a finite volume scheme for the shallow-water system with...

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).



- 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



- Generic 2nd-order FV scheme

Non-linear 1D-SW system

Discretisation: time



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


Discretisation: space



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


Hydrostatic reconstruction (Audusse et al., 2004)

Define:

Set:



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 second-order scheme (Audusse et al. methodology)

Setting the discretisation of the 1D-SW system leads to:

where



( , , ),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!



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

MUSCL in brief

Reconstructed values at the left- and right-hand side of interfaces:

where



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:



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



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)



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



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



max,min,max(| |, | |) .i Ci

max,min,0.ii

max,min,

max,min,

min(| |, | |)1 .

max(| |, | |)

iiS

ii

- MOOD vs. MUSCL

MOOD detector algorithm



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:



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



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)



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



- Numerical results

DAM BREAK ON A WET BED

MOOD MUSCL



Mesh: cells;5, 0 25,

100 3; ( ,0)1, 25 50.

xT x

x

- Numerical results


MOOD MUSCL



- Numerical results


MOOD:

MUSCL:



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



Mesh: cells; . 100 1.5T

cells10000

st1 -order

- Numerical results

DRY/WET SIMULATION (smooth bathymetry)

MOOD MUSCL




cells10000

st1 -order

- Numerical results

DRY/WET SIMULATION (discontinuous bathymetry)

MOOD MUSCL




cells10000

st1 -orderMOOD involves and !h u

- Numerical results

DRY/WET SIMULATION (discontinuous bathymetry)

MOOD MUSCL



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).



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