node (in black) and transition nodes (in white).
(a)
(b)
Figure 4: (a) Triangle as the problem domain; (b) links for graph data structure (cell node represents the refinement
of level 0 and transition nodes represent boundaries).
This scheme of control volume refinement uses bisection (to divide an element in two equal parts) of triangular
control volumes, i.e., median segment is traced from the chosen vertex to the mid-point of the opposite face. In
Figure 5, it was chosen to refine the initial control volume of Figure 4a tracing the median from superior vertex and
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the opposite mid-point of the opposite face. It can be chosen any median for refinement. Nevertheless, it is traced
median between biggest vertex angle and its opposite face (that is, also the biggest face) for avoiding a mesh
distortion.
(a)
(b)
Figure 5: (a) Refinement examples with triangular control volumes (initial refinement); (b) graph with transition
and cell nodes that form the scheme of triangular control volume refinement (numbers by graph nodes represent
the refinement level).
Both cell nodes in Figure 5b represent each triangular control volume in Figure 5a. Graph links in Figure 5b are
represented by lines. Those three additional nodes, which are pointed from cell nodes, are named transition nodes.
Both cell nodes in Figure 5b point to transition nodes. Transition nodes represent the domain boundary faces in
Figure 5b. It is created three additional nodes (white circles in Figure 5b) in order to the edges that do not point to
cell nodes make sense when indicate the boundary domain. Each transition node also points to cell nodes, which
point them. Each node has three pointers as can be seen in Figure 5b. Pointers that are not used in transition nodes
receive a null pointer. A possible mesh refinement is depicts in Figure 6, where the right-hand side triangle of
Figure 5a is adaptively refined.
Figure 6: Adaptive refinement for triangular control volumes.
7. Graph data structure in the adaptive mesh refinement formed by triangular control volumes
Parent nodes are deleted in the local refinement of each triangular control volume. It is just stored cell nodes that
represent the two new triangular control volume children that are generated and the three transition nodes required.
Children cell nodes become autonomous as their parent node is deleted. Figure 7 depicts a bunch created in righthand side from the cell node showed in left-hand side. This renders a low computational cost and flexibility when
going through cell nodes with different levels of refinement in comparison with methods that use a tree-based
refinement scheme. When a triangular control volume is refined, a parent cell node is substituted by a new subgraph, i.e., a bunch with two cell nodes and three transition nodes. Such nested refinement process permits eventual
unrefinement of the created bunch, i.e., the parent cell node is recreated and the previous stages can be easily
reached. Unrefinement process can be understood in Figure 7, where right-hand side sub-graph is a bunch that
regenerates its parent cell node represented in left-hand side.
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Figure 7: Right-hand side sub-graph created after a refinement of left-hand side parent cell node. The opposite is
accomplished in the unrefinement process.
Such as Figure 7 depicts, there is a transition node that points to both children cell nodes created. This transition
node is representatively located in the refined face of the triangle control volume. The other two transition nodes,
which represent the other two control volume faces, point to only each one of the children cell nodes just refined.
Such transition nodes indicate the refinement level of the control volume in relation to their neighbor control
volumes. Figure 8 depicts examples of adaptive refinement using such scheme.
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Figure 8: Examples of successive adaptive refinements using the proposed scheme.
8. Sierpinski curve
Sierpinski curve is a well-known fractal space-filling curve. Sierpinski, in 1912, proved that the limit of the
sequence given by curves in order of 1, 2, …, depict in Figure 9, is the curve that passes through each point in the
unit square [0,1]x[0,1], or in a closed continuous surface.
Figure 9: Circular lists formed by Sierpinski curve of orders 0 to 9, being a square the original format of the
domain.
Sierpinski curve in uniform successive refinement in equilateral triangles is depicts in Figure 10.
Figure 10: Generator process of the Sierpinski curve through equilateral triangles.
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9. Ordering the control volumes of the mesh
Sierpinski curve is used for ordering the control volumes of the mesh. It is generated by a linked list. This
refinement scheme by triangular control volumes permits straightforward update of the list in the insertion of
refined cell nodes. An ordering scheme is necessary for numerical solution of the problem. An implicit formulation
demands to solve a linear system and furnishes stability in the resolution. Therefore, linear-system solvers based on
the minimization of functionals can be easily employed, such as Gradient Conjugate method, because the
coefficient matrix of the linear system is symmetric and positive definite. Figure 11 depicts successive adaptive
refinements in a quadrangular domain with triangular control volumes ordered by the modified Sierpinski curve.
Figure 11: Successive adaptive refinements with quadrangular domain with control volumes ordered by modified
Sierpinski curve in the right-hand side of each discretized domain in left-hand side.
Figure 12 depicts other example of Sierpinski curve generation. If the domain shape is the left-hand side triangle of
Figure 12, and the triangles are uniformly refined, the thirst level of refinement and ordering will be such as the
right-hand side curve of Figure 12. Dividing control volumes by the median of the triangle from the biggest angle,
and consequently, the mid-point of the biggest face, warrants a quality mesh. This scheme permits straightforward
insertion in the list of mesh ordering.
Figure 12: Sierpinski curve generation.
Because triangular control volumes are adaptively refined, this work uses the modified space-filling curve of
Sierpinski, proposed in 1912. Modification is that control volumes are adaptively refined and it can be used any
kind of triangle shapes. Any arbitrary polygonal can be applied in the initial domain shape. Figure 13 depicts
Sierpinski curve ordering scalene triangles.
Figure 13: Sierpinski curve ordering scalene triangles.
The same domain is adaptively refined and depicted in Figure 14. Figure 14 depicts examples of adaptive
refinement of triangular control volumes ordered by modified Sierpinski curve.
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Figure 14: Examples of adaptive refinement with scalene triangle control volumes.
10. Considerations
This work proposes a triangular discretization based on FVM for solving PDE’s with unstructured meshes.
Determination of dependent variable is in the centroids of the proper control volume, without requirements of
creation of a simplex mesh. Required data for solving a PDE are stored in a graph data structure in the adaptive
mesh refinement. Graph nodes are autonomous because parent nodes in the local refinement are not stored. This
refinement process permits that the created bunch can be eventually unrefined and returned to its previous stage.
This process is simple and straightforward. It is low computational cost and flexible in ordering the mesh and
linking neighbor control volumes refined in different level in comparison to tree-based methods that use adaptive
mesh refinement.
This ordering scheme of the discretized mesh is a modified Sierpinski curve, which is generated by a linked list. It
is a modified version of the well-known curve due to the adaptive mesh refinement and it permits using several
kinds of triangles.
The refinement scheme of triangular control volume proposed allows straightforward insertion in the linked list for
mesh ordering. Additionally, linear-system solvers based on the minimization of functionals can be easily
employed, such as Gradient Conjugate method, because the coefficient matrix of the linear system is symmetric
and positive definite.
Using a triangular unstructured mesh, this adaptive refinement scheme is more efficient than ALG because it
requires less refinement to obtain the dependent variable value in an arbitrary point of the domain. This proposed
scheme allows a better adaptation to problems with complex domains. Further, in a local refinement, the proposed
scheme of dividing the triangular control volume in two new cell nodes permits more efficiency in the process of
insertion in the linked list of mesh ordering, in comparison to ALG. Besides, the here proposed technique allows
any shapes for the initial domain, such as, square, rectangles, any kind of triangles, i.e., arbitrary polygonal shapes.
Each discretized problem is evaluated with its own features, and the most appropriate linear interpolation will be
employed for the problem treated. Green-Gauss integration will be used to reconstruct the solution in respect to
required gradients. Green-Gauss reconstruction has characteristics that it can be extended to a simplified leastsquare gradient reconstruction or some vertex-centered scheme for problems that numerical oscillation occurs or if
the mesh can be distorted.
This work is in its initial stage of analysis. Development of computational implementations with classical equations
will demonstrate the efficiency of the new technique as well as its advantages in comparison to other numerical
solution methods of PDE’s.
Future research:
- Using quadrangular control volumes seems to be straightforward. It can be tested the use of hybrid meshes using
Sierpinski curve for mesh ordering;
- In several theoretical and real problems, verify the step time in relation to CFL condition and verify stability and
convergence analysis;
- Parallelization of the methods for specific types of problems, which a large linear system is required to be solved;
- Verify the three-dimensional model. A cell-centered control volume seems to be interesting, since Frink’s
reconstruction is a well-developed proposal;
- Voronoi-based scheme allows direct use of FVM basic formulation. The Delaunay triangulation could be
generated in the first time step; and for each local simplex refinement, the entire mesh is not redesigned for each
time step. In this case, Sierpinski curve would link vertices of the mesh, instead of control volume centroids;
15
- Median dual with specific numerical quadrature can allow stability and convergence analysis in the same way
that is obtained in FEM, with the earlier study in relation to Sierpinski curve;
- Simplified least-square gradient reconstruction is more appropriate for distorted meshes;
- Use a higher order accuracy to avoid numerical oscillation:
i) ENO/WENO for avoiding numerical oscillation in specific problems, when computational cost increasing is not
priority;
ii) Frink’s reconstruction is interesting because obtains higher order accuracy with little computational cost
increase, besides, it is applied to cell-centered control volume approach in three-dimensional problems;
iii) Verify limiters for MUSCL.
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