Introduction
PVM F.V. methods
Two-phase models for debris flows.
Numerical approach by PVM methods
Enrique D. Fernández-Nieto
Dpto. Matemática Aplicada I,
University of Sevilla
Multiphase flow in industrial and environmental engineering
Chambery, 2012
Numerical tests
Introduction
PVM F.V. methods
Outline
1
Introduction
2
PVM F.V. methods
3
Numerical tests
Numerical tests
Introduction
PVM F.V. methods
Numerical tests
Two-phase models for debris flows
Mathematical modelling of some geophysical mass flows containing a mixture
of solid material and interstitial fluid in order to simulate avalanches evolution.
Savage and Hutter presented in 1991 a pioneering work on the study of aerial
granular avalanches, obtaining a model of shallow water type in local
coordinates on an inclined plane.
S. B. Savage, K. Hutter. The dynamics of avalanches of granular materials
from initiation to runout. Acta Mech. 86, 201–223, 1991.
Iverson and Denlinger in 2001 proposed a model for the study of shallow
partially fluidized avalanches, a mixture of a granular material and a fluid.
R.M Iverson, R.P. Denlinger.: Flow of variably fluidized granular masses
across three-dimensional terrain 1: Coulomb mixture theory. J. Geophys.
Res. 106, 537–552, 2001.
Introduction
PVM F.V. methods
Numerical tests
Two-phase models for debris flows
The actual models that we use to describe fluid-solid mixture are mainly based
on the Jackson model. This model takes into account the solid and fluid stress,
the force interaction between the fluid and the solid phases by including
buoyancy effects, through the mass and momentum conservation of the two
phases (1D: four equations ).
T.B. Anderson, R. Jackson. A fluid mechanical description of fluidized
beds. Ind. Eng. Chem. Fundam. 6, 527–539, 1967.
Introduction
PVM F.V. methods
Numerical tests
Two-phase models for debris flows
Pitman and Le and the model proposed by Pelanti et al. authors use the two
phase approach proposed by Jackson to obtain an averaged model that aims to
solve the equation of mass and momentum conservation for the solid and fluid
phases.
E.B. Pitman, L. Le. A two-fluid model for avalanche and debris flows. Phil.
Trans. R. Soc. A 363, 1573–1601, 2005.
M. Pelanti, F. Bouchut, A. Mangeney. A Roe-type scheme for two-phase
shallow granular flows over variable topography. M2AN 42, 851–885, 2008.
Introduction
PVM F.V. methods
Numerical approach of the Pitman-Le model
Let us consider the Pitman-Le model,
∂t (hϕ) + ∂x (hϕv)
=
0;
∂t (h(1 − ϕ)) + ∂x (h(1 − ϕ)u)
=
0;
∂t (ϕhv) + ∂x (ϕhv2 )
=
∂t ((1 − ϕ)hu) + ∂x ((1 − ϕ)h u2 )
−
1
− (1 − r)gh2 cos θ ∂x ϕ
2
gh cos θ ϕ ∂x (b + h)
−
ghsen θ ϕ
+
CF (u − v);
=
−gh cos θ (1 − ϕ) ∂x (b + h)
−
ghsen θ (1 − ϕ)
1
CF (u − v).
r
−
Numerical tests
Introduction
PVM F.V. methods
Numerical tests
Hyperbolic systems with nonconservative products
The Pitman-Le model can be written in the following form:
wt + F(w)x + B(w) · wx = S(w)Hx ,
By adding the equation
Ht = 0,
(1)
the system can be rewritten in the form of a hyperbolic systems with nonconservative
products:
»
–
w
Wt + A(W) · Wx = 0, where W =
∈ Ω = O × R,
H
In this work we consider Path-conservative finite volume solvers.
They are based in the choice of a family of paths Φ(s; WL , WR ).
Introduction
PVM F.V. methods
Numerical tests
Hyperbolic systems with nonconservative products
The Pitman-Le model can be written in the following form:
wt + F(w)x + B(w) · wx = S(w)Hx ,
By adding the equation
Ht = 0,
(1)
the system can be rewritten in the form of a hyperbolic systems with nonconservative
products:
»
–
w
Wt + A(W) · Wx = 0, where W =
∈ Ω = O × R,
H
In this work we consider Path-conservative finite volume solvers.
They are based in the choice of a family of paths Φ(s; WL , WR ).
Introduction
PVM F.V. methods
Numerical tests
Hyperbolic systems with nonconservative products
The Pitman-Le model can be written in the following form:
wt + F(w)x + B(w) · wx = S(w)Hx ,
By adding the equation
Ht = 0,
(1)
the system can be rewritten in the form of a hyperbolic systems with nonconservative
products:
»
–
w
Wt + A(W) · Wx = 0, where W =
∈ Ω = O × R,
H
In this work we consider Path-conservative finite volume solvers.
They are based in the choice of a family of paths Φ(s; WL , WR ).
Introduction
PVM F.V. methods
Numerical tests
Roe matrix
• Roe matrix definition: for any WL , WR ∈ Ω,
Z 1
∂Φ
AΦ (WL , WR ) · (WR − WL ) =
A(Φ(s; WL , WR ))
(s; WL , WR ) ds.
∂s
0
• ..... but for the bilayer SWE we have
»
A(w)
A(W) =
0
−S(w)
0
–
,
(2)
Introduction
PVM F.V. methods
Numerical tests
Roe matrix
• .... Then, we consider Roe linearizations AΦ (WL , WR ) of the form:
–
»
AΦ (wL , wR ) −SΦ (wL , wR )
,
AΦ (WL , WR ) =
0
0
where
AΦ (wL , wR ) = J(wL , wR ) + BΦ (wL , wR ).
• J(wL , wR ) is a Roe linearization of the Jacobian of the flux F in the usual sense:
J(wL , wR ) · (wR − wL ) = F(wR ) − F(wL );
• BΦ (wL , wR ) is a matrix satisfying:
Z 1
∂Φ[1,··· ,4]
BΦ (wL , wR ) · (wR − wL ) =
B(Φ(s; WL , WR ))
(s; WL , WR ) ds;
∂s
0
• and SΦ (wL , wR ) is a vector satisfying:
Z 1
∂Φ5
SΦ (wL , wR )(HR − HL ) =
S(Φ(s; WL , WR ))
(s; WL , WR ) ds.
∂s
0
Introduction
PVM F.V. methods
Numerical tests
Roe method
It can be shown that Roe scheme can be written in the original variables w as follows:
wn+1
= wni −
i
´
∆t ` +
n
n
Di−1/2 (wni , wni+1 , Hi , Hi+1 ) + D−
i+1/2 (wi , wi+1 , Hi , Hi+1 ) ,
∆x
being
n
n
D±
i+1/2 (wi , wi+1 , Hi , Hi+1 ) =
and
Bi+1/2 = BΦ (wni , wni+1 ),
Si+1/2 = SΦ (wni , wni+1 ),
Ai+1/2 = AΦ (wni , wni+1 ).
1`
F(wni+1 ) − F(wni ) + Bi+1/2 (wni+1 − wni )
2
−Si+1/2 (Hi+1 − Hi )
´
±|Ai+1/2 |(wni+1 − wni − A−1
i+1/2 Si+1/2 (Hi+1 − Hi )) ,
Introduction
PVM F.V. methods
Numerical tests
Roe method
It can be shown that Roe scheme can be written in the original variables w as follows:
wn+1
= wni −
i
´
∆t ` +
n
n
Di−1/2 (wni , wni+1 , Hi , Hi+1 ) + D−
i+1/2 (wi , wi+1 , Hi , Hi+1 ) ,
∆x
being
n
n
D±
i+1/2 (wi , wi+1 , Hi , Hi+1 ) =
and
Bi+1/2 = BΦ (wni , wni+1 ),
Si+1/2 = SΦ (wni , wni+1 ),
Ai+1/2 = AΦ (wni , wni+1 ).
1`
F(wni+1 ) − F(wni ) + Bi+1/2 (wni+1 − wni )
2
−Si+1/2 (Hi+1 − Hi )
´
±|Ai+1/2 |(wni+1 − wni − A−1
i+1/2 Si+1/2 (Hi+1 − Hi )) ,
Introduction
PVM F.V. methods
Numerical tests
Absolute value of Roe matrix
Let us first observe that the matrix |AΦ (wL , wR )| can be rewritten as follows:
|AΦ (wL , wR )| =
3
X
αj AjΦ (wL , wR ),
j=0
where αj , j = 0, · · · , 3 are defined in terms of the eigenvalues λj , j = 1, · · · , 4 of
AΦ (wL , wR ), by solving the linear system:
0
10
1 0
1
1 λ1 λ21 λ31
α0
|λ1 |
2
3
B 1 λ2 λ2 λ2 C B α1 C B |λ2 | C
B
CB
C B
C
(3)
@ 1 λ3 λ23 λ33 A @ α2 A = @ |λ3 | A .
2
3
α3
|λ4 |
1 λ4 λ4 λ4
Note that:
(3) has an unique solution provided that λi 6= λj , i 6= j, i, j = 1, · · · , 4.
.... only the explicit knowledge of the eigenvalues of AΦ (wL , wR ) are needed.
.... Nevertheless, the CPU time needed to compute |AΦ (wL , wR )| in this way is
equivalent to calculate the eigenstructure of AΦ (wL , wR ).
Introduction
PVM F.V. methods
Numerical tests
PVM methods
The numerical scheme in the unknowns w can be written as follows:
wn+1
= wni −
i
´
∆t ` +
Di−1/2 + D−
i+1/2 ,
∆x
being
D±
i+1/2 =
1`
F(wi+1 ) − F(wi ) + Bi+1/2 (wi+1 − wi ) − Si+1/2 (Hi+1 − Hi )
2
”
± Qi+1/2 (wi+1 − wi − A−1
i+1/2 Si+1/2 (Hi+1 − Hi )) ,
with
Bi+1/2 = BΦ (Wi , Wi+1 ),
Si+1/2 = SΦ (Wi , Wi+1 ),
Ai+1/2 = AΦ (Wi , Wi+1 ),
Qi+1/2 = QΦ (Wi , Wi+1 ) is a numerical viscosity matrix.
Different numerical schemes can be obtained for different definitions of Qi+1/2
Introduction
PVM F.V. methods
Numerical tests
Some definitions of Qi+1/2
Roe scheme corresponds to the choice
QΦ (WL , WR ) = |AΦ (WL , WR )|,
Lax-Friedrichs scheme:
QΦ (WL , WR ) =
∆x
Id,
∆t
being Id the identity matrix.
Lax-Wendroff scheme:
QΦ (WL , WR ) =
∆t 2
AΦ (WL , WR ),
∆x
FORCE and GFORCE schemes are presented in the bibliography as a convex
combination of Lax-Friedrichs and Lax-Wendroff scheme:
∆x
∆t 2
QΦ (WL , WR ) = (1 − ω)
Id + ω
AΦ (WL , WR ),
∆t
∆x
with ω = 0.5 and ω =
1
,
1+γ
respectively, being γ the CFL parameter.
Introduction
PVM F.V. methods
Numerical tests
PVM methods
We propose a class of finite volume methods defined by
Qi+1/2 = Pl (Ai+1/2 ),
being Pl (x) a polinomial of degree l,
Pl (x) =
l
X
αj xj ,
j=0
and Ai+1/2 = AΦ (Wi , Wi+1 ) a Roe matrix.
M.J. Castro, E.D. Fernández-Nieto A class of computationally fast first order
finite volume solvers: PVM methods. SIAM J. Sci. Comput. (2012).
See also:
P. Degond, P.F. Peyrard, G. Russo, Ph. Villedieu. Polynomial upwind schemes
for hyperbolic systems. C. R. Acad. Sci. Paris 1 328, 479-483, (1999).
Introduction
PVM F.V. methods
Numerical tests
PVM methods
Taking into account the properties of Roe matrix we have the method defined by
win+1 = wni −
´
∆t ` +
Di−1/2 + D−
i+1/2 ,
∆x
where D±
i+1/2 can be rewritten as follows:
±α0
(wi+1 − wi − A−1
i+1/2 Si+1/2 (Hi+1 − Hi ))
2
„
«
l
X
δj,1 ± αj (j−1)
Ai+1/2 F(wi+1 ) − F(wi ) + Bi+1/2 (wi+1 − wi ) − Si+1/2 (Hi+1 − Hi )
+
2
j=1
D±
i+1/2 =
with

δj,1 =
1
0
if j = 1,
otherwise.
Introduction
PVM F.V. methods
Numerical tests
PVM methods. y = Pl (x)
A sufficient condition to ensure that the numerical scheme is linearly L∞ -stable
is that
Pl (x) ≥ |x| ∀x ∈ [λ1,i+1/2 , λN,i+1/2 ].
(4)
Let us consider the following notation: PVM-l(S0 , · · · , Sk ).
In practice, the parameters S0 , · · · , Sk will be related to the approximations of some
wave speeds.
Upwind methods
A PVM method is said to be upwind if

AΦ
Pl (AΦ ) =
−AΦ
and it will be denoted as PVM-lU.
if λ1 > 0
if λN < 0,
(5)
Introduction
PVM F.V. methods
Numerical tests
PVM-(N-1)(λ1 , · · · , λn ) or Roe method
QΦ (WL , WR ) = |AΦ (WL , WR )| =
N−1
X
αj AjΦ (WL , WR ),
j=0
where αj , j = 0, · · · , N − 1 are the solution of the following linear system:
0
B
B
B
@
1
1
..
.
1
λ1
λ2
..
.
λN
...
...
..
.
...
λN−1
1
λN−1
2
..
.
λN−1
N
10
CB
CB
CB
A@
α0
α1
..
.
αN−1
1
0
C B
C B
C=B
A @
λ1 , · · · , λN are the eigenvalues of the matrix AΦ (WL , WR ).
|λ1 |
|λ2 |
..
.
|λN |
1
C
C
C,
A
Introduction
PVM F.V. methods
Numerical tests
PVM-0(S0 ) methods: Rusanov, Lax-Friedrichs and modified Lax-Friedrichs
schemes
P0 (x) = S0 .
That is, y = P0 (x) is an horizontal line
PVM−0(S)
λ1
λ2
λj
...
λN
S
Introduction
PVM F.V. methods
Numerical tests
PVM-0(S0 )
Stability requirements imply that
max |λj,i+1/2 | ≤ S0 ≤
j
∆x
.
∆t
Thus, several interesting choices for S0 can be performed, taking into account that
max |λj,i+1/2 | = α
j
∆x
∆x
≤
.
∆t
∆t
Therefore, S0 can be defined by
mod
S0 ∈ {SRus , SLF , SLF
},
being
SRus = max |λj,i+1/2 |,
j
SLF =
∆x
∆t
and
mod
SLF
=α
Note that:
Rusanov scheme corresponds to the choice S0 = SRus ,
Lax-Friedrichs with S0 = SLF
mod
modified Lax-Friedrichs with S0 = SLF
.
∆x
.
∆t
Introduction
PVM F.V. methods
Numerical tests
PVM-1U(SL , SR ) or HLL method
such as P1 (SL ) = |SL |, P1 (SR ) = |SR |.
P1 (x) = α0 + α1 x
PVM−1U(SL,SR)
SL λ 1
λ2
λj
...
λN
SR
Introduction
PVM F.V. methods
Numerical tests
PVM-1U(SL , SR ) or HLL method
The definition of the classical HLL flux for a conservative system can be written as follows
”
∆t “ HLL
win+1 = wni −
φi+1/2 − φHLL
(6)
i−1/2 ,
∆x
where
φHLL
i+1/2
8
F(wi )
>
<
SR F(wi ) − SL F(wi+1 ) + SR SL (wi+1 − wi )
=
F HLL =
>
SR − SL
:
F(wi+1 )
if SL ≥ 0,
if SL ≤ 0 ≤ SR ,
if 0 ≥ SR .
(7)
Introduction
PVM F.V. methods
Numerical tests
PVM-1U(SL , SR ) or HLL method
The definition of the classical HLL flux for a conservative system can be written as follows
”
∆t “ HLL
win+1 = wni −
φi+1/2 − φHLL
(6)
i−1/2 ,
∆x
where
φHLL
i+1/2
8
F(wi )
>
<
SR F(wi ) − SL F(wi+1 ) + SR SL (wi+1 − wi )
=
F HLL =
>
SR − SL
:
F(wi+1 )
if SL ≥ 0,
if SL ≤ 0 ≤ SR ,
(7)
if 0 ≥ SR .
SL (respectively SR ) is an approximation of the minimum (respectively maximum) wave
speed.
One possible choice is to set SL = λ1,i+1/2 ,
SR = λN,i+1/2 .
Some other different possibilities have been proposed in the bibliography. For
example Davis proposes
SL = min(λ1,i+1/2 , λ1,i ),
SR = max(λN,i+1/2 , λN,i+1 ),
being λi,1 < · · · < λi,N the eigenvalues of matrix AΦ (Wi , Wi ).
Introduction
PVM F.V. methods
Numerical tests
PVM-1U(SL , SR ) or HLL method
The definition of the classical HLL flux for a conservative system can be written as follows
”
∆t “ HLL
win+1 = wni −
φi+1/2 − φHLL
(6)
i−1/2 ,
∆x
where
φHLL
i+1/2
8
F(wi )
>
<
SR F(wi ) − SL F(wi+1 ) + SR SL (wi+1 − wi )
=
F HLL =
>
SR − SL
:
F(wi+1 )
if SL ≥ 0,
if SL ≤ 0 ≤ SR ,
if 0 ≥ SR .
If the system is conservative (B = 0 and S = 0), the the conservative flux is defined by
φi+1/2 = D−
i+1/2 + F(wi ),
Then, the PVM-1U method corresponds to
„
φi+1/2 =
F(wi )(SR + |SR | − SL − |SL |) + F(wi+1 )(SR − |SR | − SL + |SL |)
«
−(SR |SL | − SL |SR |)(wi+1 − wi ) /(2SR − 2SL ),
which is a compact definition of the numerical HLL flux φHLL
i+1/2 given in (7).
(7)
Introduction
PVM F.V. methods
Numerical tests
PVM-1U(SL , SR ) or HLL method
The definition of the classical HLL flux for a conservative system can be written as follows
”
∆t “ HLL
win+1 = wni −
φi+1/2 − φHLL
(6)
i−1/2 ,
∆x
where
φHLL
i+1/2
8
F(wi )
>
<
SR F(wi ) − SL F(wi+1 ) + SR SL (wi+1 − wi )
=
F HLL =
>
SR − SL
:
F(wi+1 )
if SL ≥ 0,
if SL ≤ 0 ≤ SR ,
(7)
if 0 ≥ SR .
Remarks
The usual HLL scheme coincides with PVM-1U(SL , SR ) in the case of conservative
systems.
PVM-1U(SL , SR ) gives a natural generalization of HLL method for nonconservative
problems.
If λ1,i+1/2 = −λN,i+1/2 , then PVM-1U(SL , SR ) coincides with PVM-0(SRus ).
Introduction
PVM F.V. methods
Numerical tests
PVM-2(S0 ) methods or FORCE type methods
P2 (x) = α0 + α2 x2 , such as P2 (S0 ) = S0 , P02 (S0 ) = 1,
PVM−2(S0)
λ1
λ2
λj
...
λN
S0
Introduction
PVM F.V. methods
Numerical tests
PVM-2(S0 ) methods or FORCE type methods
P2 (x) = α0 + α2 x2 , such as P2 (S0 ) = S0 , P02 (S0 ) = 1,
where
α0 =
S0
,
2
α2 =
1
.
2S0
mod
S0 ∈ {SRus , SLF , SLF
},
being
SRus = max |λj,i+1/2 |,
j
SLF =
∆x
∆t
mod
and SLF
=α
Remarks
If S0 = SLF then we obtain FORCE method.
GFORCE scheme can be obtained by imposing
mod
mod
P2 (SLF
) = SLF
,
mod
P02 (SLF
)=
2α
,
1+α
∆x
.
∆t
Introduction
PVM F.V. methods
Numerical tests
PVM-2(S0 ) methods or FORCE type methods
P2 (x) = α0 + α2 x2 , such as P2 (S0 ) = S0 , P02 (S0 ) = 1,
where
α0 =
S0
,
2
α2 =
1
.
2S0
mod
S0 ∈ {SRus , SLF , SLF
},
being
SRus = max |λj,i+1/2 |,
j
SLF =
∆x
∆t
mod
and SLF
=α
∆x
.
∆t
Remarks
mod
S0 = SLF or S0 = SLF
, then the coefficients α0 and α2 depend on
For S0 = SLF ,
∆x
.
∆t
1 ∆x
1 ∆x 2
Id +
A
.
2 ∆t
2 ∆t i+1/2
Then, PVM-2(S0 ) can be interpreted as a combination of Lax-Friedrichs and
Lax-Wendroff schemes.
Qi+1/2 =
Introduction
PVM F.V. methods
Numerical tests
PVM-2U(SM , Sm ) method
P2 (x) = α0 + α1 x + α2 x2 ,
such as
P2 (Sm ) = |Sm |, P2 (SM ) = |SM |, P02 (SM ) = sgn(SM ),
where

SM =
λ1,i+1/2
λN,i+1/2
if |λ1,i+1/2 | ≥ |λN,i+1/2 |,
if |λ1,i+1/2 | < |λN,i+1/2 |.

Sm =
λN,i+1/2
λ1,i+1/2
PVM−2U(SL,SR)
SL λ 1
λ2
λj
...
λN
SR
if |λ1,i+1/2 | ≥ |λN,i+1/2 |
if |λ1,i+1/2 | < |λN,i+1/2 |
Introduction
PVM F.V. methods
Numerical tests
PVM-4(SM , SI ) and PVM-4(S0 ) methods
P4 (x) = α0 + α2 x2 + α4 x4 ,
P4 (SM ) = |SM |,
SI =
8
<
2≤j≤N
:
1≤j≤(N−1)
max (|λj,i+1/2 |)
max
(|λj,i+1/2 |)
P4 (SI ) = SI ,
P04 (SI ) = 1,
if |λ1,i+1/2 | ≥ |λN,i+1/2 |,
if |λ1,i+1/2 | < |λN,i+1/2 |.
PVM−4(S1,S2)
S1 = λ 1
λ2
λj
...
S2 = λ N
Introduction
PVM F.V. methods
Numerical tests
PVM-4(SM , SI ) and PVM-4(S0 ) methods
P4 (x) = α0 + α2 x2 + α4 x4 ,
P4 (SM ) = |SM |,
P4 (SI ) = SI ,
P04 (SI ) = 1,
Another version of the method correspond to set SI = SM = S.
PVM−4(S)
λ1
λ2
λj
...
λN
S
Introduction
PVM F.V. methods
Numerical tests
Numerical diffusion
Let us consider the linear advection equation
ut + λux = 0,
λ > 0.
(8)
It is easy to check that the numerical viscosity of methods PVM-l(S0 ), when they are
applied to equation (8) is given by
νN =
where
∆t
λ
∆x
∆x
(Pl (λ) − αλ) ,
2
= α, and Pl (x) are the polynomials associated to PVM-l(S0 ) methods.
Introduction
PVM F.V. methods
Numerical tests
Numerical diffusion
PVM−0(S)
PVM−2(S)
PVM−4(S)
PVM−4(S ,S )
1
λ1
λ2
2
λj
...
S1 = λ N S2 = S
Introduction
PVM F.V. methods
Numerical tests
Numerical diffusion
PVM−1U(S ,S )
L
R
PVM−2U(SL,SR)
SL λ 1
λ2
λj
...
λN
SR
Introduction
PVM F.V. methods
Numerical tests
Numerical diffusion. Comparison of p(0) = α0
1
PVM−0(S )
0
PVM−1U(SL,SR)
0.8
SL = −1.
SR ∈ [−1, 1]
PVM−2(S0)
PVM−2U(SM,Sm)
PVM−4(S )
0
PVM−4(SM,SI)
!0
0.6
0.4
0.2
0
−1
−0.8
−0.6
−0.4
−0.2
0
S
R
0.2
0.4
0.6
0.8
1
Introduction
PVM F.V. methods
Numerical tests. Pitman-Le model
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
∂hf
∂qf
+
= 0,
∂t
∂x
∂qf
∂
+
∂t
∂x
q2f
g
+ h2f
hf
2
!
+ ghf
db
∂hs
= −ghf ,
∂x
dx
>
>
>
∂qs
∂hs
>
>
+
= 0,
>
>
∂t
∂x
>
>
>
>
>
„ 2
«
>
>
∂hf
∂qs
∂
qs
g 2
1−r
db
>
>
+
+ hs + g
hs hf + rghs
= −ghs .
:
∂t
∂x hs
2
2
∂x
dx
hs = ϕh,
and
hf = (1 − ϕ)h.
The unknowns qs and qf represent the mass-flow of each phase.
Numerical tests
Introduction
PVM F.V. methods
Numerical tests
Test: LeVeque’s test for Pitman-Le model
Let us consider a flat channel in the domain I = [−15, 15]. The initial condition is
2
h(x, 0) = h0 + δe−16x ,
2
ϕ(x, 0) = ϕ0 − δe−16x ,
uf = us = 0,
where
h0 = 1,
ϕ0 = 0.6,
δ = 0.2 × 10−3 .
Free boundary conditions are set.
T = 3.5,
∆x = 0.15
A reference solution computed with Roe scheme for ∆x = 0.03
Introduction
PVM F.V. methods
Numerical tests
Test: Flow depth h. PVM-0,2,4(S0 )
8e−06
10
ROE
LF
FORCE
PVM−4(S0)
2e−05
10
ROE
LF
FORCE
PVM−4(S0)
7e−06
10
Ref. solution
Ref. solution
6e−06
10
5e−06
f
s
h=h +h
h=hs+hf
10
4e−06
10
1e−05
10
3e−06
10
2e−06
10
1e−06
10
0
10
−15
0
−10
−5
0
x
5
(a) Flow depth h
10
15
10
−6
−4
−2
0
x
2
(b) Flow depth h (zoom)
4
6
Introduction
PVM F.V. methods
Numerical tests
Test: Solid volume fraction ϕ. PVM-0,2,4 (S0 )
−0.22185
−0.22185
10
10
−0.22186
−0.22186
10
10
−0.22187
−0.22187
!
10
!
10
−0.22188
−0.22188
10
10
ROE
LF
FORCE
PVM−4(S )
−0.22189
10
ROE
LF
FORCE
PVM−4(S )
−0.22189
10
0
0
Ref. solution
−15
−10
−5
0
x
5
(c) Solid volume fraction ϕ
10
Ref. solution
15
−6
−4
−2
0
x
2
4
(d) Solid volume fraction ϕ (zoom)
6
Introduction
PVM F.V. methods
Numerical tests
Test: Phase velocity uf . PVM-0,2,4(S0 )
−4
2
−4
x 10
1
x 10
ROE
LF
FORCE
PVM−4(S0)
1.5
0.5
Ref. solution
1
0
0
uf
uf
0.5
−0.5
−0.5
−1
−1
ROE
LF
FORCE
PVM−4(S )
−1.5
−1.5
0
Ref. solution
−2
−15
−10
−5
0
5
10
15
−2
−13
−12
−11
−10
−9
−8
−7
−6
−5
−4
x
(e) Phase velocity uf
(f) Phase velocity uf (zoom)
−3
−2
Introduction
PVM F.V. methods
Numerical tests
Test: Phase velocity us . PVM-0,2,4(S0 )
−4
2
−4
x 10
1
x 10
1.5
0.5
1
0
us
us
0.5
0
−0.5
−0.5
−1
−1
ROE
LF
FORCE
PVM−4(S )
−1.5
ROE
LF
FORCE
PVM−4(S )
−1.5
0
0
Ref. solution
−2
−15
−10
−5
0
x
5
(g) Phase velocity us
10
Ref. solution
15
−2
−13
−12
−11
−10
−9
−8
−7
−6
−5
−4
x
(h) Phase velocity us (zoom)
−3
−2
Introduction
PVM F.V. methods
Numerical tests
Test: Flow depth h. PMV-1U,2U,4(SM ,SI )
1e−05
ROE
HLL
PVM−2U(SM,Sm)
10
ROE
HLL
PVM−2U(SM,Sm)
PVM−4(SM,SI)
PVM−4(SM,SI)
Ref. solution
Ref. solution
2e−05
f
s
h=h +h
h=hs+hf
10
1e−05
10
0
10
−15
0
−10
−5
0
x
(i) Flow depth h
5
10
15
10
−6
−4
−2
0
x
2
(j) Flow depth h (zoom)
4
6
Introduction
PVM F.V. methods
Numerical tests
Test: Solid volume fraction ϕ. PMV-1U,2U,4(SM ,SI )
−0.22185
−0.22185
10
10
−0.22186
−0.22186
10
10
−0.22187
−0.22187
10
−0.22188
!
!
10
10
−0.22189
−0.22188
10
−0.22189
10
10
ROE
HLL
PVM−2U(S ,S )
−0.2219
10
ROE
HLL
PVM−2U(S ,S )
−0.2219
10
M m
M m
PVM−4(SM,SI)
10
−15
PVM−4(SM,SI)
Ref. solution
−0.22191
−10
−5
0
x
5
(k) Solid volume fraction ϕ
10
Ref. solution
−0.22191
10
15
−6
−4
−2
0
x
2
4
(l) Solid volume fraction ϕ (zoom)
6
Introduction
PVM F.V. methods
Numerical tests
Test:. Phase velocity uf . PMV-1U,2U,4(SM ,SI )
−4
2.5
−4
x 10
0.5
x 10
2
0
1.5
1
−0.5
0
uf
uf
0.5
−1
−0.5
−1.5
−1
ROE
HLL
PVM−2U(SM,Sm)
−1.5
ROE
HLL
PVM−2U(SM,Sm)
−2
PVM−4(SM,SI)
−2
PVM−4(S ,S )
M I
Ref. solution
Ref. solution
−2.5
−15
−10
−5
0
x
5
(m) Phase velocity uf
10
15
−2.5
−13
−12
−11
−10
−9
−8
−7
−6
−5
−4
x
(n) Phase velocity uf (zoom)
−3
−2
Introduction
PVM F.V. methods
Numerical tests
Test: Phase velocity us . PMV-1U,2U,4(SM ,SI )
−4
2.5
−4
x 10
1.5
2
x 10
1
1.5
0.5
1
0
us
us
0.5
0
−0.5
−0.5
−1
−1
−1.5
ROE
HLL
PVM−2U(S ,S )
−1.5
M m
PVM−4(S ,S )
−2
ROE
HLL
PVM−2U(S ,S )
M m
−2
PVM−4(S ,S )
M I
M I
Ref. solution
−2.5
−15
−10
−5
0
x
5
(o) Phase velocity us
10
Ref. solution
15
−2.5
−13
−12
−11
−10
−9
−8
−7
−6
−5
−4
x
(p) Phase velocity us (zoom)
−3
−2
Introduction
PVM F.V. methods
Numerical tests
2D Accuracy test
This test is inspired in the accuracy study presented in
M. Dumbser, M.J. Castro, C. Parés, E. F. Toro. ADER schemes on unstructured
meshes for nonconservative hyperbolic systems: Applications to geophysical
flows. Computers &; Fluids, 38(9): 1731–1748, 2009.
where an unsteady two-dimensional analytical exact solution for the two-fluid flow
model of Pitman and Le is obtained.
Introduction
PVM F.V. methods
Numerical tests
2D Accuracy test
−1
10
−2
Error
10
−3
10
−4
10 −2
10
ROE 1st order
LF 1st order
FORCE 1st order
PVM−4(S0) 1st order
ROE 2nd order
LF 2nd order
FORCE 2nd order
PVM−4(S0) 2nd order
0
10
2
10
CPU time
4
10
6
10
(q) CPU time versus errors: 1st and 2nd order one-wave
PVM and Roe schemes.
Figure: Accuracy test: Efficiency and ratio of convergence of the one-wave PVM schemes:
comparison with Roe scheme.
Introduction
PVM F.V. methods
Numerical tests
2D Accuracy test
−1
10
−2
Error
10
ROE 1st order
LF 1st order
FORCE 1st order
PVM−4(S0) 1st order
−3
10
ROE 2nd order
LF 2nd order
FORCE 2nd order
PVM−4(S0) 2nd order
−4
10
1sr order
2nd order
−2
10
−1
10
0
10
!
(a) ratio of convergence: 1st and 2nd order one-wave
PVM and Roe schemes.
Figure: Accuracy test: Efficiency and ratio of convergence of the one-wave PVM schemes:
comparison with Roe scheme.
Introduction
PVM F.V. methods
Numerical tests
2D Accuracy test
−1
10
−2
Error
10
−3
10
ROE 1st order
HLL 1st order
PVM−2U(SM,Sm) 1st order
PVM−4(SM,SI) 1st order
ROE 2nd order
HLL 2nd order
PVM−2U(SM,Sm) 2nd order
−4
10 −2
10
PVM−4(SM,SI) 2nd order
0
10
2
10
CPU time
4
10
6
10
(a) CPU time versus errors: 1st and 2nd order one-wave
PVM and Roe schemes.
Figure: Accuracy test: Efficiency and ratio of convergence of the two-waves PVM schemes:
comparison with Roe scheme.
Introduction
PVM F.V. methods
Numerical tests
2D Accuracy test
−1
10
−2
Error
10
ROE 1st order
HLL 1st order
PVM−2U(SM,Sm) 1st order
−3
10
PVM−4(SM,SI) 1st order
ROE 2nd order
HLL 2nd order
PVM−2U(SM,Sm) 2nd order
−4
10
PVM−4(SM,SI) 2nd order
1st order
2nd order
−2
10
−1
10
0
10
!
(a) CPU time versus errors: 1st and 2nd order two-waves
PVM and Roe schemes.
Figure: Accuracy test: Efficiency and ratio of convergence of the two-waves PVM schemes:
comparison with Roe scheme.
Introduction
PVM F.V. methods
Numerical tests
2D accuracy test
A speed-up of about 2.6 is obtained for first order PVM schemes with respect to
Roe and of about 4.7 for second order schemes.
Concerning the ratio of convergence, second order schemes achieves the
expected ratio of convergence.
Second order PVM-4(SM , SI ) scheme is the one that performs the best results
among the PVM schemes concerning the efficiency, but no significant
differences can be found with the others PVM schemes for this test.
Introduction
PVM F.V. methods
Numerical tests
Circular dam break
Let us consider a 2D test using a first and second order extension of the
PVM-2U(SM , Sm ) scheme to two-dimensional domains. The domain is
[−2, 2] × [−2, 2].
The bottom function is given by b(x, y) = 0.5e−0.5(x
As initial condition we set ~us = ~uf = ~0 and
2
+y2 )
.
p
1 − b(x, y) + 0.5 if x2 + y2 ≤ 0.5,
1 − b(x, y)
otherwise,
p

0.1 if x2 + y2 ≤ 0.5,
ϕ(x, y, 0) =
0.9 otherwise.

h(x, y, 0) =
Wall boundary conditions are set: ~us · ~
η = ~uf · ~
η = 0, where ~
η is the unit normal
vector to the boundaries.
A mesh of 200x200 cells has been considered and a reference solution is
computed using the PVM-2U(SM ,Sm ) first order scheme with a mesh with
800x800 cells.
Introduction
PVM F.V. methods
Numerical tests
Evolution of the free surface η = h + b
Time : 0.4
Time : 0.4
2
2
1.07
1.5
1.06
1.08
1.5
1
1.05
1
0.5
1.04
0.5
1.03
0
1.06
1.04
0
1.02
1.02
−0.5
1.01
−1
1
0.99
−1.5
−2
−2
0.98
−1
0
1
2
−0.5
−1.5
−2
−2
(a) First order PVM-2U(SM , Sm )
−1
0
1
2
Time : 1
2
1
0.98
(b) Second order PVM-2U(SM , Sm )
Time : 1
1.5
1
−1
2
1.08
1.06
1.1
1.5
1
1.05
0.5
1.04
0
0.5
0
1.02
−0.5
−0.5
1
−1
−1.5
−1
0.98
−1.5
1
Introduction
PVM F.V. methods
Numerical tests
Evolution of the solid volume fraction ϕ
Time : 0.4
Time : 0.4
2
2
0.85
1.5
0.8
1
0.5
0.75
0
0.85
1.5
0.8
1
0.75
0.5
0
0.7
0.7
−0.5
−0.5
0.65
0.65
−1
−1
0.6
−1.5
0.6
−1.5
0.55
−2
−2
−1
0
1
−2
−2
2
(a) First order PVM-2U(SM , Sm )
0
1
2
(b) Second order PVM-2U(SM , Sm )
Time : 1
Time : 1
2
2
0.95
1.5
1
−1
1.5
0.9
0.5
1
0.95
0.9
0.5
0.85
0
0.85
0
0.8
−0.5
−1
0.8
−0.5
−1
0.75
Introduction
PVM F.V. methods
Numerical tests
Evolution of the free surface. 1D Sections at x = 0 and x = y.
1.25
1.25
Free surface at x=0
Free surface at x=y
Reference solution
1.2
1.15
1.15
1.1
1.1
1.05
1.05
1
1
0.95
−3
−2
−1
0
1
2
(a) First order PVM-2U(SM , Sm )
3
0.95
−3
1.1
1.1
1.08
1.06
1.06
1.04
1.04
1.02
1.02
1
1
0.98
0.98
0.94
Free surface at x=0
Free surface at x=y
Reference solution
−2
−1
0
1
2
3
(b) Second order PVM-2U(SM , Sm )
1.08
0.96
Free surface at x=0
Free surface at x=y
Reference solution
1.2
0.96
0.94
Free surface at x=0
Free surface at x=y
Reference solution
Introduction
PVM F.V. methods
Numerical tests
Evolution of the solid volume fraction. 1D Sections at x = 0 and x = y.
1
1
0.8
0.8
0.6
0.6
0.4
0.4
! at x=0
! at x=y
Reference solution
0.2
0
−3
−2
−1
0
1
2
(a) First order PVM-2U(SM , Sm )
0.95
3
0
−3
0.9
0.85
0.8
0.8
0.75
0.75
0.7
0.7
0.65
0.65
0.5
−1
0
1
2
0.95
0.85
0.6
−2
! at x=0
! at x=y
Reference solution
3
(b) Second order PVM-2U(SM , Sm )
0.9
0.55
! at x=0
! at x=y
Reference solution
0.2
0.6
0.55
0.5
! at x=0
! at x=y
Reference solution
Introduction
PVM F.V. methods
Numerical tests
Evolution of the free surface. Comparison between Roe and PVM-2U(SM , Sm )
schemes at 1D section located at x = 0.
1.25
1.25
ROE
PVM−2U(SM,Sm)
1.2
1.15
1.15
1.1
1.1
1.05
1.05
1
1
0.95
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0.95
−2
(a) First order schemes
1.1
1.1
1.08
1.06
1.06
1.04
1.04
1.02
1.02
1
1
0.98
0.98
ROE
PVM−2U(SM,Sm)
Reference solution
−1.5
−1
−0.5
0
0.5
1
1.5
(b) Second order schemes
1.08
0.96
ROE
PVM−2U(SM,Sm)
1.2
Reference solution
0.96
ROE
PVM−2U(SM,Sm)
2
Introduction
PVM F.V. methods
Test: friction effect
Landslide experiment.
Haga click para visualizar la simulacin
Numerical tests
Introduction
PVM F.V. methods
Test: friction effect (without friction)
Landslide experiment.
Haga click para visualizar la simulacin
Numerical tests
Introduction
PVM F.V. methods
Numerical tests
Conclusions
Conclusions
PVM-n(S1 ,. . . ,Sk ) methods,
Defined in terms of viscosity matrices computed by a suitable polynomial
evaluation of a Roe matrix
They only need some information about the eigenvalues of the system to be
defined, and no spectral decomposition of Roe Matrix is needed.
As consequence, they are faster than Roe method.
These methods can be seen as a generalization of the schemes introduced by
Degond, Peyrard, Russo and Villedieu.
They include upwind and centered schemes such as: Lax-Friedrichs, Rusanov,
HLL, FORCE or GFORCE method.
Some new solvers are also proposed. See also:
E.D. Fernández-Nieto, M.J. Castro, C. Parés. On an intermediate field
capturing Riemann solver based on a parabolic viscosity matrix for the
two-layer shallow water system. J. Sci. Comp. 117-140, vol. 48, 2011.
Introduction
PVM F.V. methods
Two-phase models for debris flows.
Numerical approach by PVM methods
Enrique D. Fernández-Nieto
Dpto. Matemática Aplicada I,
University of Sevilla
Multiphase flow in industrial and environmental engineering
Chambery, 2012
Numerical tests