a finite volume scheme for the shallow-water system with...
TRANSCRIPT
-
A comparison between the second-order MOOD and MUSCL methods for the 1D shallow water problem in the presence of dry/wet interfaces
J. Figueiredo, S. Clain
Centre of Mathematics, University of Minho
Financial support:
FEDER (COMPETE) FCT (Pest-C/MAT/UI0013/2014, PTDC/MAT/121185/2010, FCT-ANR/MAT-NAN/0122/2012)
-
- Presentation outline
• Motivation
• Generic second-order FV scheme
• MOOD vs. MUSCL
• Numerical results
• Conclusions and future work
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 2
-
- Motivation
o The SW problem with varying bathymetry has a wide number of applications: tsunami, flooding, coastal erosion…
o A large number of numerical schemes exists, in particular very high-order FV schemes.
o Still, new 2nd-order methods are interesting because these techniques are widely used in engineering and environmental applications due to their simplicity and computational efficiency but need to be improved.
o The MUSCL method is often used in this context, but recently a new method, MOOD, involving a different limiting approach has been proposed and tested, for example, for the Euler system and for the 2D-SW problem with varying bathymetry (up to 6th-order of accuracy on unstructured meshes).
o MOOD and MUSCL differ essentially in the limiting procedure used to prevent oscillations in the vicinity of discontinuities (a priori for MUSCL, a posteriori for MOOD).
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 3
-
- Motivation
o Aim: compare the “performance” of MOOD vs. MUSCL for the 1D-SW system with varying bathymetry.
accuracy (order of convergence for smooth solutions)
Assess: shock capturing ability (sharpness and oscillations)
behaviour of the numerical solution near dry/wet interfaces
set of numerical simulations of different nature
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 4
-
- Generic 2nd-order FV scheme
Non-linear 1D-SW system
Discretisation: time
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 5
x
2 2
0,
( ) ,2
t x
t x x
h hu
ghu hu h gh b
hydrostatic condition: h b
b
h
0 10, 0 n NT t t t t T
1
time step:
if regular subdivision
t , k 1, , . t t k kk k
Tt t N
N
-
- Generic 2nd-order FV scheme
Discretisation: space
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 6
non-overlapping cells:0, I L
0 L
1c 1ic ic 1ic Ic
( )ni ix
1ix 1ix ix
1/2ix 1/2i
x
For a function at :nt
1/2, 1/2
1/2, 1/2
( )
( )
ni L i
ni R i
x
x
i ix c
-
- Generic 2nd-order FV scheme
Hydrostatic reconstruction (Audusse et al., 2004)
Define:
Set:
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 7
1/21/2 , 1/2,max( , )
i i L i Rb b b
1/2,
1/2, 1/2, 1/2 1/2,
1/2, 1/2, 1/2
1/2,, .
max
with
(0, ),
, ,
ii K
i K i K
i K i
i i K K
K
iK L R
h
h
h b
b
b
uu
1/2,i Lb
1/2,i Rb1/2i
b
1/2,i Lh
1/2,i Lh
1/2ix
-
- Generic 2nd-order FV scheme
Generic second-order scheme (Audusse et al. methodology)
Setting the discretisation of the 1D-SW system leads to:
where
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 8
( , , ),U h hu b
1/2,1/2 1/2 1/21 , ,nn ni ii
n nni R iiLi i
ntU U tx
S
F F
, ,1/2, 1/1 2,/2 , ,n ni L Rni iU U FF
2 2
,1/2, 1/2,1/2,
, ,with( ) ( ) , 2
ni
n ni K i KK
K L Rg
h h
1/2, 1/2, 1/2, 1/2,.
2
n n n ni L i R i L i
iR
i
nh h b b
xS g
-
- MOOD vs. MUSCL
Local second-order reconstruction
Local linear reconstruction required approximation of first derivatives:
But non-limited linear reconstruction will lead to oscillations in the vicinity of
discontinuities and even in less extreme cases.
Need to implement limiting procedures loss of accuracy!
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 9
1 11/2 1/2
1 1
interfaces:
centroid:
( ) , ( ) .
( ) .2
i ii ii i
i ii
p px x
px
-
- MOOD vs. MUSCL
Limitation strategy
MUSCL (Monotonic Upstream-Centred Scheme for Conservation Laws; van Leer 1979):
slopes are reduced in order to verify some stability criterion;
corrections to slopes are made before computing the solution
(a priori).
MOOD (Multi-dimensional Optimal Order Detection; Clain, Diot, Loubère 2011):
assumes solution is smooth enough and the candidate solution is
computed without changing the slopes; corrections to slopes are
made if the candidate solution is “problematic” (a posteriori).
MOOD generally less intrusive, providing better accuracy
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 10
-
- MOOD vs. MUSCL
MUSCL in brief
Reconstructed values at the left- and right-hand side of interfaces:
where
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 11
1/2, 1/2,
1/2, 1/2, 1/2, 1/2, 1/2, 1/2,
with( ) , ( ) , , , ,2 2
, ,
i i i ii R i L
i R i R i R i L i L i L
x xq q h hu
b h b h
1/2, 1/2,( ) ( ), ( )i i R i Lq p p min-mvan-Alabada
van-
od
Leer
-
- MOOD vs. MUSCL
MOOD in brief
- Key concepts: Cell/Edge Polynomial Degree (CPD/EPD)
EPD polynomial degree used to evaluate the reconstruction at both
sides of each edge:
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 12
11/2
full slope used (second )
nul
-
first-ol slope rd
ord
used ( )
e
e r
r
EPD ( ) min CPD ( ),CPD ( )
CPD ( )
1
0
i ii
i
-
- MOOD vs. MUSCL
MOOD in brief
- MOOD loop:
know approximation at time
initialise for each cell and function
compute candidate solution from CPD/EDP map
check if each satisfies a set of conditions (detectors)
if all cells are valid then otherwise reduce CPD and
next time step
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 13
1, ,( )n n
i i IU U nt
CPD 1
U
iU
1 ,nU U
-
- MOOD vs. MUSCL
MOOD basic detectors
- Aim: determine if is eligible (physically ok, no spurious oscillations)
Physical Admissible Detector:
(PAD)
Maximum Principle Detector:
(MPD)
Extrema Detector:
(ED)
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 14
iU
0ih
1 1 1 1min( , , ) max( , , )n n n n n n
i i ii i i i
1 1 1 1min( , ) max( , )ii i i i
-
- MOOD vs. MUSCL
MOOD relaxation detectors
- Aim: distinguish real (smooth) extrema from extrema caused by local
spurious oscillations (Gibbs phenomenon)
Local curvature indicator
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 15
1 11 2 12
2, , 12
( ) , ; ( ) ( ), ( ) ( ).( )
ii ii I Ii IC C C C C
x
max,1 1 1 1min,
max, max,min, min,
2, , 1min( , , ), max( , , ), .
( ), ( ).
i i ii i i ii
i ii i
i IC C C C C C
-
- MOOD vs. MUSCL
MOOD relaxation detectors
Small Curvature Detector:
(SCD)
numerical solution is locally linear
Local Oscillation Detector:
(LOD)
local oscillation due to variation of curvature sign
Smoothness Detector:
(SD)
numerical solution is locally smooth
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 16
max,min,max(| |, | |) .i Ci
max,min,0.ii
max,min,
max,min,
min(| |, | |)1 .
max(| |, | |)
iiS
ii
-
- MOOD vs. MUSCL
MOOD detector algorithm
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 17
PAD
U
ED
SCD
LOD
SD
0CPD
1CPD
no
yes
no
yes
1CPD
yes
no
0CPD
noyes
no
0CPD
1CPD yes
-
- Numerical results
General aspects
Time scheme: Conservative flux scheme:
Meshes: Dry/wet threshold:
MUSCL limiter procedure: MOOD limiter procedure:
Convergence assessment:
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 18
TVD-RK2, 0.4CFL
ic x
HLL
, ,h hu h
610
3with , ; 0.5C sx L
1
1
1- error: , - error: max N ex N ex
I
i i i iii
L LI
-
- Numerical results
LAKE AT REST
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 19
Meshes: cells; up to time steps. 50,100, 200 1 22 15T
15 141
C -property satisfie- error - error d10 , 10L L
-
- Numerical results
SMOOTH SOLUTION (steady-state supercritical flow)
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 20
2
Meshes: cells; . 100, ,1600 10 ( ) 0.2exp( 5( 4) ); T b x x
(0) 2
( , ) 13.29q x t
detector only oMUSCL MOn deteO ct r.D o; h
-
- Numerical results
SMOOTH SOLUTION (steady-state supercritical flow)
MUSCL
MOOD
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 21
-
- Numerical results
DAM BREAK ON A WET BED
MOOD MUSCL
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 22
Mesh: cells;5, 0 25,
100 3; ( ,0)1, 25 50.
xT x
x
-
- Numerical results
DAM BREAK ON A WET BED
MOOD MUSCL
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 23
-
- Numerical results
DAM BREAK ON A WET BED
MOOD MUSCL
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 24
-
- Numerical results
DAM BREAK ON A WET BED
MOOD:
MUSCL:
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 25
1
Overall convergence:
err ( )C x
0.037
0.072u
C
C
0.055
0.107u
C
C
-
- Numerical results
DRY/WET SIMULATION (smooth bathymetry)
MOOD MUSCL
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 26
Mesh: cells; . 100 1.5T
cells10000
st1 -order
-
- Numerical results
DRY/WET SIMULATION (smooth bathymetry)
MOOD MUSCL
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 27
Mesh: cells; . 100 1.5T
cells10000
st1 -order
-
- Numerical results
DRY/WET SIMULATION (discontinuous bathymetry)
MOOD MUSCL
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 28
Mesh: cells; . 100 2.75T
cells10000
st1 -orderMOOD involves and !h u
-
- Numerical results
DRY/WET SIMULATION (discontinuous bathymetry)
MOOD MUSCL
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 29
Mesh: cells; . 100 2.75T cells10000
st1 -order
-
- Conclusions and future work
• We performed comparisons between the 2nd-order MOOD and the MUSCL methods for the non-linear shallow water system with varying bathymetry.
• Both methods comply with the C-property, but the MOOD method is less
diffusive than the MUSCL method (shock, rarefaction propagation), and
is more accurate when dealing with smooth solutions.
• Dry/wet interfaces with smooth/discontinuous bathymetry are reasonably well handled by MOOD technique (better velocity / kinetic energy estimates).
• Still, new detectors and schemes have to be proposed to improve the treatment of wet/dry interfaces (location, height and velocity profiles).
MOOD vs. MUSCL for the dry/wet SW problem
SHARK-FV 2015, 18-22 May, Ofir J. Figueiredo, S. Clain 30