Geometric Error of Finite Volume Schemes for Conservation Laws on Evolving Surfaces

1.4 MB·English

by  Jan Giesselmann

GEOMETRIC ERROR OF FINITE VOLUME SCHEMES FOR
CONSERVATION LAWS ON EVOLVING SURFACES
JAN GIESSELMANN∗, THOMAS MU¨LLER∗∗
∗WEIERSTRASSINSTITUTE,
MOHRENSTR.39,D-10117BERLIN,GERMANY
Email: [email protected]
∗∗ABTEILUNGFU¨RANGEWANDTEMATHEMATIK,UNIVERSITA¨TFREIBURG,
3
HERMANN-HERDER-STR.10,D-79104FREIBURG,GERMANY
1
Email: [email protected]
0
2
n Abstract. Thispaperstudiesfinitevolumeschemesforscalarhyperbolicconservationlaws
a on evolving hypersurfaces of R3. We compare theoretical schemes assuming knowledge of
J all geometric quantities to (practical) schemes defined on moving polyhedra approximating

7 the surface. For the former schemes error estimates have already been proven, but the
implementationofsuchschemesisnotfeasibleforcomplexgeometries. Thelatterschemes,

] in contrast, only require (easily) computable geometric quantities and are thus more useful
A
foractualcomputations. Weprovethatthedifferencebetweenapproximatesolutionsdefined
N by the respective families of schemes is of the order of the mesh width. In particular, the
. practicalschemeconvergestotheentropysolutionwiththesamerateasthetheoreticalone.
h
Numerical experiments show that the proven order of convergence is optimal.
t
a
m
[

1. Introduction
1
v
Hyperbolic conservation laws serve as models for a wide variety of applications in contin-
7
8 uum dynamics. In many applications the physical domains of these problems are stationary
2 or moving hypersurfaces. Examples of the former are in particular geophysical problems [27]
1
and magnetohydrodynamics in the tachocline of the sun [15, 24]. Examples of the latter in-
.
1 cludetransportprocessesoncellsurfaces[22], surfactantflowoninterfacesinmultiphaseflow
0
[5] and petrol flow on a time dependent water surface. There are several recent approaches
3
1 to the numerical computation of such equations. Numerical schemes for the shallow water
: equations on a rotating sphere can be found in [6, 16, 23]. For the simulation of surfactant
v
i flow on interfaces we refer to [1, 4, 17]. As we are interested in numerical analysis we focus
X
on nonlinear scalar conservation laws as a model for these systems. The intense study of
r
a conservation laws posed on fixed Riemannian manifolds started within the last years. There
are results about well-posedness [3, 10, 20] of the differential equations and about the conver-
gence of appropriate finite volume schemes [2, 13, 14, 19]. For recent developments on finite
volume schemes for parabolic equations we refer to [21].
1991 Mathematics Subject Classification. 65M08 and 35L65 and 58J45.
Key words and phrases. hyperbolic conservation laws and finite volume schemes and curved surfaces and
error bound.
1
2 JanGiesselmann,ThomasMu¨ller
In the previous error analysis for finite volume schemes approximating nonlinear conser-
vation laws on manifolds the schemes were defined on curved elements lying on the curved
surface and it was assumed that geometric quantities like lengths, areas and conormals are
known exactly. While this is a reasonable assumption for schemes defined on general Rie-
mannian manifolds or even more general structures [18] with no ambient space, most engi-
neering applications involve equations on hypersurfaces of R3 and one aims at computing the
geometry with the least effort. This is in particular important for moving surfaces where
the geometric quantities have to be computed in each time step. Now the question arises to
which extent an approximation of the geometry influences the order of convergence of the
scheme.
We consider the following initial value problem, posed on a family of closed, smooth
hypersurfaces Γ = Γ(t) ⊂ R3. For a derivation cf. [11, 25]. For some T > 0, find
u : G := (cid:83) Γ(t)×{t} → R with
T t∈[0,T]
(1.1) u˙ +u∇ ·v+∇ ·f(u,·,·) = 0 in G ,
Γ Γ T
(1.2) u(·,0) = u on Γ(0),
0
where v is the velocity of the material points of the surface and u : Γ(0) → R are initial
0
data. For every u¯ ∈ R, t ∈ [0,T] the flux f(u¯,·,t) is a smooth vector field tangential to Γ(t),
which depends Lipschitz on u¯ and smoothly on t. Moreover, we impose the following growth
condition
(1.3) |∇ ·f(u¯,x,t)| ≤ c+c|u¯| ∀u¯ ∈ R,(x,t) ∈ G
Γ T
for some constant c > 0. By u˙ we denote the material derivative of u which is given by
d
u˙(Φ (x),t) := u(Φ (x),t),
t t
dt
where Φ : Γ(0) → Γ(t) is a family of diffeomorphisms depending smoothly on t, such that Φ
t 0
is the identity on Γ(0). Obviously this excludes changes of the topology of Γ. We will assume
that the movement of the surface and also the family Φ is prescribed. A main result of this
t
paper is a bound for the difference between two approximations of u. In particular, we will
give an estimate for the difference between the flat approximate and the curved approximate
solution. By curved approximate solution we refer to a numerical solution given by a finite
volumeschemedefinedonthecurvedsurface,cf. Section2.2,andbyflatapproximatesolution
we refer to a numerical solution given by a finite volume scheme defined on a polyhedron
approximating the surface, cf. Section 2.3. We will see that the arising geometry errors can
be neglected compared to the error between the curved approximate solution and the exact
solution, i.e. both approximate solutions converge to the entropy solution with the same
convergence rate. We will present numerical examples showing that the proven convergence
rateisoptimalundertheassumptionsforthenumericalanalysis. However,formostnumerical
experiments we observe higher orders of convergence.
Our analysis also indicates that the geometry error poses an obstacle to the construction
of higher order schemes. To this end we perform numerical experiments underlining in which
mannertheorderofconvergenceofthehigherorderschemeisrestrictedbytheapproximation
of the geometry. This shows that to obtain higher order convergence also the geometry of
the manifold has to be approximated more accurately, cf. [9] in a finite element context.
GeometricErrorofFVSchemesforConservationLawsonEvolvingSurfaces 3
Theoutlineofthispaperisasfollows. InSection2wereviewthedefinitionoffinitevolume
schemes on moving curved surfaces and define finite volume schemes on moving polyhedra
approximatingthesurfaces. Theapproximationerrorsforgeometricquantitiesareestablished
in Section 3. Section 4 is devoted to estimating the difference between the curved and the
flat approximate solution. Finally, numerical experiments are given in Section 5.
2. The Finite Volume Schemes
ThissectionisdevotedtotheconstructionofafamilyoftriangulationsT (t)ofthesurfaces
h
suitably linked to polyhedral approximations Γ (t) of the surfaces. Afterwards we will recall
h
the definition of a finite volume scheme on T (t) which was considered in the hitherto error
h
analysis and define a finite volume scheme on Γ (t) which is an algorithm only relying on
h
easily computable quantities. We mention that our triangulation as well as the definition of
the finite volume scheme on Γ is in the same spirit as the one Lenz et al. [21] used for the
h
diffusion equation on evolving surfaces.
2.1. Triangulation. We start by mentioning that there are neighbourhoods N(t) ⊂ R3 of
Γ(t) such that for every x ∈ N(t) there is a unique point a(x,t) ∈ Γ(t) such that
(2.1) x = a(x,t)+d(x,t)ν (a(x,t)),
Γ(t)
where d(·,t) denotes the signed distance function to Γ(t) and ν (a(x,t)) the unit normal
Γ(t)
vectortoΓ(t)pointingtowardsthenon-compactcomponentofR3\Γ(t). See[12]forexample.
Let us choose a polyhedral surface Γ (0) ⊂ N(0) which consists of flat triangles such that
h
the vertices of Γ (0) lie on Γ(0), and h is the length of the longest edge of Γ (0). In addition
h h
we impose that the restriction of a| : Γ (0) → Γ(0) is one-to-one. We define Γ (t) as the
Γ (0) h h
h
polyhedral surface that is constructed by moving the vertices of Γ (t) via the diffeomorphism
h
Φ and connecting them with straight lines such that all triangulations share the same grid
t
topology. A triangulation T¯ (t) of Γ (t) is automatically given by the decomposition into
h h
faces. We define the triangulation T (t) on Γ(t) as the image of T¯ (t) under a(·,t)| .
h h Γ (t)
h
We will denote the curved cells with K(t) and the curved faces with e(t). A flat quantity
corresponding to some curved quantity is denoted by the same letter and a bar, e.g. let
e(t) ⊂ Γ(t) be a curved face then e¯(t) = (a(·,t)| )−1(e(t)). In order to reflect the fact that
Γ (t)
h
all triangulations share the same grid topology we introduce the following misuse of notation.
We denote by K the family of all curved triangles relating to the same triangle K¯(0) on
Γ (0). We do the same for e,K¯,e¯. Analogously by T we denote the family of such families
h h
of triangles K.
For later use we state the following Lemma summarizing geometric properties, whose
derivation can be found in [12].
Lemma 2.1. Let Γ (t) be a polyhedral approximation of Γ(t) as described above then there
h
exists C = C(T) such that for all t ∈ [0,T]
(1) ν = ∇d(·,t),
Γ(t)
(2) (cid:107)d(·,t)| (cid:107) ≤ Ch2 .
Γ (t) L∞(Γ (t))
h h
We will use the following notation. By h := diam(K(t)) we denote the diameter of
K(t)
each cell, furthermore h := max max h and |K(t)|, |∂K(t)| are the Hausdorff
t∈[0,T] K(t) K(t)
4 JanGiesselmann,ThomasMu¨ller
measures of K(t) and the boundary of K(t) respectively. When we write e(t) ⊂ ∂K(t) we
mean e(t) to be a face of K(t).
We need to impose the following assumption uniformly on all triangulations T¯ (t). There
h
is a constant number α > 0 such that for each flat cell K¯(t) ∈ T¯ (t) we have
h
αh2K¯(t) ≤ (cid:12)(cid:12)K¯(t)(cid:12)(cid:12),
(2.2)
α(cid:12)(cid:12)∂K¯(t)(cid:12)(cid:12) ≤ hK¯(t).
Later on, we will see that (2.2) implies the respective estimate for the curved triangulation,
cf. Remark 3.4. A consequence of (2.2) is that 2α2h is a lower bound of the radius of
K¯(t)
the inner circle of K¯(t), which implies that the sizes of the angles in K¯(t) are bounded from
below. Furthermore we denote by κ(x,t) the supremum of the spectral norm of ∇ν (x).
Γ(t)
By straightforward continuity and compactness arguments κ is uniformly bounded in space
and time.
2.2. The Finite Volume Scheme on Curved Elements. In this section we will briefly
reviewthenotionoffinitevolumeschemesonmovingcurvedsurfaces. Weconsiderasequence
of times 0 = t < t < t < ... and set I := [t ,t ]. Moreover we assign to each n ∈ N
0 1 2 n n n+1
and K ∈ T the term un approximating the mean value of u on (cid:83) K(t)×{t} and to each
h K t∈In
K ∈ T and face e ⊂ ∂K a numerical flux function fn : R2 → R, which should approximate
h K,e

(2.3) (cid:104)f(u(x,t),x,t),µ (x,t)(cid:105)de(t)dt,
K,e
In e(t)
where de(t) is the line element, µ (x,t) is the unit conormal to e(t) pointing outwards from
K,e
K(t) and (cid:104)·,·(cid:105) is the standard Euclidean inner product. Please note that µ (t) is tangential
K,e
to Γ(t). Then the finite volume scheme is given by

u0 := u (x)dΓ(0),
K 0
K(0)
(2.4) un+1 := |K(tn)| un − |In| (cid:88) |e(t )|fn (un ,un ),
K |K(t )| K |K(t )| n K,e K Ke
n+1 n+1
e⊂∂K
uh(x,t) := un for t ∈ [t ,t ),x ∈ K(t),
K n n+1
where K denotes the cell sharing face e with K and dΓ(0) is the surface element. For
e
the convergence analysis it was usually assumed [13, 19] that the used numerical fluxes are
uniformly Lipschitz, consistent, conservative and monotone. Additionally, the CFL condition
α2h
(2.5) t −t ≤
n+1 n
8L
has to be imposed to ensure stability, where L is the Lipschitz constant of the numerical
fluxes. Lax-Friedrichs fluxes satisfying this condition are usually defined by

1
(2.6) fn (u,v) := (cid:104)f(u,·,t) + f(v,·,t),µ (t)(cid:105)de(t)dt + λ(u − v),
K,e 2 K,e
In e(t)
where λ = 1(cid:107)∂ f(cid:107) is an artificial viscosity coefficient ensuring the monotonicity of fn and
2 u ∞ K,e
stabilizing the scheme.
GeometricErrorofFVSchemesforConservationLawsonEvolvingSurfaces 5
2.3. The Finite Volume Scheme on Flat Elements. In this section we define a finite
volume scheme on T¯ which is in the same spirit as (2.4) but only relies on easily accessible
h
geometrical information. We assume that f is smoothly extended from G to the whole
T
(cid:83)
of N(t) × {t}. We want to point out that the calculation of areas and lengths is
t∈[0,T]
straightforward for flat elements. As well, the approximation of integrals can be achieved
using quadrature formulas by mapping cells and edges to a standard triangle and the unit
interval, respectively, using affine linear maps. In this fashion we obtain for every time
t ∈ [0,T] quadrature operators Q : C0(K¯(t)) → R, and Q : C0(e¯(t)) → R of order
K¯(t) e¯(t)
p ,p ∈ N,respectively. InadditionforanycompactintervalI ⊂ [0,T]thetermQ : C0(I¯) →
1 2 I
R denotes a quadrature operator of order p ∈ N.
3
Beforewecanusethequadratureoperatorstodefinenumericalfluxesweneedtodetermine
the ”discrete” conormals. To each flat triangle K¯(t) we fix a unit normal ν¯ by imposing
K¯(t)
(2.7) (cid:104)ν¯ ,ν (y)(cid:105) > 0,
K¯(t) Γ(t)
where y is the barycentre of K(t). We will see in Lemma 3.2 that ν¯ converges to ν (y)
K¯(t) Γ(t)
for h → 0. To each face e¯(t) and adjacent cell K¯(t) there is a unique unit tangent vector
¯t such that ν¯ ×¯t is a conormal to e¯(t) pointing outward from K¯(t). Hence
K¯(t),e¯(t) K¯(t) K¯(t),e¯(t)
this vector product is one candidate for µ¯ . However in general
K¯(t),e¯(t)
(2.8) ν¯ ×¯t (cid:54)= ±(ν¯ ×¯t )
K¯(t) K¯(t),e¯(t) K¯e¯(t) K¯e¯(t),e¯(t)
such that a choice like
µ¯ = ν¯ ×¯t
K¯(t),e¯(t) K¯(t) K¯(t),e¯(t)
would lead to a loss of conservativity of the resulting numerical fluxes. Therefore we choose
1 (cid:16) (cid:17)
µ¯ := ν¯ ×¯t +ν¯ ×¯t .
K¯(t),e¯(t) 2 K¯(t) K¯(t),e¯(t) K¯e¯(t) K¯(t),e¯(t)
We define a numerical Lax-Friedrichs flux and a finite-volume scheme:
(cid:20) (cid:21)
1 1 (cid:16) (cid:17)
f¯ (u,v) := Q Q (cid:104)f(u,·,·)+f(v,·,·),µ¯ (cid:105)
K¯,e¯ |I | In 2|e¯(·)| e¯(·) K¯(·),e¯(·)
n
+λ(u−v),
1
u¯0 := Q (u ),
(2.9) K¯ |K¯(0)| K¯(0) 0
u¯n+1 := |K¯(tn)| u¯n − |In| (cid:88) |e¯(t )|f¯n (u¯n ,u¯n ),
K¯ |K¯(tn+1)| K¯ |K¯(tn+1)| n K¯,e¯ K¯ K¯e¯
e¯⊂∂K¯
u¯h(x,t) := u¯n , for t ∈ [t ,t ), x ∈ K(t),
K¯ n n+1
for some sufficiently large λ ≥ 0. Note that by (2.9) the function u¯h is defined on G .
4 T
3. Geometrical Estimates
Inthissectionwederiveestimatesfortheapproximationerrorsofthegeometricquantities.
Throughout this section we suppress the time dependence of all quantities. All the estimates
can be derived uniformly in time. To obtain the geometrical estimates, we introduce the
following lift operator.
6 JanGiesselmann,ThomasMu¨ller
Definition 3.1. Let U¯ ⊂ Γ and g¯ a function on U¯ then we define a function g¯l on a| (U¯)
h Γh
as
g¯l = g¯◦a|−1.
Γ
h
Similarly we define the inverse of this lift operator by
g−l = g◦a|
Γ
h
for a function g defined on some U ⊂ Γ.
We begin our investigation with the differences between the normal vectors of the flat and
curved elements.
Lemma 3.2. There is a constant C such that for all flat cells K¯ and every y ∈ K¯ we have
(cid:13) (cid:13)
(3.1) (cid:13)ν−l(y)−ν¯ (cid:13) ≤ Ch.
(cid:13) Γ K¯(cid:13)
The constant C depends on derivatives of d, in particular on κ.
Proof. WLOG we can assume that K¯ is a subset of {(x,y,0) ∈ R3|y < 0} such that e¯ =
{(s,0,0) ∈ R3|s ∈ [0,h ]} is one of its faces and (ν )−l(y) > 0 for some y ∈ K¯. We start by
e¯ Γ 3
showing that there exists some constant C > 0 such that
(3.2) |(ν ) | ≤ Ch, for i = 1,2.
Γ i
We recall that ν = ∇d, where d is the signed distance function to Γ. As the vertices of Γ
Γ h
lie on Γ we know that there exists (x,y,0) ∈ K¯ such that
d(0,0,0) = 0, d(h ,0,0) = 0, d(x,y,0) = 0.
e¯
Hence, the directional derivatives of d with respect to (x,y,0) and (1,0,0) need to van-
ish somewhere in K¯. Thus their absolute value is of order O(h) on K¯. Due to the angle
condition (2.2) an analogous inequality also holds for the directional derivative of d with
respect to (0,1,0). As the directional derivative of d with respect to (1,0,0), (0,1,0) co-
incides with (ν ) , (ν ) , respectively, this proves (3.2). This immediately implies (ν ) =
Γ 1 Γ 2 Γ 3
(cid:112)
± 1−O(h2) = ±1+O(h2). By assumption (ν ) = 1+O(h2) everywhere and by (2.7) we
Γ 3
have ν¯ = (0,0,1) which proves (3.1). (cid:3)
K¯
Lemma 3.3. For the difference between the length of a curved edge e and the corresponding
flat edge e¯ we have
(cid:12) (cid:12)
(3.3) (cid:12)(cid:12)|e| −1(cid:12)(cid:12) ≤ Ch2,
(cid:12)|e¯| (cid:12)
and for the difference between the area of a curved cell K and the corresponding flat cell K¯
we have
(cid:12) (cid:12)
(cid:12)|K| (cid:12)
(3.4) (cid:12) −1(cid:12) ≤ Ch2,
(cid:12)(cid:12)(cid:12)(cid:12)K¯(cid:12)(cid:12) (cid:12)(cid:12)
where C does not depend on h but on κ.
Furthermore let c be the parametrization of e over e¯ given by a| then we have
e e¯
(3.5) (cid:12)(cid:12)(cid:13)(cid:13)c(cid:48)(s)(cid:13)(cid:13)−1(cid:12)(cid:12) ≤ Ch2.
e
GeometricErrorofFVSchemesforConservationLawsonEvolvingSurfaces 7
Proof. We assume without loss of generality that K¯ ⊂ R2×{0}. For small enough h we can
parametrizethecurvedcellK accordingto(2.1)byaparametrizationc = a| : K¯ → K ⊂ R3
K¯
with
c(x ,x ) = (x ,x ,0)−d(x ,x ,0)ν (c(x ,x )),
1 2 1 2 1 2 Γ 1 2
where we suppressed the third coordinate in K¯. The ratio of volume elements of K and K¯
with respect to the parametrization c is given by
(cid:112) (cid:112)
|g| := det(g),
where the matrix g is defined by
g = (g ) := ((cid:104)∂ c,∂ c(cid:105)) .
ij 1≤i,j,≤2 i j 1≤i,j,≤2
For the parametrization c of K we have
∂ c = e −(cid:104)∇d,e (cid:105)ν ◦c−d ∂ c (∇ν )T ◦c for i = 1,2,
i i i Γ i Γ
where e denotes the i-th standard unit vector. Due to the bounded curvature of Γ and
i
Lemma 2.1 we can show that
(3.6) ∂ c = e −((ν ) ν )◦c+O(h2) for i = 1,2.
i i Γ i Γ
Applying (3.1) we see that
ν = ±(0,0,1)+O(h) and (cid:104)e ,ν (cid:105) = (ν ) = O(h) for i = 1,2.
Γ i Γ Γ i
Thus, for the matrix g we have
(cid:18)1+O(h2) O(h2) (cid:19)
g =
O(h2) 1+O(h2)
which implies for the volume element
(3.7) dK = (cid:112)|g|dK¯ = (cid:112)1+O(h2)dK¯ = dK¯ +O(h2)dK¯.
Therefore, we arrive at
ˆ
(cid:12) (cid:12)
(cid:12)(cid:12)|K|−(cid:12)(cid:12)K¯(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) (cid:112)|g|−1dK¯(cid:12)(cid:12) ≤ C(cid:12)(cid:12)K¯(cid:12)(cid:12)h2
(cid:12) K¯ (cid:12)
for the error of the cell area which proves (3.4).
To prove (3.3) and (3.5) we consider WLOG an edge e¯ = {(s,0,0)|0 ≤ s ≤ h } ⊂ ∂K¯,
e¯
where h denotes the length of e¯. The corresponding curved edge e is parametrized by
e¯
(3.8) c (s) = c(s,0) = (s,0,0)−d(s,0,0)ν (c (s)).
e Γ e
Due to the bounded curvature of Γ we get for the derivative
(3.9) c(cid:48)(s) = (1,0,0)−ν (c (s))(ν ) (c (s))+O(h2).
e Γ e Γ 1 e
Applying (3.1) we get
(3.10) (cid:13)(cid:13)c(cid:48)(s)(cid:13)(cid:13) = 1+O(h2)
e
and therefore
ˆ
||e|−|e¯|| = (cid:12)(cid:12)(cid:12) he¯(cid:13)(cid:13)c(cid:48)(s)(cid:13)(cid:13)−1 ds(cid:12)(cid:12)(cid:12) ≤ C|e¯|h2.
e
(cid:12) (cid:12)
0
(cid:3)
8 JanGiesselmann,ThomasMu¨ller
Remark 3.4. Let us note that an analogous estimate to (2.2) for curved elements is an easy
consequence of (2.2), (3.3), (3.4) and the fact |h −h | ≤ Ch2, which is a consequence of
K¯ K
Lemma 3.3.
Lemma 3.5. There is a constant C (depending on κ) such that for all flat cells K¯, all flat
edges e¯⊂ ∂K¯ and every x ∈ e¯ we have
(cid:12) (cid:12)
(3.11) (cid:12)(cid:104)µ¯ ,t−l(x)(cid:105)(cid:12) ≤ Ch2,
(cid:12) K¯,e¯ (cid:12)
(cid:12) (cid:12)
(3.12) (cid:12)(cid:104)µ¯ ,ν−l(x)(cid:105)(cid:12) ≤ Ch,
(cid:12) K¯,e¯ Γ (cid:12)
(cid:12) (cid:12)
(3.13) (cid:12)(cid:104)µ¯ ,µ−l (x)(cid:105)−1(cid:12) ≤ Ch2,
(cid:12) K¯,e¯ K,e (cid:12)
where t denotes a unit tangent vector to e. We want to point out that this estimate is
independent of the sign of t.
Proof. Itissufficienttoshowversionsof (3.11)-(3.13)whereµ¯ issubstitutedbyν¯ ×¯t .
K¯,e¯ K¯ K¯,e¯
Then analogous results for ν¯ ×¯t are immediate. Indeed, estimates (3.11) - (3.13) follow
K¯e¯ K¯,e¯
because µ¯ is the mean of the vectors ν¯ ×¯t and ν¯ ×¯t . Firstly, we address the
K¯,e¯ K¯e¯ K¯,e¯ K¯ K¯,e¯
proof of (3.11). Let the same assumptions as in the proof of Lemma 3.2 hold and in addition
let e¯be given by {(x,0,0) ∈ R3|x ∈ [0,h ]}. We obviously have
e¯
(3.14) ν¯ ×¯t = (0,1,0).
K¯ K¯,e¯
Note that the assumptions of the proof of Lemma 3.3 are satisfied. Hence we can use (3.9)
i.e. the parametrization of e given by c satisfies
(3.15) c(cid:48)(s) = (1,0,0)−ν (c(s))(ν ) (c(s))+O(h2),
Γ Γ 1
and coincides with t(c(s)) up to standardization. Hence, in view of (3.5) we obtain
(3.16) t−l(x) = (1,0,0)−ν (c(s))(ν ) (c(s))+O(h2)
Γ Γ 1
for some s ∈ [0,h ]. Combining (3.14) and (3.16) we find using (3.2)
e¯
(cid:12) (cid:12)
(cid:12)(cid:104)ν¯ ×¯t ,t−l(x)(cid:105)(cid:12) = |(ν ) (c(s))(ν ) (c(s))|+O(h2) ≤ Ch2,
(cid:12) K¯ K¯,e¯ (cid:12) Γ 2 Γ 1
which is (3.11). Concerning (3.12),
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)(cid:104)ν¯ ×¯t ,ν−l(x)(cid:105)(cid:12) ≤ (cid:12)(ν−l) (x)(cid:12) ≤ Ch
(cid:12) K¯ K¯,e¯ Γ (cid:12) (cid:12) Γ 2 (cid:12)
holdsbecauseof (3.14)and(3.2). Thus,itremainstoshow(3.13). Bydefinitiont−l(x),ν−l(x),µ−l (x)
Γ K,e
form an orthonormal basis of R3 and the vector ν¯ ×¯t is of unit length. This means that
K¯ K¯,e¯
for every x¯ in e¯ there exist b (x¯),b (x¯),b (x¯) ∈ R satisfying b2(x¯)+b2(x¯)+b2(x¯) = 1 such
1 2 3 1 2 3
that
(3.17) ν¯ ×¯t = b (x¯)t−l(x¯)+b (x¯)ν−l(x¯)+b (x¯)µ−l (x¯).
K¯ K¯,e¯ 1 2 Γ 3 K,e
We know from (3.11) and (3.12) that |b (x¯)|,|b (x¯)| ≤ Ch for some C > 0, which implies
1 2
using Taylor expansion
(cid:112)
(3.18) b (x¯) = ± 1+O(h2) = ±1+O(h2).
3
GeometricErrorofFVSchemesforConservationLawsonEvolvingSurfaces 9
Note that it only remains to show that in (3.18) the + holds. As b depends continuously
3
on x¯ it is sufficient to find one (x¯ ,0,0) ∈ K¯ such that b (x¯ ) = 1+O(h2). To that end we
1 3 1
consider some x¯ ,y¯ > 0 such that
1 1
γ : (−y¯ ,0] −→ K¯, s (cid:55)→ (x¯ ,s,0)
1 1
is a curve leaving K¯ through e¯. By definition the curve γ˜ given by
γ˜(s) := γ(s)−ν−l(γ(s))d(γ(s))
Γ
is a curve in K leaving through e. This means we have
(3.19) 0 < (cid:104)γ˜(cid:48)(0),µ (γ˜(0))(cid:105).
K,e
Due to (3.17), (3.18) and the fact that µ is of unit length we already know that
K,e
(3.20) µ ≡ ±(0,1,0)+O(h).
K,e
We are able to compute
(3.21) γ˜(cid:48)(s) = (0,1,0)−(0,1,0)(∇ν−l(γ(s)))Td(γ(s))−ν (γ˜(s))(cid:104)ν (γ˜(s)),(0,1,0)(cid:105)
Γ Γ Γ
= (0,1,0)+O(h),
because ∇ν−l is bounded, Lemma 2.1 and (3.2). Inserting (3.20) and (3.21) in (3.19) we find
Γ
(3.22) 0 < ±1+O(h),
where ± is the sign from (3.18). Obviously for h sufficiently small (3.22) only holds for “+”,
which finishes the proof. (cid:3)
4. Estimating the Difference Between Both Schemes
This section is devoted to establishing a bound for the difference between the curved and
flat approximate solutions. To start with we investigate the difference between the numerical
fluxes defined on the flat and the curved triangulation respectively.
Lemma 4.1. Let K be some compact subset of R2. Provided the quadrature operators Q
e¯(t)
and Q are of order at least 1, then there is a constant C depending only on G and K such
In T
that for the Lax-Friedrichs fluxes (2.6) and (2.9) with the same diffusion rate λ the following
1
inequality holds
(cid:12) (cid:12)
(cid:12)fn (u,v)−f¯n (u,v)(cid:12) ≤ Ch2 ∀ (u,v) ∈ K, K ∈ T , e ⊂ ∂K.
(cid:12) K,e K¯,e¯ (cid:12) h
10 JanGiesselmann,ThomasMu¨ller
Proof. We start by observing that the diffusive terms drop out, such that
(cid:12)
(cid:12) (cid:12) (cid:12)
2(cid:12)fn (u,v)−f¯n (u,v)(cid:12) =(cid:12) (cid:104)f(u,x,t),µ (x)(cid:105)de(t)dt
(cid:12) K,e K¯,e¯ (cid:12) (cid:12) K(t),e(t)
(cid:12) In e(t)
(cid:20) (cid:21)
1 1
− Q Q [(cid:104)f(u,·,·),µ¯ (cid:105)]
|I | In |e¯(·)| e¯(·) K¯(·),e¯(·)
(4.1) n
+ (cid:104)f(v,x,t),µ (x)(cid:105)de(t)dt
K(t),e(t)
In e(t)
(cid:20) (cid:21)(cid:12)
1 1 (cid:12)
−|I |QIn |e¯(·)|Qe¯(·)[(cid:104)f(v,·,·),µ¯K¯(·),e¯(·)(cid:105)] (cid:12)(cid:12).
n
Asuandvappearsymmetricallywewillomitalltermscontainingthelatterinoursubsequent
analysis. We now add zero several times in (4.1) and get

(cid:12) (cid:12)
(cid:12) (cid:12)
(4.2) 2(cid:12)(cid:12)fKn,e(u,v)−f¯Kn¯,e¯(u,v)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) T1+T2+T3+T4+T5 dt(cid:12)(cid:12)(cid:12)
In
with

T (t) := (cid:104)f(u,x,t),µ (x)(cid:105)de(t)− (cid:104)f−l(u,x,t),µ−l (x)(cid:105)de¯(t),
1 K(t),e(t) K(t),e(t)
e(t) e¯ (t)
T (t) := (cid:104)f−l(u,x,t),µ−l (x)(cid:105)de¯(t)− (cid:104)f−l(u,x,t),µ¯ (cid:105)de¯(t)
2 K(t),e(t) K¯(t),e¯(t)
e¯(t) e¯(t)
T (t) := (cid:104)f−l(u,x,t),µ¯ (cid:105)de¯(t)− (cid:104)f(u,x,t),µ¯ (cid:105)de¯(t)
3 K¯(t),e¯(t) K¯(t),e¯(t)
e¯(t) e¯(t)
(cid:34) (cid:35)
1
T (t) := (cid:104)f(u,x,t),µ¯ (cid:105)de¯(t)− Q (cid:104)f(u,x,·),µ¯ (cid:105)de¯(·)
4 K¯(t),e¯(t) |I | In K¯(·),e¯(·)
e¯(t) n e¯(·)
(cid:34) (cid:35)
1 1 (cid:104)(cid:68) (cid:69)(cid:105)
T := Q (cid:104)f(u,x,·),µ¯ (cid:105)de¯(·)− Q f(u,·,·),µ¯ .
5 |I | In K¯(·),e¯(·) |e¯(·)| e¯(·) K¯(·),e¯(·)
n e¯(·)
In the following we will estimate the summands one by one. First, by properties of the
quadrature operators Q , Q and the CFL condition (2.5)
In e¯(t)

(cid:12) (cid:12)
(4.3) (cid:12)(cid:12) T4(t) dt(cid:12)(cid:12) ≤ Chp3+1, |T5| ≤ Chp2+1,
(cid:12) (cid:12)
In
as the integrands are sufficiently smooth. In particular, we use the fact that the surface
evolves smoothly. Addressing the estimates for T ,T ,T we will omit the time dependency
1 2 3
as all three estimates are uniform in time. To establish an estimate for T we recall that we
1
can parametrize e over e¯such that for the parametrisation c inequality (3.5) holds. We have
e
(cid:12) (cid:12)
(cid:12) (cid:12)
(4.4) |T | = (cid:12) (cid:104)f−l(u,x),µ−l (x)(cid:105)(cid:0)(cid:107)c(cid:48)(s)(cid:107)−1(cid:1)de¯(cid:12) ≤ (cid:107)f(cid:107) Ch2,
1 (cid:12) K,e e (cid:12) ∞
(cid:12) e¯ (cid:12)
where (cid:107)f(cid:107) denotes the supremum of f(u,x,t) for (x,t) ∈ G and u ∈ K. Next we turn to
∞ T
T . Its estimate is based on the assumption that we have extended f(u,·) to N smoothly and
3