Background One-dimensional models MSM Conclusions

Some mathematical problems in floodmodelling

Gavin Esler and Oliver Osvald

Mathematics, University College London

September 5, 2016

Background One-dimensional models MSM Conclusions

A flood

Background One-dimensional models MSM Conclusions

Some issues in flood modelling

State-of-the-art flood models (e.g. LISFLOOD-FP, UIM,Hi-PIMS) have certain features:

• “... simplified models may reproduce numerical results comparable to afull model [however] they are unable to simulate supercritical flowsaccurately.” [Liang and Smith, 2015, J. Hydroinfomatics].

• “For an explicit code this [time-step restriction] means that thecomputational cost will increase as (1/∆x)4.” [Bates et al., 2010, J.Hydrology].

• “The optimum time step is determined ... to avoid the ‘chequer board’oscillations.” [Chen et al., 2012, J. Hydrology].

• “... grid resolutions below the length scales of building size and streetwidth [are] required to provide consistent and accurate estimates ofurban flooding.” [Neal et al. 2007, J. Flood Risk. Man.].

Background One-dimensional models MSM Conclusions

Some issues in flood modelling

These raise certain questions...

• When are approximate (‘diffusion wave’) modelsacceptable? When are the full shallow water equationsrequired?

• How can onerous time-step restrictions be circumvented?

• How are numerical instabilities best suppressed?

• How can coarse resolution models be constructed whichtake account of small-scale structure?

Can mathematics help...?

Background One-dimensional models MSM Conclusions

Shallow water equations

Consider the nondimensional 1D SWE with drag:

ut + uux + hx + bx = −Ch−4/3|u|u SWEht + (uh)x = 0.

Here u(x , t) velocity, h(x , t) layer thickness, b(x) topography.

• Units: h,b ∼ H, u ∼ (gH)1/2.

Leaves

C =n2gH1/3

LH, n : Manning coefficient (sm−1/3).

(L: typical horizontal scale).

Background One-dimensional models MSM Conclusions

Friction dominated flows

Alternatively

C =L

LF, LF =

H4/3

n2g, Friction length

• Flows with L & LF will be friction dominated.

• Flows with L . LF will exhibit SWE phenomenology(shocks, hydraulic control etc.).

Typical values: n ≈ 0.02 (road surface) to 0.1 (floodplain withheavy brush) sm−1/3, g ≈ 10ms−2, H ≈ 1m, gives

LF ≈ 10− 250m.

Background One-dimensional models MSM Conclusions

Friction dominated flows: Asymptotics

Write C = ε−1 where ε� 1. Then expand

u(x , t , ε) = ε1/2 (u0(x , t) + εu1(x , t) + ...) .

=⇒ hierarchy of models (!?).

Leading order: Find u0 = U(h, s) where s = hx + bx (interfaceslope), and

U(h, s) = −sgn(s)|s|1/2h2/3.

Leads to the diffusion wave model (with b = 0 here)

hτ = (sgn(hx )|hx |1/2h5/3)x , DIFFWAVE

where τ = ε1/2t is a rescaled time.

Background One-dimensional models MSM Conclusions

Friction dominated flows: Asymptotics

First order: Obtain

hτ = − (U(h, s)h + εu1h)x ,

where

u1 = −|s|1/2h2

2

(sx

6s− 4hx

9h− h

2s

(5hxx

3h− 5h2

x

9h2 +5hx sx

3hs+

sxx

2s− s2

x

4s2

)).

• Well-posed (sign of hyperdiffusion term always negative).• Ill-conditioned as s → 0.• Needs to be regularised to be useful!

In fact: need to solve DIFFWAVE regularisation problem first.

Background One-dimensional models MSM Conclusions

DIFFWAVE versus SWE

Notice that:

• The SWE are hyperbolic (c.f. wave equation)=⇒ information propagates along characteristics1.

• DIFFWAVE is parabolic (c.f. diffusion equation)=⇒ information transmitted instantaneously everywhere.

Important influence on numerics:

• In hyperbolic system CFL criterion ∆t . ∆x/cmax.

• In parabolic system ∆t . (∆x)2/κ (κ diffusivity).

Worse news: In DIFFWAVE κ ∼ |s|−1/2 =⇒ ∆t → 0 as theinterface flattens (s → 0).

1(with max. speed cmax = |u|+ h1/2).

Background One-dimensional models MSM Conclusions

Friction in the SWE

SWE numerical solutions (CLAWPACK) with increasing C.

0 1 2 3 4 5 6 7 8 9 10

Distance

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Inte

rface h

eig

ht

SWE, C=0

SWE, C=1/4

SWE, C=1

SWE, C=4

SWE, C=16

T=4

Friction: • damps shocks and • does not affect propagation ofinformation in rarefactions.

Background One-dimensional models MSM Conclusions

Dam-breaks in DIFFWAVE

The diffwave (partial) dam-break (hl < 1) problem:

hτ = (h1/2x h5/3)x , h(x ,0) =

{hl x < 01 x ≥ 0

.

has a similarity solution:

h(x , τ) = F (s), where s =xτ2/3 .

Here F (s) satisfies the 2nd-order ODE boundary value problem

F ′′(s) +10F ′(s)2F (s)2/3 + 4sF ′(s)3/2

3F (s)5/3 = 0,F (−∞) = hl ,F (+∞) = 1.

Can be solved numerically using the shooting method.

Background One-dimensional models MSM Conclusions

Dam-breaks in DIFFWAVE

Dam-break solution has a universal profile:

-5 -4 -3 -2 -1 0 1 2 3 4 5

Distance

0

0.2

0.4

0.6

0.8

1

Inte

rfa

ce

he

igh

tSimilarity solution

Numerical calculation

t=0.2t=1

t=0.5

• Notice that h(0, t) = F (0) is constant.• Flux of fluid across x = 0 is F ′(0)1/2F (0)5/3t−1/3.

Background One-dimensional models MSM Conclusions

DIFFWAVE versus SWE

Comparison with DIFFWAVE - need to rescale time.

0 1 2 3 4 5 6 7 8 9 10

Distance

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Inte

rface h

eig

ht

SWE C=4, T=1

SWE C=16, T=2

SWE C=64, T=4

DIFFWAVE, t=0.5

In fact: all SWE dam-breaks converge→ the DIFFWAVE profile(timescale LF/

√gH, non-dimensional C−1).

Background One-dimensional models MSM Conclusions

Multiscale methods and flooding

• Multiscale methods (MSM): techniques used to obtainaveraged equations, accounting for small-scale structure inphysical problems.

• For example: used to model flow through porous media,properties of metamaterials, crystallography.

• Small-scale structure can be regular (e.g. periodic) orrandom.

• How can this be applied to flooding?

Background One-dimensional models MSM Conclusions

Textbook example problem(e.g. Holmes, Introduction to perturbation methods, Ch. 5)• Consider the diffusion equation

ut = (κux )x

where κ(x) is rapidly-varying with small-scale structure.• Naive approach: Use a coarse-grain average

ut = (〈κ〉ux )x , 〈f 〉 =1L

∫ x+L/2

x−L/2f (x̄) dx̄

• MSM result:

ut = (κeffux )x , κeff = 〈κ−1〉−1.

Background One-dimensional models MSM Conclusions

Multiscale methods and flooding: Example

As a (simple) model problem consider flooding through anetwork of streets.

n

W

Ln

Le

e

W

Can we obtain a model for which we do not need to resolveevery street and junction?

Background One-dimensional models MSM Conclusions

Model Equations

Flow in channels satisfies (generalised) DIFFWAVE equations

het + (F (he,he

x ))x = 0,hn

t +(G(hn,hn

y ))

y = 0.

Heights: he(x , t) (east-west roads), hn(y , t) (north-south).

F (h, s), G(h, s) differentiable functions.

Junction boundary conditions (network limit):

he = hn, [F (he,hex )]+− + [G(he,he

y )]+− = 0.

[·]+− - difference between quantity on positive x (or y ) side of thejunction and negative side.

Background One-dimensional models MSM Conclusions

Multiscale analysis

• Choose length L� Le,Ln (junction spacings) for (x , y).

• Small parameter ε = Le/L� 1. Junction ratio β = Ln/Le.

• Seek solution using multi-scale ansatz

he(x , y , t) = h0(x , y , t) +∞∑

k=1

εkhek (X , x , y , t)

hn(x , y , t) = h0(x , y , t) +∞∑

k=1

εkhnk (Y , x , y , t)

• Here (X ,Y ) = (x/ε, y/βε) are the ‘cell-scale’ variables.

• Cell-scale solutions are periodic on X ∈ [0,1).

Background One-dimensional models MSM Conclusions

Multiscale analysis

• Insert ansatz into DIFFWAVE equations. Use multi-scaleformalism ∂x → ε−1∂X + ∂x .

• Leading order:

∂sF (h0,h0x + he1X ) he

1XX = 0∂sG(h0,h0y + hn

1Y ) hn1YY = 0.

• Apply periodic boundary conditions:

he1 = he

1(x , y , t), hn1 = hn

1(x , y , t).

Background One-dimensional models MSM Conclusions

Multiscale analysis

• Next order (cell problem):

h0t + ∂hF (h0,h0x ) h0x + ∂sF (h0,h0x ) (h0xx + he2XX ) = 0

h0t + ∂hG(h0,h0y ) h0y + ∂sG(h0,h0y )(h0yy + hn

2YY)

= 0.

• Integrate over edges of ‘block’ to get

∂sF (h0,h0x )[he2X ]+− = h0t + ∂xF (h0,h0x )

∂sG(h0,h0y )[hn2Y ]+− = β

(h0t + ∂yG(h0,h0y )

)• Finally observe that

[F ]+− = ε∂sF (h0,h0x )[he2X ]+−, [G]+− = ε∂sG(h0,h0y )[hn

2Y ]+−,

which allows use of the flux b.c.

Background One-dimensional models MSM Conclusions

Multiscale equation

Using the flux boundary condition:

Homogenised eqn.

h0t +1

1 + β(F (h0,h0x ))x +

β

1 + β

(G(h0,h0y )

)y = 0.

• Describes large-scale (slow) evolution at leading order.

• Includes effect of the street network without resolving it.

• Can therefore be integrated on much coarser grid.

Background One-dimensional models MSM Conclusions

Model comparison

Comparison between full model (resolves network) andhomogenised equations:

Initial Full Network

Homog. Eqn.

Square network: captures anisotropic diffusion.

Background One-dimensional models MSM Conclusions

Model comparison

Comparison between full model (resolves network) andhomogenised equations:

Initial Full Network

Homog. Eqn.

Rectangular network: diffusion faster in x-direction.

Background One-dimensional models MSM Conclusions

Model comparison: computational savings

• Full model: needs to resolve junctions. 10 grid pointsbetween junctions =⇒ 500 × 100 grid points.∆t ∝ (1/500)2.

• Homogenised model: Resolution 1 point per street =⇒50 × 50 grid. ∆t ∝ (1/50)2.

Saving factor (approx): 2000.

Background One-dimensional models MSM Conclusions

Conclusions and outlook

Mathematics will (should?) be useful for:

• Modifying DIFFWAVE models to:1. Track SWE solutions more accurately.2. Minimize expensive CFL restrictions.3. Optimize time-steps to suppress numerical instabilities.

• Providing (semi-)analytical results to calibrate models.

• Identifying conditions to switch between DIFFWAVE andSWE in regions where the latter are needed.

• Deriving coarse-grain homogenized models including theeffects of small-scale structure.