Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 1 / 60 Numerical methods for shallow water wave equations Th. Katsaounis Dept. of Applied Math. Univ. of Crete, IACM, FORTH, Crete, GREECE Supported by ACMAC project EU-FP7 joint work with A. Delis TUC, Greece
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Numerical methods shallow water wave equations · 2010-11-18 · Numerical methods for shallow water wave equations Th. Katsaounis Dept. of Applied Math. Univ. of Crete, IACM, FORTH,
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Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 1 / 60
Numerical methodsfor
shallow water wave equations
Th. Katsaounis
Dept. of Applied Math. Univ. of Crete,IACM, FORTH, Crete, GREECE
Supported by ACMAC project EU-FP7
joint work withA. Delis TUC, Greece
Overview
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 2 / 60
Finite volume schemes for Conservation Laws
Relaxation Model for Conservation Laws
– Scalar and Systems of Cons. Laws
– Cons. Laws with Source Terms
Shallow Water Equations
– Relaxation Model - Schemes
– Numerical Results
Dispersive water wave models
– Finite volume schemes
– Numerical Results
Finite volume schemes for Conservation Laws
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 3 / 60
Scalar Conservation Laws
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 4 / 60
We consider the initial value problem for scalar conservation laws
ut + f(u)x = 0, x ∈ R, t > 0
u(x, 0) = u0(x), x ∈ R
Consider a uniform partition of R× R+ in cells [xi− 1
2
, xi+ 1
2
]× [tn, tn+1]
xi− 1
2
xi+ 1
2
tn
tn+1
xi−1 xi xi+1 xi = i∆x, i ∈ Z, xi+ 1
2
=xi+xi+1
2 , tn = n∆t
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 5 / 60
0 =
∫ tn+1
tn
∫ xi+1
2
xi− 1
2
[ut + f(u)x] dx dt
=
∫ xi+1
2
xi− 1
2
[u(x, tn+1)− u(x, tn)
]dx+
∫ tn+1
tn
[f(u(xi+ 1
2
, t))− f(u(xi− 1
2
, t))]dt
= Change of Mass+ Difference of Fluxes in cell [xi− 1
2
, xi+ 1
2
]× [tn, tn+1]
Uni ∼ 1
∆x
∫ xi+1
2
xi− 1
2
u(x, tn)dx, Fni+ 1
2
:= F (U+,ni , U−,n
i+1 ) ∼1
∆t
∫ tn+1
tnf(u(xi+ 1
2
, t))dt
Uni approximates the average of u in Ci = [xi− 1
2
, xi+ 1
2
] at time tn
Fni+ 1
2
approximates the average in [tn, tn+1] at x = xi+ 1
2
F is a numerical flux functionU+,ni some approximation of u(xi+ 1
2
− 0, tn)
U−,ni+1 some approximation of u(xi+ 1
2
+ 0, tn)
Basic FV scheme
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 6 / 60
3. Higher order polynomials are constructed using the cell averages Ui.
Reconstruction process
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 7 / 60
Constant xi− 3
2
xi− 1
2
xi+ 1
2
xi+ 3
2
Ui−1Ui U
+i
Ui+1U−i+1
Linear
b
b
b
xi− 3
2
xi− 1
2
xi+ 1
2
xi+ 3
2
Ui
U+i
Ui+1
U−i+1
Relaxation model for Conservation Laws
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 8 / 60
Relaxation Model for Scalar CL
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 9 / 60
ut + f(u)x = 0, x ∈ R, t > 0,
u(x, 0) = u0(x), x ∈ R.(1)
Relaxation system proposed by Jin & Xin 1995
ut + vx = 0,
vt + c2ux = −1
ǫ(v − f(u)), ǫ → 0
(2)
This system can be viewed as a regularization of (1) by the wave operator
ut + f(u)x = −ǫ(utt − c2uxx) +O(ǫ2).
Applying the Champan-Enskog expansion we get
ut + f(u)x = ǫ∂x((c2 − f ′(u)2
)∂xu
)+O(ǫ2).
If the subcharacteristic condition : |f ′(u)| < c holds then a rigorousconvergence analysis, for 1D scalar case, can be applied yielding at therelaxation limit ǫ → 0 the conservation law (1). (JX, 1995)
Relaxation Model with Source term
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 10 / 60
For a conservation law with a source term
ut + f(u)x = q(u), x ∈ R, t > 0,
u(x, 0) = u0(x), x ∈ R,(3)
a relaxation system considered takes the form
ut + vx = q(u),
vt + c2ux = −1
ǫ(v − f(u)),
(4)
yielding the following regularization of (3),
ut + f(u)x = q(u) + ǫq(u)t − ǫ(utt − c2uxx).
Remarks1) Same subcharacteristic condition2) Extra term : ǫq(u)t.3) In general (4) does not preserve the steady states.4) The time discretization of (4)
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 11 / 60
Un+1 − Un
∆t+ V n
x = q(Un+1),
V n+1 − V n
∆t+ c2Un
x = −1
ǫ(V n+1 − f(Un+1)),
(5)
is fully coupled system, not the case for the corresponding time discetizationof (2).An alternative approach : we consider the following relaxation system
ut + vx = 0,
vt + c2ux = −1
ǫ(v − f(u))− 1
ǫR(u),
(6)
where R(u) is an antiderivative of q(u),
R(u(x)) =
∫ x
q(u(s))ds .
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 12 / 60
In this case (6) provides exactly a wave-type regularization of (3),
ut + f(u)x = q(u)− ǫ(utt − c2uxx). (7)
Also an implicit-explicit time discretization is now possible when we treat thesource terms implicitly:
Un+1 − Un
∆t+ V n
x = 0,
V n+1 − V n
∆t+ c2Un
x =− 1
ǫ
(V n+1 − f(Un+1)
)− 1
ǫR(Un+1).
(8)
If |f ′(u)| < c from (7) we recover formally (3)
System (6) preserves steady states
Initial, Boundary Cond. : v0 = f(u0), vb = f(ub)
System of CL
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 13 / 60
∂tu+d∑
j=1
∂xjFj(u) = 0, x ∈ R
d, u = u(x, t) ∈ Rn, t > 0
u(·, 0) = u0(·)
Relaxation model
∂tu+d∑
j=1
∂xjvj = 0,
∂tvi +Ai∂xiu = −1
ǫ(vi − Fi(u)) , i = 1, . . . , d
it’s a regularization by a wave operator of order ǫ, and Ai are symmetricpositive definite matrices with constant coefficients that are selected to satisfythe corresponding sub-characteristic conditions.
Shalllow Water Equations (SWE)
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 14 / 60
Shallow water eqns (1D)
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 15 / 60
ht + (hu)x = 0,
(hu)t + (hu2 +g
2h2)x = −ghZ ′,
(9)
General steady states :Q = hu = Cnst u2
2 + g(h+ Z) = CnstSWE is a hyperbolic system with source term
SW Relaxation Models
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 16 / 60
Relaxation Model A
ht + vx = 0
Qt + wx = −ghZ ′
vt + c21hx = −1
ǫ(v −Q)
wt + c22Qx = −1
ǫ(w − (
Q2
h+
g
2h2))
Relaxation Model B
ht + vx = 0
Qt + wx = 0
vt + c21hx = −1
ǫ(v −Q)
wt + c22Qx = −1
ǫ(w − (
Q2
h+
g
2h2)) +
1
ǫR(Z;h)
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 17 / 60
R(Z;h)(x) =
∫ x
g(hZ ′)(y)dy
c1, c2 are chosen according to sub-characteristic condition :
|λi(F′)| < ci, i = 1, 2, F ′ = Jacobian of flux vector
For Z ≡ 0, (A) ≡ (B)
For ǫ → 0 we recover the original SW system
Both relaxation systems have linear principal part
Implicit-explicit time discretizations for (B)
System (B) have same steady states as the continuous problem
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 18 / 60
We consider the Relaxation Model B and let
u =
[hQ
], v =
[vw
],
our system can be rewritten as
ut + vx = 0,
vt +C2ux = −1
ǫ(v − F(u))− 1
ǫS(u),
F(u) = (Q,Q2
h+
g
2h2)T , S(u) = (0,−
∫ x
gh(y)Z ′(y)dy)T
where u,v ∈ R2 and C
2 ∈ R2×2 is a positive matrix.
Relaxation Schemes for SWE (1D)
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 19 / 60
Upwind Scheme
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 20 / 60
We assume a uniform spaced grid with ∆x = xi+ 1
2
− xi− 1
2
and a uniform time
step ∆t = tn+1 − tn, n = 0, 1, 2, . . ..
uni ∼ 1
∆x
∫ xi+1
2
xi− 1
2
u(x, tn)dx, uni+ 1
2
∼ u(xi+ 1
2
, tn)
We start by considering the following one-step conservative system for thehomogeneous case (no source term present)
∂
∂tui +
1
∆x(vi+ 1
2
− vi− 1
2
) = 0,
∂
∂tvi +
1
∆xC
2(ui+ 1
2
− ui− 1
2
) = −1
ǫ(vi − F(ui)).
The linear hyperbolic part has two Riemann invariants (characteristic speeds)v ±Cu associated with the characteristic fields ±C respectively. The firstorder upwind approximation of v ±Cu is
(v +Cu)i+ 1
2
= (v +Cu)i, (v −Cu)i+ 1
2
= (v −Cu)i+1.
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 21 / 60
Hence,
ui+ 1
2
=1
2(ui + ui+1)−
1
2C
−1(vi+1 − vi),
vi+ 1
2
=1
2(vi + vi+1)−
1
2C(ui+1 − ui).
First order upwind semi-discrete approximation of the relaxation scheme :
∂
∂tui +
1
2∆x(vi+1 − vi−1)−
1
2∆xC(ui+1 − 2ui + ui−1) = 0,
∂
∂tvi +
1
2∆xC
2(ui+1 − ui−1)−1
2∆xC(vi+1 − 2vi + vi−1) = −1
ǫ(vi − F(ui))
− 1
ǫS(ui),
where
S(ui) =
[0
−∫ x
i+12 gh(y)Z ′(y)dy
].
MUSCL Scheme
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 22 / 60
We replace the piecewise constant approximation by a MUSCL piecewiselinear interpolation : for the k−th component of v ±Cu we have:
(v + cku)i+ 1
2
= (v + cku)i +1
2∆xs+i ,
(v − cku)i+ 1
2
= (v − cku)i+1 −1
2∆xs−i+1,
where u, v are the k−th components of v,u and the slopes s± in the i−thcell :
s±i =1
∆x(vi+1 ± ckui+1 − vi ∓ ckui)φ(θ
±i )
θ±i =vi ± ckui − vi−1 ∓ ckui−1
vi+1 ± ckui+1 − vi ∓ ckui,
where φ is a limiter function satisfying 0 ≤ φ(θ) ≤ minmod(2, 2θ).MinMod (MM) : φ(θ) = max(0,min(1, θ)),
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 28 / 60
We consider a channel of length L = 2000m. A dam is located atx0 = 1000m and at time t = 0 the dam collapses. We compute the solutionfor time T = 50s with initial conditions:
u(x, 0) = 0, h(x, 0) =
h1 x ≤ 1000,
h0 x > 1000,
with h1 > h0. This is the Riemann problem for the homogeneous problem.The flow consists of a shock wave (bore) travelling downstream and ararefaction wave (depression wave) travelling upstream. The upstream depthh1 was kept constant at 10m, while the downstream depth h0 was differentfor each problem.
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 29 / 60
0 200 400 600 800 1000 1200 1400 1600 1800 20004
5
6
7
8
9
10h
(m)
x (m)
Dam-break flow, h0/h1=0.5, (x) Upwind and (o)MUSCL with MC limiter.
Dam Break Flow, Supercritical
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 30 / 60
0 200 400 600 800 1000 1200 1400 1600 1800 20000
1
2
3
4
5
6
7
8
9
10
h (m
)
x (m)
Dam-break flow, h0/h1=0.05, (x)Upwind and (o)MUSCL with MC limiter.
Dam Break Flow, Strongly Supercritical
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 31 / 60
0 200 400 600 800 1000 1200 1400 1600 1800 20000
1
2
3
4
5
6
7
8
9
10
h (m
)
x (m)
Dam-break flow, h0/h1=0.005, (x)Upwind and (o)MUSCL with MC limiter.
Dry Bed problem, h0 = 0
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 32 / 60
This a challenging problem as a result of the singularity that occurs at thetransition point of the advancing front. We compute the solution at T = 40s
No modifications to the scheme to incorporate the dry area
Globally accurate results free of oscillations
The water height and discharge remain positive
The transition point between the wet and the dry zone is close to theexact one, but some difficulties appear on the velocity.
Overall the solution is stable, monotone with no special front trackingtechniques
CFL = 0.5 ∆x = 10m, c1 = 18, c2 = 16, ǫ = 1.E − 4
Dry Bed problem, Water Height
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 33 / 60
0 200 400 600 800 1000 1200 1400 1600 1800 20000
1
2
3
4
5
6
7
8
9
10
x (m)
h (m
)
Dry bed dam-break flow (h), (o)MUSCL with MM limiter.
Dry Bed problem, Discharge
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 34 / 60
0 200 400 600 800 1000 1200 1400 1600 1800 20000
5
10
15
20
25
30
x (m)
q (m
2 /s)
Dry bed dam-break flow (q), (o)MUSCL with MM limiter.
Dry Bed problem, Velocity
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 35 / 60
0 200 400 600 800 1000 1200 1400 1600 1800 20000
2
4
6
8
10
12
14
16
18
20
u (m
/s)
x (m)
Dry bed dam-break flow (u), (o)MUSCL with MM limiter.
Flow at Rest, Z 6= 0
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 36 / 60
We consider at channel of length L = 25m with a non-trivial bathymetry Z,with initial conditions
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 37 / 60
0 5 10 15 20 250
0.5
1
1.5
2
2.5
Hei
gh t(
m)
Distance (m)
Flow at rest (water height): (+) standard source, (o) integral source
Flow at Rest, Discharge
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 38 / 60
0 5 10 15 20 25−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
x (m)
q (m
2 /s)
Flow at rest (discharge): (+) standard source, (o) integral source
Flow at Rest, Discharge (zoom)
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 39 / 60
5 10 15 20
−0.01
−0.005
0
0.005
0.01
x (m)
q (m
2 /s)
Flow at rest: Magnified view of the discharge.
Non trivial steady states
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 40 / 60
We consider with the convergence towards steady flow over the parabolichump in a channel of length L = 25m.Depending on the boundary conditions the flow maybe subcritical, transcriticalwith a shock or without a shock. In all cases we use MUSCL scheme with
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 41 / 60
0 5 10 15 20 250
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
h (m
)
x (m)
Transcritical flow with shock (h)
Transcritical Flow with shock : Discharge
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 42 / 60
0 5 10 15 20 250
0.05
0.1
0.15
0.2
0.25
0.3
x (m)
q (m
2 /s)
Transcritical flow with shock (q)
Drain on a non-flat bottom.
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 43 / 60
Difficult problem since it involves the calculation of dry areas.
BC’s : Upstream reflective, Downstream dry bed
IC’s : h+ Z = 0.5m and q = 0m3/s
Solution : a state at rest, on the left part of the hump withh+ Z = 0.2m with q = 0m3/s and a dry state (i.e. h = 0 andq = 0m3/s) on the right of the hump.
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 51 / 60
0
50
100
150
200
0
50
100
150
200
5
6
7
8
9
10
Y(m)X(m)
Wat
er D
epth
(m)
2D Partial Dam-Break, hd = 0.1m
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 52 / 60
0
50
100
150
200
0
50
100
150
200
0
2
4
6
8
10
Y(m)X(m)
Wat
er D
epth
(m)
2D Partial Dam-Break, hd = 0m
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 53 / 60
0
50
100
150
200
0
50
100
150
200
0
2
4
6
8
10
Y(m)
X(m)
Wat
er D
epth
(m)
Circular Dam Break
Dept. of Applied Mathematics Univ. of Crete Numerical methods for shallow water waves – 54 / 60
A two dimensional Riemann problem for the 2D SWEs
Two regions of still water separated by a cylindrical wall with radius 11mcentered in a channel. The water depth within the cylinder is 10m and1m outside.
The wall is removed instantaneously, the bore waves will spread andpropagate radially and symmetrically
There is a transition from subcritical to supercritical flow.