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Slope limiting the velocity field in a discontinuous
Galerkin divergence free two-phase flow solver
Tormod Landet∗
, Kent-Andre Mardal and Mikael Mortensen
Matematisk Institutt, Moltke Moes vei 35, 0851 Oslo, Norway
March 20, 2018
Abstract
Solving the Navier-Stokes equations when the density field contains a large sharp
discontinuity—such as a water/air free surface—is numerically challenging. Con-
vective instabilities cause Gibbs oscillations which quickly destroy the solution. We
investigate the use of slope limiters for the velocity field to overcome this problem
in a way that does not compromise on the mass conservation properties. The equa-
tions are discretised using the interior penalty discontinuous Galerkin finite element
method that is divergence free to machine precision.
A slope limiter made specifically for exactly divergence free (solenoidal) fields
is presented and used to illustrated the difficulties in obtaining convectively stable
fields that are also exactly solenoidal. The lessons learned from this are applied in
constructing a simpler method based on the use of an existing scalar slope limiter
applied to each velocity component.
We show by numerical examples how both presented slope limiting methods are
vastly superior to the naive non-limited method. The methods can solve difficult
two-phase problems with high density ratios and high Reynolds numbers—typical
for marine and offshore water/air simulations—in a way that conserves mass and
stops unbounded energy growth caused by the Gibbs phenomenon.
Keywords: DG FEM; divergence free; solenoidal; Navier-Stokes; two-phase;
slope limiter; Gibbs oscillations; density jump
1 Introduction
The incompressible and variable density Navier-Stokes equations for the unknown velocity
uand pressure p, with gravity gand spatially varying density ρand viscosity µ,
ρ∂u
∂t + (u· ∇)u=∇ · µ∇u+ (∇u)T− ∇p+ρg,(1)
∇ · u= 0,(2)
∂ρ
∂t +u· ∇ρ= 0,(3)
∗Corresponding author. Email: tormodla@math.uio.no
1
arXiv:1803.06976v1 [physics.flu-dyn] 19 Mar 2018
are used in applications where the fluid velocity is much smaller than the Mach number
and where gravity effects are important such as for internal and surface gravity waves.
Unfortunately, numerical problems will immediately occur when using equations (1) to (3)
to study air/water free surface physics with a higher order solution method. These
problems are due to the sharp factor 1000 jump in momentum across the liquid/gas
interface which causes Gibbs oscillations in the velocity field, even though the true velocity
has no discontinuity at the interface. The energy in the velocity field will eventually blow
up to destroy the solution if this non-linear convective instability is not handled with
care.
The convergence of numerical approximations to the Navier-Stokes equations with
variable and potentially discontinuous density and viscosity was studied by Liu and Walk-
ington (2007). They show that if the continuous problem has an unique solution then a
stable discontinuous Galerkin (DG) discretisation with a piecewise constant density field
will converge to that solution. This work presents such a stable DG discretisation. Stable
fractional step methods for solving the equations have been studied by Guermond and
Quartapelle (2000); Pyo and Shen (2007); Guermond and Salgado (2009). Our results
are computed using a direct solver on the coupled velocity and pressure system in order
to eliminate any fractional step splitting errors from influencing the conclusions.
In this work we will use a piecewise constant density field and the volume of fluid, VOF,
method by Hirt and Nichols (1981) for evolving this density field in time while maintaining
a sharp interface. In the VOF method a transport equation for the density is solved for
a normalised so called colour function,C∈[0,1], which is linearly related to the density.
Among the most commonly used variations are the algebraic VOF schemes CICSAM
(Ubbink, 1997) and HRIC (Muzaferija et al., 1998). The level set method (Osher and
Sethian, 1988) is another alternative for tracking free surfaces. Modern versions exist
that significantly improve on the historical mass conservation problems of the level set
method—see, e.g., Olsson and Kreiss (2005); Tour´e, Fahsi, and Soula¨ımani (2016).
For most industrial applications the above methods are implemented in solvers based
on the finite volume method where it is possible to ensure convective stability by appro-
priate use of flux limiters, see, e.g., the illustrative discussions and diagrams of Sweby
(1984) and Leonard (1988). Flux limiters based on TVD or ENO/WENO schemes are
applied in the solution of the momentum equation, and some sort of stable interface
sharpening scheme such as CICSAM or HRIC is used for the flux of density. Finite
volume methods can be partially extended to higher order by use of larger geometrical
stencils, but this approach is not trivial to implement when using irregular grids, so in
practice it is common that only immediate neighbours are used and most schemes are
hence low order.
For irregular geometries there are two major research directions that look to en-
able higher order methods: immersed boundary methods and finite element methods.
Immersed boundary methods use regular background grids and perform special treat-
ment of grid cells near or inside objects embedded in the computational domain (Peskin,
2002). Finite element methods use basis functions with local support to enable high or-
der approximating functions on irregular meshes. This work is based on a discontinuous
Galerkin (DG) finite element method (FEM). DG-FEM has two main advantages over
continuous Galerkin methods, the ease of using an upwind flux limiter for stabilising the
linear convective instabilities and the option to create exactly divergence free and mass
2
conserving numerical schemes (Cockburn, Kanschat, and Sch¨otzau, 2004, 2005). There
is one important drawback, and that is the significantly increased number of degrees
of freedom. A variation of the scheme from Cockburn, Kanschat, and Sch¨otzau (2005)
is used, where instead of using a local discontinuous Galerkin, LDG, treatment of the
elliptic term, a symmetric interior penalty, SIP, treatment (Arnold, 1982) is employed.
When using higher order approximating polynomials it is no longer sufficient to use
only a flux limiter to obtain convective stability—a slope limiter must also be included
in the method (Cockburn and Shu, 1998, 2001; Kuzmin, 2010). The flux limiter ensures
that the cell average values are bounded. When using higher order basis functions, the
solution can go out of bounds in localized regions due to steep gradients inside each cell.
A slope limiter prevents this by flattening steep slopes near discontinuities while leaving
the solution untouched near smooth maxima so that the method’s high order accuracy
is retained. While a flux limiter is an implicit part of the equation system and has its
stabilising effect included in the results from the linear equation solver, a slope limiter is
typically applied to the resulting function as an explicit post-processing operator. Other
options could be to use non-linear diffusion to combat the non-linear instability, or to
apply spectral filtering, see, e.g., Michoski et al. (2016); Zingan et al. (2013).
This paper starts with a description of the Gibbs instability and the discontinuous
Galerkin method employed for solving the variable density Navier-Stokes equations in
sections 2.1 to 2.4. Slope limiting is then presented; first the hierarchical Taylor based
slope limiter by Kuzmin (2010, 2013) in section 3.1, and then the possibility of construct-
ing a vector field slope limiter that leaves the resulting velocity field both solenoidal and
free from local maxima is explored in section 3.2. After showing that it is likely not
possible to obtain a single field that is both solenoidal and stable, separate limiting of the
convecting and the convected velocity fields is introduced. Two alternatives are presented
in section 3.3; both ensure solenoidal convecting velocities while keeping convective sta-
bility. Readers familiar with DG methods and slope limiting can start at section 3.2,
though the presentation builds directly on the preceding methods, so some referring back
may be needed. Results from numerical tests are shown in section 4 and discussion and
concluding remarks can be found in section 5.
2 The numerical method
2.1 Instabilities
The most common numerical instabilities related to handling of large density jumps are
illustrated in figures 1a to 1c. A ‘block’ of water starts at rest in a box filled with
air. Already in the very first time step (figure 1a) the solution will start to blow up
if one does not either apply smoothing to the density field or stabilise the numerical
scheme. Smoothing the initial field is not sufficient as the interface may pinch and
become non smooth (figure 1b), so continuous smoothing of the density field is necessary
if this approach is selected. A level set method is a natural way to implement a smoothed
density field, see, e.g., Unverdi and Tryggvason (1992); Sussman, Smereka, and Osher
(1994). In this work we will not apply stabilisation through smoothing. Our aim is to
decouple the stability of the method from the treatment of the density field.
Figure 1c illustrates a situation where the solenoidal properties of uare important.
3
(a) Initial condition (b) Pinching (c) Corner impact
Figure 1: Illustration of typical numerical problems
A divergence free convecting velocity is required for the stability of the density transport
equation (3) and any divergence quickly becomes a problem in difficult situations such as
the corner impact where in our experience either mass loss or unbounded densities will
occur if care is not taken to ensure that the convecting velocity used for density transport
is solenoidal.
2.2 Preliminaries
The notation used in this paper is relatively standard. When looking at a facet between
two cells (finite elements) we will denote one of the cells K+and the other K−in an
arbitrary, but repeatable manner. Function values in each cell will be given the same
superscripts to distinguish between the values on opposite sides of the facet in the dis-
continuous space. The average and jump operators across an internal facet are defined
as
{{u}} =1
2(u++u−),(4)
JuK=u+−u−,(5)
JuKn=u+·n++u−·n−.(6)
Vector terms such as the velocity uand gravity gwill be denoted with a bold font
while scalars such as the pressure p, the density ρ, and the dynamic viscosity µare
shown in italics. Projection operators are written in blackboard bold. An example is the
projection Dinto a solenoidal vector space. Sets are written in a calligraphic typeface.
Fluxes of a quantity, used in DG facet integrals, are marked with a circumflex accent over
the quantity, e.g., ˆ
u, ˆp. An additional superscript is added when different definitions of
the flux are used in different parts of the weak form, e.g., ˆ
uwand ˆ
up, the flux of velocity
related to convection and the flux of velocity related to the incompressibility constraint.
Let ∂K denote the boundary of element K. ΓIis the inlet portion of the domain
boundary ∂Ω where u·n<0 for an outwards pointing normal, n. The set of all grid
cells is T(the tessellation), and the set of all facets is S(the mesh skeleton). The set
of outside facets is SO=S ∩ ∂Ω and the set of inside facets shared between two cells is
SI=S \ SO.
For a facet on the boundary, Fb∈ SO, let the boundary cell be denoted K+with
regards to the facet and hence n+·JvK=n+·v+=n·von Fb. We set terms related
to K−to zero and also set the average to be the K+value {{v}} =von SO.
4
Let Pk(K) denote the space of polynomials of order kon an element and Pk(F)
denote the space of polynomials of order kon a facet. The basis functions in Pk(K)
are discontinuous across facets and functions in Pk(F) are discontinuous across edges
(vertices in 2D). Let the superscript •n+1 denote a value at time step t= ∆t(n+ 1). For
nabla the conventions (∇u)ij =∂juiand (∇ · σ)i=∂jσij are used.
2.3 Discretisation
We will approximate the unknown functions in space by using discontinuous Lagrange
polynomial function spaces with polynomial order k= 2 for the velocity in d= 2 spatial
dimensions. Let the Galerkin test functions for {u, p, ρ}be denoted respectively {v, q, r}.
The discontinuous function spaces are not restricted at the boundaries—all boundary
conditions will be imposed weakly—so the trial and test functions share spaces,
u,v∈[Pk(K)]d,
p, q ∈Pk−1(K),(7)
ρ, r ∈P0(K).
2.3.1 Two-phase density transport
The transport equation (3) for the density is modified by the introduction of a solenoidal
convecting velocity field wwhich is close to u, but may not be identical as will be
explained in sections 2.4 and 3.3. Boundary conditions are needed only on the inlet. The
strong form can then be written
∂ρ
∂t +w· ∇ρ= 0 in Ω,(8)
ρ=ρIon ΓI.
In VOF methods the fluid properties are expressed in terms of an indicator function,
C∈[0,1]. The true density and viscosity fields can easily be recovered from Cwhen the
density and kinematic viscosity properties of the two fluids are known,
ρ=Cρwater + (1 −C)ρair,(9)
µ= [Cνwater + (1 −C)νair]ρ. (10)
In order to compute Cthe transport equation (8) for the density is modified by inserting
equation (9). Now the solution Cn+1 ∈P0(K) can be found by expressing the resulting
equation on weak form and integrating by parts,
ZT
1
∆t(γ1Cn+1 +γ2Cn+γ3Cn−1)rdx(11)
−ZT
Cn+1w· ∇rdx+ZS
ˆ
Cn+1 w·n+JrKds= 0,
where we have assumed that the convecting velocity wis Hdiv-conforming such that the
flux is continuous across facets, JwKn= 0.
5
A second order backwards differencing formulation, BDF2, is used for time integration.
The parameters are {γ1, γ2, γ3}={3/2,−2,1/2}. The BDF2 method is monotonicity
preserving when started using a backward Euler step (Hundsdorfer, Ruuth, and Spiteri,
2003), though the time step required to preserve monotonicity is half that of backward
Euler. The benefit is that the method is second order—and small time steps are anyhow
required to keep the interface sharp (Muzaferija et al., 1998). Second order extrapolation
is used for the convective velocity,
w= 2wn−wn−1,(12)
which makes ρn+1 independent of the computed velocity at time t= (n+ 1)∆t. The
density transport equation is hence uncoupled from the momentum equation.
For the density flux, ˆ
Cn+1, the most stable choice is to take the upwind value, which
means that the boundary condition ˆρ=ρIis used to determine ˆ
Cn+1 on inlet facets,
S ∩ ΓI. For the internal facets the term related to the flux can be written on upwind
form as
ˆ
Cn+1 w·n+=sCn+1 1
2(w·n+|w·n|){,(13)
and by replacing the plus by a minus in equation (13) the downwind flux can similarly
be computed. Using the upwind and downwind fluxes on each facet a downwind-blended
compressive interface flux can be applied to the density transport such as CICSAM
(Ubbink, 1997) or HRIC (Muzaferija et al., 1998). Such blended fluxes sharpen the
interface between the two fluid layers, but remain convectively stable unlike downwind
or central fluxes. Both CICSAM and HRIC are algebraic VOF methods which define
facet-wise blending factors to combine the upwind and downwind fluxes into one linearly
stable flux. The results in section 4 are computed using the HRIC method.
One important note is that standard VOF flux limiters from finite volume methods
ensure that the Cfield remains sharp and bounded based on a convecting velocity field
that is piecewise constant on each facet. Such a field can easily be computed from wand
as long as wis solenoidal, so is the piecewise constant field. It is this field that is used
when solving for Cn+1 since wis only needed on the facets—for r∈P0(K) the volume
integral term containing win equation (11) is identically zero.
2.3.2 The variable density Navier-Stokes equations
The strong form of the variable density Navier-Stokes equations, where the convecting
velocity is replaced by w, can be written
ρ∂u
∂t + (w· ∇)u=∇ · µ(∇u+∇u)T− ∇p+ρgin Ω,(14)
∇ · u= 0 in Ω,
u=uDon ΓD,
∂u
∂n =aon ΓN,
where ΓDand ΓNare the parts of the boundary where Dirichlet and Neumann boundary
conditions are applied respectively; ΓN=∂Ω\ΓD. Dirichlet boundary conditions can
6
be enforced on external facets due to the elliptic viscosity term, ∇ · µ(∇u+∇u)T.
The Navier-Stokes equations will be written on weak form, see equation (19), but first
we will briefly discuss the symmetric interior penalty (SIP) method. First, the boundary
condition, u=uD, is written on weak form on an external facet F,
ZF
u·vds=ZF
uD·vds, (15)
and then—using the stabilisation scheme proposed by Nitsche (1971)—the test function
vis replaced by a Petrov-Galerkin test function ˜
v=κµv−µ∇v+ (∇v)T·n. This
method is extended to enforce continuity across internal facets which is necessary for
stability (Arnold, 1982). This gives
ZF
κµJuK·JvKds−ZF
(µ∇v+ (∇v)T·n+)·JuKds= 0.(16)
where κµis a penalty parameter which must be sufficiently large to ensure stability.
The analyses in Epshteyn and Rivi`ere (2007) and Shahbazi, Fischer, and Ethier (2007)
guide us in defining the penalty parameter as a function of the minimum and maximum
dynamic viscosities, µmin and µmax, the order kof the approximating polynomials, and
the surface area SKand volume VKof each cell K,
κµ= 3 µ2
max
µmin
k(k+ 1) max
KSK
VK.(17)
The same scheme is used for the left-hand side of the momentum equation as for
the density transport equation, but a pure upwind flux is used without any downwind
blending for the convective term. To avoid overloading the notation we now drop the
•n+1 superscript on the unknown quantities. The upwind flux related to convection can
then be written
ˆ
uww·n+=su1
2(w·n+|w·n|){.(18)
Both the pressure gradient and the viscosity on the right-hand side of the momentum
equation are integrated by parts. The resulting weak form is a direct combination of
the LDG Navier-Stokes method by Cockburn, Kanschat, and Sch¨otzau (2005) and the
SIP diffusion method by Arnold (1982). Both these references contain more details and
proofs of stability. The stability of the convective and the diffusive terms are not inter-
connected, so replacing the LDG elliptic operator with the SIP version is unproblematic.
The treatment of the momentum transport and the inter-cell continuity, both described
above, are easily recognisable in the complete weak form,
ZT
ρ
∆t(γ1u+γ2un+γ3un−1)vdx(19)
−ZT
u·∇·(ρv⊗w) dx+ZS
w·n+ˆ
uw·JρvKds
+ZT
µ∇u+ (∇u)T:∇vdx+ZSI
κµJuK·JvKds
7
−ZS
(µ∇u+ (∇u)T·n+)·JvKds
−ZSI
(µ∇v+ (∇v)T·n+)·JuKds
−ZT
p∇ · vdx+ZS
ˆpn+·JvKds=ZT
ρgdx.
where the flux of pressure is taken as ˆp={{p}}.
The continuity equation (2) is also integrated by parts using ˆ
up={{u}} as the flux
related to the incompressibility constraint,
ZS
ˆ
up·n+JqKds−ZT
u· ∇qdx= 0.(20)
Dirichlet boundaries On the inflow part of the Dirichlet boundary take ˆ
uw=uD,
and on the outflow part take ˆ
uw=u, i.e., the upwind values are used. On the whole
Dirichlet boundary let ˆ
up=uDand ˆp=p. The viscous penalty and symmetrisation
terms from equation (16) can be used also on the domain boundary by replacing u−by
uD. As noted in the references used to define κµabove, the best choice is to use twice
the amount of penalisation on external facets compared to the interior facets. This leads
to external boundary integrals
ZΓD
2κµ(u−uD)·vds−ZΓD
(µ∇v·n)·(u−uD) ds= 0.(21)
Neumann boundaries For the Neumann boundaries we take ˆ
uw=u,ˆ
up=uand
ˆp=p. The extra viscous terms for symmetrisation and penalty are removed and only
the normal integration by parts terms are left. The surface term becomes
−ZΓN
µa·vds. (22)
Pure Neumann boundary conditions will not be used, but we will use Dirichlet for
one velocity component and Neumann for the other to implement free-slip boundary
conditions on planes that are parallel to the axes. The definitions above can easily be
split into component-wise treatment of boundary conditions.
Solution algorithm
The Navier-Stokes equations and the density transport equation are solved in a de-
coupled manner for each time step in the following order:
1. Compute an explicit convecting velocity wn+1 by use of equation (12).
2. Find Cn+1 ∈P0(K) such that equation (11) is satisfied.
3. Compute the density ρn+1 and viscosity µn+1 by use of equations (9) and (10).
4. Use the computed coefficient fields wn+1,µn+1 and ρn+1 to find un+1 ∈[Pk(K)]d
and pn+1 ∈Pk−1(K) from equations (19) and (20).
8
2.4 Hdiv projection of the velocity field
In finite element methods for solving the Navier-Stokes equations it is common to impose
the incompressibility criterion, ∇ · u= 0, weakly by multiplying with a scalar test
function, q, and integrating over the domain. This term, RΩ∇·uqdx, will appear directly
in a coupled solver and as the right-hand side in the Poisson equation for the pressure
in a pressure correction fractional step scheme such as the commonly used incremental
pressure correction scheme.
Imposing the incompressibility criterion weakly in the space of the pressure is sufficient
for stability, but one must require exact incompressibility to locally conserve mass and
momentum. Below we explain the core ideas of the method presented in Cockburn,
Kanschat, and Sch¨otzau (2005). By using this method the resulting divergence will be
zero almost to machine precision, approximately 10−13 on each cell in our tests. We
calculate the cell-wise error by computing the integrated absolute value of the divergence
internally in each cell, RK|∇ · u|dx, and add to it the error in flux continuity between
cells on each connected facet, R∂K |Ju·nK|ds.
Let us start by noting that Hdiv-conforming finite elements exist, and that such ele-
ments of a given polynomial order will be subspaces of the fully discontinuous elements of
the same order with the same cell geometry. Such elements impose continuity of normal
fluxes across facets and hence have fewer global degrees of freedom. We follow Cockburn,
Kanschat, and Sch¨otzau (2005) and define a projection from our fully discontinuous ve-
locity uinto a velocity w=Puthat exist in a space of polynomials that are consistent
with the Hdiv-conforming elements.
The projection operator w=Puis defined in a cell-wise manner. This local projection
is hence very fast and consists of finding w∈[Pk(K)]dsuch that
ZF
w·nv1ds=ZF
ˆ
up·nv1ds∀v1∈Pk(F), F ∈∂K, (23)
ZK
w·v2dx=ZK
u·v2dx∀v2∈Nk−1(K).(24)
The first equation ensures the continuity of the normal velocities across each facet. The
continuity stems from using a single valued flux which is here ˆ
up={{u}} on internal
facets and ˆ
up=uDon external facets (ˆ
up=uon ΓN). This flux is consistent for
continuous velocity fields.
In equation (24) the space Nk−1(K) is the N´ed´elec H(curl) element of the first kind
of order k−1, see, e.g., Kirby et al. (2012); N´ed´elec (1986). The dimension of the
Brezzi-Douglas-Marini (BDM) element Pk(F)×Nk−1(K) is the same as that of the
Discontinuous Lagrange DGkelement on each cell (Brezzi and Fortin, 1991). It is hence
possible to form square projection matrices between the two spaces in each cell. The
projection Puis not square globally since the degrees of freedom related to v1are shared
between exactly two cells on all internal facets. This leads to the test space having fewer
global degrees of freedom than the fully discontinuous trial space.
The properties of this BDM-like projection used as a velocity post-processing step in
a discontinuous Galerkin method is given by Cockburn, Kanschat, and Sch¨otzau (2005).
The projection gives both a continuous flux w·non all inter-element facets and also
ensures that the total flux across each individual cell’s facets is zero. Fulfilling these two
criteria is what we mean by exact incompressibility.
9
3 Slope limiting
The numerical diffusion due to upwinding in the spatial DG scheme is sufficient to stabilise
convective operators and avoid spurious oscillations when piecewise constant approximat-
ing functions are employed. One way to stabilise convective operators when using higher
order approximating functions is to employ slope limiters (Cockburn and Shu, 1998).
3.1 The hierarchical Taylor polynomial based slope limiter
The hierarchical vertex based slope limiter by Kuzmin (2010, 2013) is used to remove
high frequency oscillations near discontinuities when solving an equation containing a
convective term for a scalar quantity φ. This slope limiter is based on using discontinuous
Taylor function spaces. We are using Lagrange polynomials, so the first step is to project
the unknown function φfrom the discontinuous Lagrange function space to a function φt
in the discontinuous Taylor function space. This projection φt=Tφis local to each cell
and can be applied and inverted exactly by a single matrix vector product in each cell,
so the cost of converting back and forth is negligible.
The discontinuous Taylor function space is slightly altered from the standard definition
by use of cell averages. The expansion
φt(x, y) = ¯
φ+∂φ
∂x c
α1(x−xc) + ∂φ
∂y c
α1(y−yc)+ (25)
∂2φ
∂x2c
α2"(x−xc)2
2−(x−xc)2
2#+∂2φ
∂y2c
α2"(y−yc)2
2−(y−yc)2
2#+
∂2φ
∂x∂ y c
α2"(x−xc)(y−yc)
2−(x−xc)(y−yc
2#
is used to describe a second order polynomial function on a triangle where the six co-
efficients are {¯
φ, ∂φ/∂ x|c,· · · ,∂2φ/∂x∂y|c}. The number of coefficients is the same as the
number of nodes in a second order Lagrangian function space on a triangle which ensures
that the transformation matrix resulting from Tis square on each cell. The same is also
true for tetrahedra and for higher and lower polynomial orders.
Over-lined terms such as ¯
φand (x−xc)2/2denote cell averages and the restriction ·|c
signifies evaluation in the cell centre. Slope limiters αi∈[0,1] will be applied to the i’th
derivative terms as shown in equation (25) using the same α-factors for all derivatives of
the same order. How to compute αiis described below. As long as no slope limiting is
performed (αi= 1), we have φ=T−1Tφ.
To find αiwe follow Kuzmin (2013) and define three bi-linear functions on each cell,
˜
φ0(x, y) = ¯
φ+ ˜α1∂φ
∂x c
(x−xc) + ∂φ
∂y c
(y−yc),(26)
˜
φx(x, y) = ∂φ
∂x c
+ ˜α2x∂2φ
∂2xc
(x−xc) + ∂2φ
∂x∂ y c
(y−yc),(27)
˜
φy(x, y) = ∂φ
∂y c
+ ˜α2y∂2φ
∂2yc
(y−yc) + ∂2φ
∂x∂ y c
(x−xc).(28)
10
The parameters ˜αiare intermediate slope limiters for the approximate function ˜
φi. Equa-
tions (26) to (28) are used to linearly approximate φand its first derivatives at each of
the three vertices of the triangular cells. These approximate vertex values should not
form local maxima or minima when compared to the cell centre values of the linear
representations in the surrounding cells.
Pick any one of the above linear approximations in equations (26) to (28) and call
it ˜
φ(x, y). The corresponding value in the cell centre is denoted ˜
φc=˜
φ(xc, yc). Iterate
through all cells in the mesh and for each cell consider each vertex and record the extremal
˜
φcin the cells that share the vertex. This gives allowable bounds ˜
φmin
iand ˜
φmax
iat vertex
ifor the selected linear approximation in the given cell. For each vertex ilocated at
(xi, yi) one must ensure that ˜
φi=˜
φ(xi, yi) is bounded by the surrounding cell values,
˜
φmin
i≤˜
φi≤˜
φmax
i. To find the maximum admissible value of ˜αthat ensures this, first for
each vertex iset
˜αi=
min{1,˜
φmax
i−˜
φc
˜
φi−˜
φc}if ˜
φi−˜
φc>0,
1 if ˜
φi−˜
φc= 0,
min{1,˜
φmin
i−˜
φc
˜
φi−˜
φc}if ˜
φi−˜
φc<0,
(29)
and then take the minimum of the calculated ˜αi-values in each vertex to find ˜α1, ˜α2x
and ˜α2yfor each cell by performing the above calculations for each of the three linear
approximations in equations (26) to (28). The last step is to calculate the final slope
limiter coefficients for the second order derivatives,
α2= min{˜α2x,˜α2y},(30)
and, since one can expect higher regularity of the first order derivatives than the second
order derivatives, take
α1= max{˜α1, α2},(31)
which at a smooth extremal point will stop any limiting from happening since α2= 1
here, even if it is likely that ˜α1<1.
Having found the slope limiter coefficients α1and α2for a given cell one can go back
to the discontinuous Lagrange function space with the help of the projection T−1. If there
were no spurious oscillations in the cell then the slope limiters coefficients should end up
as αi= 1 and the slope limiter projection—which we call S—will hence be an identity
transform, and we get φlim =T−1S T φ=T−1Tφ=φ. This ensures that the order of
spatial convergence is kept the same as the underlying DG scheme.
3.2 On slope limiting of solenoidal fields
Slope limiting of solenoidal vector fields is more complex than limiting scalar fields due to
an increased number of invariants. For scalar fields the only invariant is that the average
value in each cell must be unchanged after limiting. For vector fields the (i) divergence
inside each element and the (ii) flux between neighbouring elements are new invariants
in addition to the (iii) average value of each of the velocity component in each cell.
11
Breaking these invariants means that the result is not solenoidal (i), the slope limiting
post processor is not a non-local operation (ii) or that momentum is not preserved (iii).
In the following sections we will describe how these invariants can be used to reduce
the number of unknowns in the cell-wise limiting problem from 12 to only 4 for a DG2
vector field, u∈[P2(K)]2, on a triangle. We will then describe an optimisation method
which can be used to decide the value of the remaining four unknowns in a way that
minimises the tendency to produce local extrema. Removing all local extrema would
render the method stable. We do not claim that the below method is optimal, but it
does give strong indications that the four remaining unknowns in the cell-wise limiting
problem are not sufficient to fully control the local extrema. Creating a single velocity
field that is both solenoidal and stable is hence likely not possible by use of a cell-wise
slope limiting process.
3.2.1 A reduced basis for the solenoidal slope limiting problem
Let the cell volume be denoted VKand the area of facet ibe denoted Li. Preserving the
facet average of the flux between neighbouring cells gives a result which is Hdiv-compatible
in a DG0 sense, which is what is needed for mass conservation. The cell-wise invariants
we require to be left unchanged by the slope limiting procedure are then
Ri=1
LiZFi
u·nds∀i∈ {1,2,3}, Uj=1
VKZK
ujdx∀j∈ {1,2}.(32)
When applying this to a DG2 vector field on a triangle there are five invariants, three
average fluxes, Ri, and two component averages, Uj. By not changing the average flux
on each facet the total flux for each cell will still sum to zero, and it is still possible to
have a solenoidal description of the vector field.
The 12 degrees of freedom in a DG2 vector field on a triangle can be reduced. The
divergence free solution will never span the full Lagrange space, but be restricted to the
solenoidal subspace which has only 9 degrees of freedom, see e.g. Baker, Jureidini, and
Karakashian (1990). This space is spanned by
1
0,0
1;y
0,0
x,x
−y;y2
0,0
x2,x2
−2xy,−2xy
y2; (33)
which is here shown grouped into the two zeroth order, three first order and four second
order vector valued polynomials.
A projection from the Lagrangian description of uto the solenoidal description us
can be implemented as us=Duon a cell by cell level. To do this, first define local x
and ycoordinates with the origin in the centre of each cell and then form the cell-wise
rectangular inverse projection u=D−1usby evaluating the solenoidal polynomials from
equation (33) in the location of the Lagrangian nodes. The least squares pseudo-inverse
of this matrix is then D. It is important to note that in our simulations the projection is
lossless such that u=D−1Dusince the Hdiv projection along with the DG weak form of
the momentum and continuity equations causes uto be fully described by the solenoidal
subspace. In general, we have D−1D6=I, so this identity property does not hold for a
generic vector field.
12
The velocity vector field can now be described by nine degrees of freedom in each
cell, coefficients s1· · · s9. Each of the nine coefficients is multiplied by the corresponding
vector valued polynomials in equation (33), numbered from left to right. The sum of
these products form the complete vector valued polynomial field.
We will further restrict the number of degrees of freedom by making use of the five
invariants from equation (32). Let the coefficients for the constant and linear polynomials
be determined by the five invariants. The four quadratic terms are now the only degrees
of freedom. Picking four arbitrary numbers, σ1to σ4, for the coefficients s6to s9, one
can find corresponding constant and linear weights that make the final vector field keep
the selected invariants. This is done by forming a 9x9 matrix system,
q11 q12 q13 q14 q15 q16 q17 q18 q19
q21 q22 q23 q24 q25 q26 q27 q28 q29
q31 q32 q33 q34 q35 q36 q37 q38 q39
q41 q42 q43 q44 q45 q46 q47 q48 q49
q51 q52 q53 q54 q55 q56 q57 q58 q59
000001000
000000100
000000010
000000001
·
s1
s2
s3
s4
s5
s6
s7
s8
s9
=
U1
U2
R1
R2
R3
σ1
σ2
σ3
σ4
,(34)
where the first five rows contain appropriate weights qki such that the correct integral
from equation (32) is computed when the row is multiplied by the vector of coefficients,
si. These weights can easily be computed by quadrature. By solving equation (34) the
full set of nine solenoidal coefficients, si, can be recovered from the reduced set of four
degrees of freedom, σj. The original 12 degrees of freedom of the Lagrangian velocity
field ucan be recovered by use of the inverse projection u=D−1us.
We should note that we have no proof that the matrix in equation (34) is invertible for
all possible cells, but in our extensive testing we have never found a cell where this was
not the case. For other possible matrices where σjdo not control the quadratic terms in
the solenoidal basis—but instead let’s say that σjare identical to the first four si—then
it is easy to find cells where the condition number of the resulting matrix is very high, so
the choice of the limited degrees of freedom σjis important.
3.2.2 Optimisation and cost functions
We have used optimisation to determine the remaining four degrees of freedom by min-
imising a cost function in R4. This optimisation is performed for each cell to find the
coefficients resulting in the lowest cost, which should correspond to the most stable solu-
tion with an appropriate choice of cost function. For each choice of cost function we get
a new method with unique properties.
An ideal cost function will keep the limited solution very close to the original non-
limited solution while still damping non-physical oscillations. To achieve this we have
selected to first calculate a component-wise slope limited solution uHT by applying the hi-
erarchical Taylor slope limiter described in section 3.1 to each of the velocity components.
This makes the result frame dependent, which is not desirable, but sadly unavoidable
when using a scalar field slope limiter for a vector field.
13
The uHT field is used as the target in the optimisation since it maintains the correct
convergence order. This target field does not keep all the solenoidal invariants, so it is
always slightly out of reach. The minimum and maximum allowable velocity component
values are computed for a set of points along each cell boundary by considering the
average values in the neighbouring cells just like in the hierarchical Taylor slope limiter.
The range of allowable values is extended to include the already limited field uHT in order
to avoid limiting at smooth maxima.
The cost function is implemented in two steps. First take the coefficients sicorre-
sponding to the non-limited velocity field and use these in equation (34) to calculate the
five invariants from equation (32). Then, for a set of points Pi∈∂K along the perime-
ter of a cell K, compute for each velocity component uj(j∈[1,2]) the current value
uij =uj(Pi), the allowable minimum uij,min, the preferred target uij,HT, and the allowable
maximum uij,max. The cost at point Pifor velocity component ujis then the squared
distance from the target normalised by the range of allowed values,
Cij =uij −uij,HT
uij,max −uij,min 2
.(35)
As the second step, if the current value uij is higher than uij,max, add
Cij,κ =κLuij −uij,max
uij,max −uij,min 2
+κC,(36)
and similar if uij < uij,min. If the current value is inside the stable bounds, then Cij,κ = 0.
We have used κL= 1.0 and κC= 1.0 as additional penalties for going outside the stable
zone. We have used the six Lagrangian nodes as the points Pi, i.e., the vertices and facet
midpoints. The total cost for the cell Kto be optimised is then calculated as
CK=
6
X
i=1
2
X
j=1
(Cij +Cij,κ).(37)
3.2.3 Optimized velocity fields
The cost function in equation (37) is unfortunately not able to steer even a perfect
optimisation routine towards a solution that is completely free from spurious local maxima
in all cells. There are three times more contributions to the cost function than the
degrees of freedom σ1· · · σ4in each cell—and that is when the cost function is looking
for overshoots only in the Lagrange nodes, there might still be local maxima between the
Lagrange nodes.
Detailed studies of the cost functions for some selected problematic cells show that
there are cells where there does not exist any point in the four dimensional coefficient
space σjwhere the cost is below κC= 1.0. Using a brute force optimisation algorithm
that explores the entire solution space would not help for those cells. In conclusion, it
is likely not possible—by cell-wise slope limiting—to reconstruct a velocity field that is
both solenoidal and free from local maxima based on the presented reduced basis and the
chosen criteria to detect local overshoots.
14
3.3 A split solenoidal slope limiting algorithm
Motivated by the above findings we introduce separate limiters for the convected velocity
uand the convecting velocity w. Looking at the convected and convecting velocity fields
as separate, but related, is inspired by the common practice of using an explicit convecting
velocity to linearise the Navier-Stokes equations—which is also what we have done in
equation (14). A similar splitting is done in cell-centered finite volume schemes where
the convecting velocity field is interpolated to the facets while the unknown convected
velocity field consists of cell averages. The convecting velocity can be partially slope
limited in a way that does not compromise on the solenoidal properties, or left entirely
unlimited. The convected velocity field must be slope limited to avoid instabilities. The
overall algorithm is shown in figure 2.
DG solver
u
BDM proj.
u
Slope lim.
w u
un-1
un-2
un-3
...
wn-1
wn-2
wn-3
...
Figure 2: The slope limit-
ing algorithm splits the ve-
locity field into two separate
time histories. Shaded veloc-
ities are solenoidal when the
box boundary is a continuous
line (grey background) and
free from local maxima when
the line is dotted (light green
background).
In the following results section we have applied three different slope limiting proce-
dures for the velocity field. The first is the naive unlimited method, the second applies the
hierarchical Taylor polynomial based limiter to each velocity component of the convected
velocity, and applies no limiting to the convecting velocity. The third method applies
slightly different versions of the solenoidal slope limiter from section 3.2.2 to the two
velocity fields. The convecting velocity field is limited without regards for the resulting
cost function value in each cell; hence, for some cells the suppression of local maxima
will not be successful. The convected velocity field is treated similarly, but here the cell
velocity fields are replaced with the component-wise limited fields from the hierarchical
Taylor polynomial based limiter, uHT, in the cells where the local maxima suppression
was unsuccessful.
4 Results
The methods described above have been implemented in Ocellaris (Landet, 2017), a two-
phase solver framework which is built on top of FEniCS (Logg, Mardal, and Wells, 2012)
and implemented in a mix of Python and C++. The optimiser employed is the BFGS
15
algorithm as implemented in SciPy (Jones et al., 2001), but for efficiency reasons we have
written the optimisation algorithm along with the cost function in C++ to decrease the
running time of the solver. The hierarchical Taylor based slope limiter for scalar fields
has been implemented in C++ and does not contribute significantly to the running time
of the solver. The rest of the Ocellaris solver is implemented in Python and depends
on the C++ code generation facilities in FEniCS to utilize the computational resources
optimally. The source code, input files, and scripts to reproduce the figures shown below
can be found in Landet (2017).
In the following text we will compare three numerical algorithms, (i) a naive DG imple-
mentation without any slope limiters, (ii) a simple velocity slope limiting implementation
using the scalar Taylor based limiter on each of the convected velocity components and no
limiting of the convecting velocity (referred to as ‘Hierarchical Taylor’), and (iii) the more
involved cell-wise optimisation described above where both the convected and convecting
velocity fields are optimised with slightly different criteria (referred to as ‘Solenoidal’).
To properly test the presented limiters it is important to ensure that artificially high
viscosity is not a contributing factor to the method’s stability. For the air/water free sur-
face test cases we have applied ν= 1.0×10−6m2s−1for both phases, which is artificially
low for the air phase. With this choice of viscosity the non-linear convective instability im-
pacts the results early in the simulations. This is also true for, e.g., ν= 1.0×10−4m2s−1,
but not for very high kinematic viscosities, when νapproaches 1.0. Note that the dy-
namic viscosity µ—the parameter that goes into the weak form in equation (19)—is not
the same in the two phases, it has the same factor 1000 jump as the density.
4.1 Taylor-Green vortex
The first test case is a study of the effect of slope limiting on the spatial convergence of
the numerical scheme by considering a Taylor-Green vortex where the density is constant
and the solution u= [u, v], p is given by,
u=−sin(πy) cos(πx) exp(−2π2νt)
v= sin(πx) cos(πy) exp(−2π2νt) (38)
p=−1/4ρ(cos 2πx + cos 2πy) exp(−4π2νt)
for t∈[0,1] on a domain Ω = {(x, y)∈[0,2] ×[0,2]}. This test case is often solved on
a periodic domain, but we use Dirichlet boundary conditions for uto test the effect of
the boundary. Initial conditions are given both at t= 0 and t=−∆tto be able to use
second order time stepping from the start. A constant time step of ∆t= 0.01 is used and
the kinematic viscosity is ν= 0.005.
A slope limited solution should be identical to the non-limited solution for this test
case—there are no non-smooth maxima in the true solution. Still, exact equivalence is
not obtained since the DG FEM vector field is only a numerical approximation. The
main discrepancy is in the handling of boundary conditions. Vertices on the boundary
will be missing neighbour cells and may hence be unnecessarily slope limited. Figure 3
shows that the results converge towards the non-limited solution when slope limiting is
avoided in the boundary facing cells. When boundary cells are included in the limiter
the expected third order L2convergence rate is still obtained, but the constant is larger.
16
Figure 3: Spatial convergence rate on the Taylor-Green vortex test case. The rate of
convergence is as expected, but missing neighbour cell info near boundaries in the limiters
gives a larger constant in the L2error plot.
4.2 Dam break
The first two-phase flow example is a classic a dam break in a box simulation as illustrated
in figure 4 based on the experiments by Martin and Moyce (1952). This is one of the
most commonly used test cases in the high Reynolds number, low surface tension regime.
This is the regime most interesting for studying marine and offshore structures, which is
the ultimate goal of the method.
5a
2a
a
3a
g
Figure 4: Dam break simulation geometry
The 2D rectangular water column is 1awide and 2ahigh and starts at rest in the
lower left corner of a box filled with air which is 5awide and 3ahigh. The size of the
column is the same as in the experiments, a= 2.25 in = 0.05715 m. The two phases are
water and air with densities 1000 kg m−3and 1.0 kg m−3respectively. This corresponds to
tables 2 and 6 in Martin and Moyce (1952). The acceleration of gravity is g= 9.81 m s−2
in the negative y-direction and the kinematic viscosity is ν= 1.0×10−6m2s−1for both
phases.
The total kinematic energy, Ek=RΩ1/2ρu·udx, is shown in figure 5 as a function of
time for the slope limited and the non-limited methods. From the start, with both fluids
at rest, the kinetic energy increases as the water mass starts to flow down and towards
the right wall. There is a slight reduction as the water hits this wall at approximately
t= 0.19 s. The Gibbs oscillations start to dominate the non-limited solution after a
short time and from t= 0.04 s the solution is non-physical and completely dominated by
17
numerical errors. In the rest of this paper we will remove the non-limited method from
the results as it leads to non-physical solutions in all cases.
Figure 5: Kinetic energy as a function of time for three slope limiting strategies
The results of the two slope limited methods are compared to the experimental results
in figure 6. The experimental data points are the ensemble averages of the results reported
for the same geometry by Martin and Moyce (1952). We have employed free slip boundary
conditions, and hence the surge front moves slightly faster than in the experimental
results, but the qualitative behaviour is correct. The maximum height of the water
column matches very well until the water hits the domain boundary and creates a jet
shooting up along the right wall at t≈0.19 s which corresponds to T≈3.5 and τ≈2.8.
The right wall was placed much further away in the experimental setup.
As can be seen in figures 5 and 6, the two investigated limiters perform very simi-
larly on the dam break test case. The ‘Solenoidal’ method requires significantly more
computation per time step than the component-wise Taylor based limiter and is more
complex to implement. One reason why one could still consider using this method is
that the optimisation may keep the difference between the convected and the convect-
ing velocity fields smaller and hence closer to the true solution. To test how much this
optimisation contributes we have calculated the difference between the velocity fields,
∆U=ku−wk/kuk, for all time steps and the results can be seen in figure 7.
The difference is large in the beginning since both uand ware close to zero at this
time. After the initial phase the difference is low for both limiters. We can see that
right before the water impacts the wall at approximately t= 0.19 s the slope limiters
start to perform differently for some time until the impact is over. The reason is that
the ‘Hierarchical Taylor’ slope limiter introduces a very high divergence in the convected
velocity field in the tank corner. The optimisation based limiter performs as expected
and the difference between the two fields is lower, but not entirely removed.
Later in the time series there is a new event when the water is shooting up in a thin
jet along the right wall at approximately t= 0.27 s. Here we can see that the optimised
slope limiter is introducing slightly higher ∆Udifferences than the Taylor based slope
limiter, quite contrary to the intention. We have no direct explanation of this, except
that it seems related to the locally high shear in the vertical velocity field.
18
(a) Surge front position (b) Height of water column
Figure 6: Comparison of the numerical results with experimental results by Martin and
Moyce (1952). The vertical axis scaling is Z=z/a and H=η/2a. Measured from the
lower left corner, zis the surge front position and ηis the height of the water column.
The horizontal axis scaling is T=tp2g/a and τ=tpg/a.
Figure 7: Dam break; comparison of the two velocity slope limiters in terms of, ∆U, the
scaled L2distance between convected and convecting velocity fields.
19
Figure 8: Tank filling results. The first row shows the density field at t= 0.167, 0.333,
0.500, 0.667, 0.833, and 1.000, the second and third row have the same temporal spacing,
hence the lower rightmost image shows the simulation end point, t= 3 s. The finest
resolution is shown, using the ‘Hierarchical Taylor’ slope limiter.
4.3 Tank filling
To provide a more challenging test of the method’s stability with a large and complex
free surface we have performed a set of tank filling simulations inspired by Guermond,
de Luna, and Thompson (2017), but unlike in their work we have used physical properties
closer to normal water and air, and the results are hence more energetic. The physical
properties are the same as in the dam break test case.
The geometry of the tank can be seen along with the resulting density field in figure 8.
The domain is the unit square and the inlet is located between 0.5 m and 0.625 m above
the floor on the left wall, 1/8 m wide. The outlet is located in the roof between 0.5 m
and 0.625 m from the left wall and is also 1/8 m wide. The inlet and outlet velocities are
prescribed as 2 m s−1, constant for all time. At t=0 the tank is completely filled with air,
the velocity inside is zero, and the pressure is hydrostatic.
The results are computed on a regular mesh with triangular cells and no mesh grading.
There are eight cells across the inlet and 64x64x2 cells in total. The domain is first
divided into squares and then these are subdivided into triangles. The time step is
∆t= 0.0001 s. The solution has also been calculated on a finer mesh with 16 elements
across the inlet, 128x128x2 cells in total. The time step was then ∆t= 5.0×10−5s
and only the ‘Hierarchical Taylor’ slope limiter was tested. The results from all three
simulations qualitatively exhibit the same physics and look similar. The density field
from the simulations on the fine mesh is shown in figure 8 at regular intervals up to the
maximum simulation time of 3s.
The performance of the slope limiters is compared in terms of stability and conserva-
tion. We have calculated the total mass and energy inside the tank and the results can be
seen in figures 9a and 9b. The analytical solutions in the figures are computed under the
assumption that only air exits through the outlet, which is not true in the second half of
the simulation. When looking at conservation of mass in the first part of the simulations,
only very minor differences can be seen between the methods; both methods preserve
20
(a) Total mass (b) Total energy
Figure 9: Tank filling: comparison of the two velocity slope limiters in terms of conser-
vation of mass and energy
Figure 10: Tank filling: comparison of the two velocity slope limiters in terms of, ∆U,
the scaled L2distance between convected and convecting velocity fields.
mass very well. This is exactly as expected since mass conservation is the key invariant
in both slope limiters—the convecting velocity field is always solenoidal.
The loss of mass and energy through the outlet—starting at approximately 1.7 seconds—
can also be seen in the time history of the total energy in figure 9b. Up to that point the
optimisation based solenoidal slope limiter can be seen to preserve total energy better
than the component-wise hierarchical Taylor based limiter. Refining the mesh improves
the results from the hierarchical Taylor based limiter, but the optimised solenoidal limiter
still outperforms it, even if it is running on a much coarser mesh.
Both methods are stable for the duration of the simulation, but some spurious in-
creases in total energy can be seen in the ‘Solenoidal’ method. The ‘Hierarchical Taylor’
method has no noticeable spurious increases, at the expense of slowly loosing energy.
The reason for the spurious increase in energy in the ‘Solenoidal’ method around 1.2 s in
figure 9b may be the fact that local maxima are allowed to occur in between the Lagrange
polynomial nodes.
The difference between the convected and the convecting velocity, ∆U, has been
calculated in the same manner as in the dam break test case and the results can be
seen in figure 10. Both methods preserve a low difference between the velocity fields for
most of the time steps, but both have instances when the difference grows larger. This is
particularly noticeable in the ‘Solenoidal’ method which has some large spikes. In general
21
the optimised method maintains a lower difference, on par with the refined hierarchical
Taylor based method. Both methods are shown to recover from temporary spikes and
return to a low difference, which is the correct solution.
5 Discussion
We have shown how convective instabilities in higher order two-phase flow simulations
can be stabilised without sacrificing mass conservation. Our efforts have been focused on
the use of slope limiters to achieve this stabilisation. Using optimisation on a solenoidal
reduced basis to create one single velocity field that is both stable and solenoidal was
investigated. The results show strong indications that this may in fact not be possible.
To overcome this problem, two methods that treats the convected velocity field differently
from the convecting velocity field were proposed. These methods combine exact mass and
momentum conservation with convective stability for two-phase simulations containing
large density jumps inside the computational domain.
The suggested methods have been tested to see if there were any negative impact
on the solution of smooth problems; and optimal convergence on a Taylor-Green vortex
problem was show. Further, a two-dimensional dam break with a factor 1000 sharp den-
sity jump in the computational domain was tested. Such problems are not possible to
handle without some form of higher order convective stabilisation. Both slope limiting
strategies handled this problem well. Finally, we have studied mass and energy conser-
vation in a very energetic flow—a tank filling simulation. A summary of our results is
that slope limiting is a possible way to stabilise a mass conserving discontinuous Galerkin
method for the incompressible Navier-Stokes equation with large density jumps.
There are other methods than slope limiting for stabilising high order convective in-
stabilities in discontinuous Galerkin methods. Instead of using an explicit post processing
operator one can add specifically tailored implicit damping to the cells near the free sur-
face with a method such as the entropy viscosity method, (Zingan et al., 2013). Another
method that has been used is to selectively reduce the polynomial approximation order in
cells near the free surface, a special use of p-adaptivity (Robert J. Labeur, private com-
munication, 2017). The advantages of the explicit slope limiting method is the decoupling
of the convective stabilisation from the definition of the weak form and the assembly of
the equation system. The relative ease of implementation is also a large advantage of the
component-by-component velocity limiting method. Some type of scalar slope limiter is
likely to be already present in a DG FEM code. Both the presented methods require no
extra method dependent viscosity parameters to be derived, and no selective p-adaptivity
is needed in the software. There is no need to tune parameters in the methods to achieve
stability. Extension to higher order is also straight forward.
Acknowledgements
The authors are thankful to Miroslav Kuchta for proofreading and valuable discussion
related to this work.
22
References
Arnold, Douglas Norman. 1982. “An interior penalty finite element method with discon-
tinuous elements.” SIAM journal on numerical analysis 19 (4): 742–760.
Baker, Garth A., Wadi N. Jureidini, and Ohannes A. Karakashian. 1990. “Piecewise
Solenoidal Vector Fields and the Stokes Problem.” SIAM Journal on Numerical Anal-
ysis 27 (6): 1466–1485.
Brezzi, Franco, and Michel Fortin. 1991. Mixed and Hybrid Finite Element Methods.
no. 15 In Springer Series in Computational Mathematics. Springer-Verlag, New York.
Cockburn, Bernardo, Guido Kanschat, and Dominik Sch¨otzau. 2004. “The local discon-
tinuous Galerkin method for the Oseen equations.” Mathematics of Computation 73
(246): 569–594.
Cockburn, Bernardo, Guido Kanschat, and Dominik Sch¨otzau. 2005. “A locally conser-
vative LDG method for the incompressible Navier-Stokes equations.” Mathematics of
Computation 74 (251): 1067–1096.
Cockburn, Bernardo, and Chi-Wang Shu. 1998. “The Runge–Kutta Discontinuous
Galerkin Method for Conservation Laws V.” Journal of Computational Physics 141
(2): 199–224.
Cockburn, Bernardo, and Chi-Wang Shu. 2001. “Runge-Kutta discontinuous Galerkin
methods for convection-dominated problems.” Journal of scientific computing 16 (3):
173–261.
Epshteyn, Yekaterina, and B´eatrice Rivi`ere. 2007. “Estimation of penalty parameters for
symmetric interior penalty Galerkin methods.” Journal of Computational and Applied
Mathematics 206 (2): 843–872.
Guermond, Jean-Luc, Manuel Quezada de Luna, and Travis Thompson. 2017. “An con-
servative anti-diffusion technique for the level set method.” Journal of Computational
and Applied Mathematics 321: 448–468.
Guermond, Jean-Luc, and Luigi Quartapelle. 2000. “A Projection FEM for Variable
Density Incompressible Flows.” Journal of Computational Physics 165 (1): 167–188.
Guermond, Jean-Luc, and Abner Salgado. 2009. “A splitting method for incompressible
flows with variable density based on a pressure Poisson equation.” Journal of Compu-
tational Physics 228 (8): 2834–2846.
Hirt, Cyril W., and Billy D. Nichols. 1981. “Volume of fluid (VOF) method for the
dynamics of free boundaries.” Journal of computational physics 39 (1): 201–225.
Hundsdorfer, Willem, Steven J. Ruuth, and Raymond J. Spiteri. 2003. “Monotonicity-
Preserving Linear Multistep Methods.” SIAM Journal on Numerical Analysis 41 (2):
605–623.
23
Jones, Eric, Travis Oliphant, Pearu Peterson, and others. 2001. “SciPy: Open source
scientific tools for Python.” http://www.scipy.org/.
Kirby, Robert C., Anders Logg, Marie E. Rognes, and Andy R. Terrel. 2012. “Common
and unusual finite elements.” In Automated Solution of Differential Equations by the
Finite Element Method, Lecture Notes in Computational Science and Engineering. 95–
119. Springer, Berlin, Heidelberg.
Kuzmin, Dmitri. 2010. “A vertex-based hierarchical slope limiter for -adaptive discon-
tinuous Galerkin methods.” Journal of Computational and Applied Mathematics 233
(12): 3077–3085.
Kuzmin, Dmitri. 2013. “Slope limiting for discontinuous Galerkin approximations with
a possibly non-orthogonal Taylor basis.” International Journal for Numerical Methods
in Fluids 71 (9): 1178–1190.
Landet, Tormod. 2017. “Ocellaris DG-FEM software and input files to reproduce results.”
Zenodo. http://doi.org/10.5281/zenodo.845352.
Leonard, Brian Phillip. 1988. “Simple high-accuracy resolution program for convective
modelling of discontinuities.” International journal for numerical methods in fluids 8
(10): 1291–1318.
Liu, Chun, and Noel J. Walkington. 2007. “Convergence of Numerical Approximations
of the Incompressible Navier–Stokes Equations with Variable Density and Viscosity.”
SIAM Journal on Numerical Analysis 45 (3): 1287–1304.
Logg, Anders, Kent-Andre Mardal, and Garth Wells. 2012. Automated Solution of Dif-
ferential Equations by the Finite Element Method: The FEniCS Book. Springer Science
& Business Media.
Martin, John Christopher, and William James Moyce. 1952. “Part IV. An Experimental
Study of the Collapse of Liquid Columns on a Rigid Horizontal Plane.” Philosophical
Transactions of the Royal Society of London A: Mathematical, Physical and Engineer-
ing Sciences 244 (882): 312–324.
Michoski, C., C. Dawson, E. J. Kubatko, D. Wirasaet, S. Brus, and J. J. Westerink. 2016.
“A Comparison of Artificial Viscosity, Limiters, and Filters, for High Order Discontinu-
ous Galerkin Solutions in Nonlinear Settings.” Journal of Scientific Computing 66 (1):
406–434. https://link.springer.com/article/10.1007/s10915-015-0027-2.
Muzaferija, Samir, Milovan Peric, Pierre C. Sames, and Thomas E. Schellin. 1998. “A
Two-Fluid Navier-Stokes Solver to Simulate Water Entry.” In Proceedings from the
22nd symposium on naval hydrodynamics, 277–289. Washington, DC.
Nitsche, Joachim A. 1971. “ ¨
Uber ein Variationsprinzip zur L¨osung von Dirichlet-
Problemen bei Verwendung von Teilr¨aumen, die keinen Randbedingungen unterworfen
sind.” Abhandlungen aus dem Mathematischen Seminar der Universit¨at Hamburg 36
(1): 9–15.
24
N´ed´elec, Jean-Claude. 1986. “A new family of mixed finite elements in R3.” Numerische
Mathematik 50 (1): 57–81.
Olsson, Elin, and Gunilla Kreiss. 2005. “A conservative level set method for two phase
flow.” Journal of Computational Physics 210 (1): 225–246.
Osher, Stanley, and James A. Sethian. 1988. “Fronts propagating with curvature-
dependent speed: algorithms based on Hamilton-Jacobi formulations.” Journal of com-
putational physics 79 (1): 12–49.
Peskin, Charles S. 2002. “The immersed boundary method.” Acta Numerica 11: 479–517.
Pyo, Jae-Hong, and Jie Shen. 2007. “Gauge–Uzawa methods for incompressible flows
with variable density.” Journal of Computational Physics 221 (1): 181–197.
Shahbazi, Khosro, Paul F. Fischer, and Christopher Ross Ethier. 2007. “A high-order dis-
continuous Galerkin method for the unsteady incompressible Navier-Stokes equations.”
Journal of computational physics 222 (1): 391–407.
Sussman, Mark, Peter Smereka, and Stanley Osher. 1994. “A Level Set Approach for
Computing Solutions to Incompressible Two-Phase Flow.” Journal of Computational
Physics 114 (1): 146–159.
Sweby, Peter K. 1984. “High resolution schemes using flux limiters for hyperbolic conser-
vation laws.” SIAM Journal on Numerical Analysis 21 (5): 995–1011.
Tour´e, Mamadou Kabirou, Adil Fahsi, and Azzeddine Soula¨ımani. 2016. “Stabilised finite-
element methods for solving the level set equation with mass conservation.” Interna-
tional Journal of Computational Fluid Dynamics 30 (1): 38–55.
Ubbink, Onno. 1997. “Numerical prediction of two fluid systems with sharp interfaces.”
Ph.D. thesis. Imperial College, University of London.
Unverdi, Salih Ozen, and Gr´etar Tryggvason. 1992. “A front-tracking method for viscous,
incompressible, multi-fluid flows.” Journal of Computational Physics 100 (1): 25–37.
Zingan, Valentin, Jean-Luc Guermond, Jim Morel, and Bojan Popov. 2013. “Implemen-
tation of the entropy viscosity method with the discontinuous Galerkin method.” Com-
puter Methods in Applied Mechanics and Engineering 253: 479–490.
25
