Gene Golub SIAM Summer School 2012 Numerical Methods for Wave - - PowerPoint PPT Presentation

▶
gene golub siam summer school 2012 numerical methods for
SMART_READER_LITE
LIVE PREVIEW

Gene Golub SIAM Summer School 2012 Numerical Methods for Wave - - PowerPoint PPT Presentation

Mar 17, 2023 45 likes •682 views

Gene Golub SIAM Summer School 2012 Numerical Methods for Wave Propagation Finite Volume Methods Lecture 3 Randall J. LeVeque Applied Mathematics University of Washington R.J. LeVeque, University of Washington Gene Golub SIAM Summer School

slide-1
SLIDE 1

Gene Golub SIAM Summer School 2012 Numerical Methods for Wave Propagation Finite Volume Methods Lecture 3

Randall J. LeVeque Applied Mathematics University of Washington

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-2
SLIDE 2

Outline

  • Shallow water equations with topography
  • Approximate Riemann solvers
  • f-wave formulation of wave-propagation method.
  • Well-balanced methods to preserve ocean-at-rest.
  • Dry state Riemann solvers

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-3
SLIDE 3

Great Tohoku Tsunami, 11 March 2011

Modeling and Simulating Tsunamis with an Eye to Hazard Mitigation, RJL and J. Behrens, SIAM News, May, 2011

http://www.siam.org/news/news.php?id=1882

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-4
SLIDE 4

Great Tohoku Tsunami, 11 March 2011

Modeling and Simulating Tsunamis with an Eye to Hazard Mitigation, RJL and J. Behrens, SIAM News, May, 2011

http://www.siam.org/news/news.php?id=1882

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-5
SLIDE 5

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-6
SLIDE 6

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-7
SLIDE 7

Great Tohoku Tsunami, 11 March 2011

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-8
SLIDE 8

Shallow water equations

h(x, t) = depth u(x, t) = velocity (depth averaged, varies only with x) Conservation of mass and momentum hu gives system of two equations. mass flux = hu, momentum flux = (hu)u + p where p = hydrostatic pressure ht + (hu)x = 0 (hu)t +

  • hu2 + 1

2gh2

  • x

= 0 Jacobian matrix: f′(q) =

  • 1

gh − u2 2u

  • ,

λ = u ±

  • gh.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-9
SLIDE 9

Shallow water equations

Hydrostatic pressure: Pressure at depth z > 0 below the surface is gz from weight of water above. Depth-averaged pressure is p = h gz dz = 1 2gz2

  • h

= 1 2gh2.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-10
SLIDE 10

The Riemann problem

Dam break problem for shallow water equations ht + (hu)x = 0 (hu)t +

  • hu2 + 1

2gh2

x = 0

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-11
SLIDE 11

The Riemann problem

Dam break problem for shallow water equations ht + (hu)x = 0 (hu)t +

  • hu2 + 1

2gh2

x = 0

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-12
SLIDE 12

The Riemann problem

Dam break problem for shallow water equations ht + (hu)x = 0 (hu)t +

  • hu2 + 1

2gh2

x = 0

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-13
SLIDE 13

The Riemann problem

Dam break problem for shallow water equations ht + (hu)x = 0 (hu)t +

  • hu2 + 1

2gh2

x = 0

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-14
SLIDE 14

The Riemann problem

Dam break problem for shallow water equations ht + (hu)x = 0 (hu)t +

  • hu2 + 1

2gh2

x = 0

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-15
SLIDE 15

Riemann solution for the SW equations in x-t plane

Similarity solution: Solution is constant on any ray: q(x, t) = Q(x/t) Riemann solution can be calculated for many problems. Linear: Eigenvector decomposition. Nonlinear: more difficult. In practice “approximate Riemann solvers” used numerically.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-16
SLIDE 16

An isolated shock

If an isolated shock with left and right states ql and qr is propagating at speed s then the Rankine-Hugoniot condition must be satisfied: f(qr) − f(ql) = s(qr − ql) For a system q ∈ l Rm this can only hold for certain pairs ql, qr: For a linear system, f(qr) − f(ql) = Aqr − Aql = A(qr − ql). So qr − ql must be an eigenvector of f′(q) = A.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-17
SLIDE 17

An isolated shock

If an isolated shock with left and right states ql and qr is propagating at speed s then the Rankine-Hugoniot condition must be satisfied: f(qr) − f(ql) = s(qr − ql) For a system q ∈ l Rm this can only hold for certain pairs ql, qr: For a linear system, f(qr) − f(ql) = Aqr − Aql = A(qr − ql). So qr − ql must be an eigenvector of f′(q) = A. A ∈ l Rm×m = ⇒ there will be m rays through ql in state space in the eigen-directions, and qr must lie on one of these.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-18
SLIDE 18

An isolated shock

If an isolated shock with left and right states ql and qr is propagating at speed s then the Rankine-Hugoniot condition must be satisfied: f(qr) − f(ql) = s(qr − ql) For a system q ∈ l Rm this can only hold for certain pairs ql, qr: For a linear system, f(qr) − f(ql) = Aqr − Aql = A(qr − ql). So qr − ql must be an eigenvector of f′(q) = A. A ∈ l Rm×m = ⇒ there will be m rays through ql in state space in the eigen-directions, and qr must lie on one of these. For a nonlinear system, there will be m curves through ql called the Hugoniot loci.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-19
SLIDE 19

Riemann solution for the SW equations in x-t plane

Nonlinear Riemann solution: If we know the 1-wave is a rarefaction wave, 2-wave is a shock, Can find h∗ by solving: ul + 2(

  • ghl −
  • gh∗) = ur + (h∗ − hr)
  • g

2 1 h∗ + 1 h r

  • .

Expensive to do at every cell interface!

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-20
SLIDE 20

HLL Solver

Harten – Lax – van Leer (1983): Use only 2 waves with s1 =minimum characteristic speed s2 =maximum characteristic speed W1 = Q∗ − Qℓ, W2 = Qr − Q∗ Conservation implies unique value for middle state Q∗: s1W1 + s2W2 = f(Qr) − f(Qℓ) = ⇒ Q∗ = f(Qr) − f(Qℓ) − s2Qr + s1Qℓ s1 − s2 .

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-21
SLIDE 21

HLL Solver

Harten – Lax – van Leer (1983): Use only 2 waves with s1 =minimum characteristic speed s2 =maximum characteristic speed W1 = Q∗ − Qℓ, W2 = Qr − Q∗ Conservation implies unique value for middle state Q∗: s1W1 + s2W2 = f(Qr) − f(Qℓ) = ⇒ Q∗ = f(Qr) − f(Qℓ) − s2Qr + s1Qℓ s1 − s2 . Choice of speeds:

  • Max and min of expected speeds over entire problem,
  • Max and min of eigenvalues of f′(Qℓ) and f′(Qr).

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-22
SLIDE 22

HLLE Solver

Einfeldt: Choice of speeds for gas dynamics (or shallow water) that guarantees positivity. Based on characteristic speeds and Roe averages: s1

i−1/2 = min p (min(λp i , ˆ

λp

i−1/2)),

s2

i−1/2 = max p (max(λp i+1, ˆ

λp

i−1/2)).

where λp

i is the pth eigenvalue of the Jacobian f′(Qi),

ˆ λp

i−1/2 is the pth eigenvalue using Roe average f′( ˆ

Qi−1/2)

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-23
SLIDE 23

Approximate Riemann Solvers

Approximate true Riemann solution by set of waves consisting

  • f finite jumps propagating at constant speeds.

Local linearization: Replace qt + f(q)x = 0 by qt + ˆ Aqx = 0, where ˆ A = ˆ A(ql, qr) ≈ f′(qave). Then decompose qr − ql = α1ˆ r1 + · · · αmˆ rm to obtain waves Wp = αpˆ rp with speeds sp = ˆ λp.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-24
SLIDE 24

Approximate Riemann Solvers

How to use? One approach: determine Q∗ = state along x/t = 0, Q∗ = Qi−1 +

  • p:sp<0

Wp, Fi−1/2 = f(Q∗), A−∆Q = Fi−1/2 − f(Qi−1), A+∆Q = f(Qi) − Fi−1/2.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-25
SLIDE 25

Approximate Riemann Solvers

How to use? One approach: determine Q∗ = state along x/t = 0, Q∗ = Qi−1 +

  • p:sp<0

Wp, Fi−1/2 = f(Q∗), A−∆Q = Fi−1/2 − f(Qi−1), A+∆Q = f(Qi) − Fi−1/2. Wave-propagation algorithm uses: A−∆Q =

  • p:sp<0

spWp, A+∆Q =

  • p:sp>0

spWp. Conservative only if A−∆Q + A+∆Q = f(Qi) − f(Qi−1). This holds for Roe solver.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-26
SLIDE 26

Roe Solver

Solve qt + ˆ Aqx = 0 where ˆ A satisfies ˆ A(qr − ql) = f(qr) − f(ql). Then:

  • Good approximation for weak waves (smooth flow)
  • Single shock captured exactly:

f(qr) − f(ql) = s(qr − ql) = ⇒ qr − ql is an eigenvector of ˆ A

  • Wave-propagation algorithm is conservative since

A−∆Qi−1/2+A+∆Qi−1/2 =

  • sp

i−1/2Wp i−1/2 = A

  • Wp

i−1/2.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-27
SLIDE 27

Roe Solver

Solve qt + ˆ Aqx = 0 where ˆ A satisfies ˆ A(qr − ql) = f(qr) − f(ql). Then:

  • Good approximation for weak waves (smooth flow)
  • Single shock captured exactly:

f(qr) − f(ql) = s(qr − ql) = ⇒ qr − ql is an eigenvector of ˆ A

  • Wave-propagation algorithm is conservative since

A−∆Qi−1/2+A+∆Qi−1/2 =

  • sp

i−1/2Wp i−1/2 = A

  • Wp

i−1/2.

Roe average ˆ A can be determined analytically for many important nonlinear systems (e.g. Euler, shallow water).

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-28
SLIDE 28

Roe solver for Shallow Water

Given hl, ul, hr, ur, define ¯ h = hl + hr 2 , ˆ u = √hlul + √hrur √hl + √ hr Then ˆ A = Jacobian matrix evaluated at this average state satisfies A(qr − ql) = f(qr) − f(ql).

  • Roe condition is satisfied,
  • Isolated shock modeled well,
  • Wave propagation algorithm is conservative,
  • High resolution methods obtained using corrections with

limited waves.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-29
SLIDE 29

Roe solver for Shallow Water

Given hl, ul, hr, ur, define ¯ h = hl + hr 2 , ˆ u = √hlul + √hrur √hl + √ hr Eigenvalues of ˆ A = f′(ˆ q) are: ˆ λ1 = ˆ u − ˆ c, ˆ λ2 = ˆ u + ˆ c, ˆ c =

  • g¯

h. Eigenvectors: ˆ r1 =

  • 1

ˆ u − ˆ c

  • ,

ˆ r2 =

  • 1

ˆ u + ˆ c

  • .

Examples in Clawpack 4.3 to be converted soon!

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-30
SLIDE 30

The Riemann problem over topography

ht + (hu)x = 0 (hu)t +

  • hu2 + 1

2gh2

x = −ghBx(x)

With piecewise constant B(x), source term is delta function.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-31
SLIDE 31

The Riemann problem over topography

ht + (hu)x = 0 (hu)t +

  • hu2 + 1

2gh2

x = −ghBx(x)

With piecewise constant B(x), source term is delta function.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-32
SLIDE 32

The Riemann problem over topography

ht + (hu)x = 0 (hu)t +

  • hu2 + 1

2gh2

x = −ghBx(x)

With piecewise constant B(x), source term is delta function.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-33
SLIDE 33

Tsunami from 27 Feb 2010 quake off Chile

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-34
SLIDE 34

Cross section of Atlantic Ocean & tsunami

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-35
SLIDE 35

Transect of 27 February 2010 tsunami

Bathymetry, depth change by > 1000 m from one cell to next, Surface elevation changes on order of a few cm.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-36
SLIDE 36

Source terms and quasi-steady solutions

qt + f(q)x = ψ(q) Steady-state solution:

qt = 0 = ⇒ f(q)x = ψ(q) Balance between flux gradient and source.

Quasi-Steady solution:

Small perturbation propagating against steady-state background. qt ≪ f(q)x ≈ ψ(q) Want accurate calculation of perturbation.

Examples:

  • Shallow water equations with bottom topography and flat surface
  • Stationary atmosphere where pressure gradient balances gravity

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-37
SLIDE 37

Fractional steps for a quasisteady problem

Alternate between solving homogeneous conservation law qt + f(q)x = 0 (1) and source term qt = ψ(q). (2) When qt ≪ f(q)x ≈ ψ(q):

  • Solving (1) gives large change in q
  • Solving (2) should essentially cancel this change.

Numerical difficulties:

  • (1) and (2) are solved by very different methods. Generally will

not have proper cancellation.

  • Nonlinear limiters are applied to f(q)x term, not to

small-perturbation waves. Large variation in steady state hides structure of waves.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-38
SLIDE 38

Incorporating source term in f-waves

qt + f(q)x = ψ with f(q)x ≈ ψ. Concentrate source at interfaces: Ψi−1/2 δ(x − xi−1/2) Split f(Qi) − f(Qi−1) − ∆xΨi−1/2 =

p Zp i−1/2

Use these waves in wave-propagation algorithm. Steady state maintained: (Well balanced) If f(Qi)−f(Qi−1)

∆x

= Ψi−1/2 then Zp ≡ 0 Near steady state: Deviation from steady state is split into waves and limited.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-39
SLIDE 39

Shallow water equations with bathymetry B(x)

ht + (hu)x = 0 (hu)t +

  • hu2 + 1

2gh2

x = −ghBx(x)

Ocean-at-rest equilibrium: ue ≡ 0, he(x) + B(x) ≡ ¯ η = sea level.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-40
SLIDE 40

Shallow water equations with bathymetry B(x)

ht + (hu)x = 0 (hu)t +

  • hu2 + 1

2gh2

x = −ghBx(x)

Ocean-at-rest equilibrium: ue ≡ 0, he(x) + B(x) ≡ ¯ η = sea level. Using Ψi−1/2 = −g 2(hi−1 + hi) gives exactly well-balanced method, but only because hydrostatic pressure is quadratic function of h:

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-41
SLIDE 41

Shallow water equations with bathymetry B(x)

ht + (hu)x = 0 (hu)t +

  • hu2 + 1

2gh2

x = −ghBx(x)

Ocean-at-rest equilibrium: ue ≡ 0, he(x) + B(x) ≡ ¯ η = sea level. Using Ψi−1/2 = −g 2(hi−1 + hi) gives exactly well-balanced method, but only because hydrostatic pressure is quadratic function of h: f(Qi) − f(Qi−1) − Ψi−1/2(Bi − Bi−1) = = 1 2gh2

i − 1

2gh2

i−1

  • + g

2(hi−1 + hi)(Bi − Bi−1) = g 2(hi−1 + hi)((hi + Bi) − (hi−1 + Bi−1)) = 0 if hi + Bi = hi−1 + Bi−1 = ¯ η.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-42
SLIDE 42

Flux-based wave decomposition (f-waves)

Choose rp (e.g. eigenvectors of linearized Jacobian). Then decompose flux difference: fr(qr) − fl(ql) =

m

  • p=1

βprp ≡

m

  • p=1

Zp rather than jump in q: qr − ql =

m

  • p=1

αprp ≡

m

  • p=1

Wp For linear system or Roe solver, Zp = spWp, sp = eigenvalue.

Bale, RJL, Mitran, Rossmanith, SISC 2002 [link] RJL, Pelanti, JCP 2001 [link]

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-43
SLIDE 43

Wave-propagation algorithm using waves

Qn+1

i

= Qn

i − ∆t

∆x[A+∆Qi−1/2 + A−∆Qi+1/2] − ∆t ∆x[ ˜ Fi+1/2 − ˜ Fi−1/2] Standard version: Qi − Qi−1 = m

p=1 Wp i−1/2

A−∆Qi+1/2 =

m

  • p=1

(sp

i+1/2)−Wp i+1/2

A+∆Qi−1/2 =

m

  • p=1

(sp

i−1/2)+Wp i−1/2

˜ Fi−1/2 = 1 2

m

  • p=1

|sp

i−1/2|

  • 1 − ∆t

∆x|sp

i−1/2|

  • Wp

i−1/2.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-44
SLIDE 44

Wave-propagation algorithm using f-waves

Qn+1

i

= Qn

i − ∆t

∆x[A+∆Qi−1/2 + A−∆Qi+1/2] − ∆t ∆x[ ˜ Fi+1/2 − ˜ Fi−1/2] Using f-waves: fi(Qi) − fi−1(Qi−1) = m

p=1 Zp i−1/2

A−∆Qi−1/2 =

  • p:sp

i−1/2<0

Zp

i−1/2,

A+∆Qi−1/2 =

  • p:sp

i−1/2>0

Zp

i−1/2,

˜ Fi−1/2 = 1 2

m

  • p=1

sgn(sp

i−1/2)

  • 1 − ∆t

∆x|sp

i−1/2|

  • Zp

i−1/2

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-45
SLIDE 45

f-wave approximate Riemann solver

Let ˆ A be any averaged Jacobian matrix, e.g. ˆ A = f′((ql + qr)/2). Use eigenvectors of ˆ A to do f-wave splitting. Then A−∆Qi−1/2 + A+∆Qi+1/2 = f(Qi) − f(Qi−1) and so method is conservative. If ˆ A = Roe average, then this is equivalent to usual Roe Riemann solver, and Zp = spWp. Clawpack: Use library routine step1fw.f instead of step1.f.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-46
SLIDE 46

Dry cells and inundation

Use regular grid (e.g. Latitude–Longitude). Finite volume cells can be wet (h > 0) or dry (h = 0). Allow state to change dynamically from one step to next. Need Riemann solver that handles dry states.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-47
SLIDE 47

Dry cells and inundation

Use regular grid (e.g. Latitude–Longitude). Finite volume cells can be wet (h > 0) or dry (h = 0). Allow state to change dynamically from one step to next. Need Riemann solver that handles dry states. Use adaptive mesh refinement (AMR) to resolve shoreline. AMR algorithms have to interact well with wetting/drying. Also with well-balancing.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-48
SLIDE 48

Inundation of Hilo, Hawaii from 27 Feb 2010 event

Resolution ∆y ≈ 160 km on Level 1 (covering Pacific Ocean), ∆y ≈ 10m on Level 5 (shown below). Using 5 levels of refinement with ratios 8, 4, 16, 32. Total refinement factor: 214 = 16, 384 in each direction. With 15 m displacement at fault:

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-49
SLIDE 49

27 February 2010 tsunami

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-50
SLIDE 50

27 February 2010 tsunami

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-51
SLIDE 51

27 February 2010 tsunami

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-52
SLIDE 52

27 February 2010 tsunami

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-53
SLIDE 53

27 February 2010 tsunami

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-54
SLIDE 54

27 February 2010 tsunami

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-55
SLIDE 55

Inundation of Hilo, Hawaii

Using 5 levels of refinement with ratios 8, 4, 16, 32. Resolution ≈ 160 km on Level 1 and ≈ 10m on Level 5. Total refinement factor: 214 = 16, 384 in each direction. With 15 m displacement at fault (27 Feb 2010):

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-56
SLIDE 56

Inundation of Hilo, Hawaii

Using 5 levels of refinement with ratios 8, 4, 16, 32. Resolution ≈ 160 km on Level 1 and ≈ 10m on Level 5. Total refinement factor: 214 = 16, 384 in each direction. With 90 m displacement at fault (1960?):

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-57
SLIDE 57

The Riemann problem with dry state

ht + (hu)x = 0 (hu)t +

  • hu2 + 1

2gh2

x = −ghBx(x)

For small velocity uℓ > 0, the shore acts as solid wall:

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-58
SLIDE 58

The Riemann problem with dry state

ht + (hu)x = 0 (hu)t +

  • hu2 + 1

2gh2

x = −ghBx(x)

For small velocity uℓ > 0, the shore acts as solid wall:

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-59
SLIDE 59

The Riemann problem with dry state

ht + (hu)x = 0 (hu)t +

  • hu2 + 1

2gh2

x = −ghBx(x)

For small velocity uℓ > 0, the shore acts as solid wall:

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-60
SLIDE 60

The Riemann problem with dry state

ht + (hu)x = 0 (hu)t +

  • hu2 + 1

2gh2

x = −ghBx(x)

For large velocity uℓ > 0, water intrudes into dry cell:

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-61
SLIDE 61

The Riemann problem with dry state

ht + (hu)x = 0 (hu)t +

  • hu2 + 1

2gh2

x = −ghBx(x)

For large velocity uℓ > 0, water intrudes into dry cell:

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-62
SLIDE 62

The Riemann problem with dry state

ht + (hu)x = 0 (hu)t +

  • hu2 + 1

2gh2

x = −ghBx(x)

For large velocity uℓ > 0, water intrudes into dry cell:

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012

slide-63
SLIDE 63

Some references

GeoClaw: www.clawpack.org/geoclaw contains links to the recent paper with references and codes: Tsunami modeling with adaptively refined finite volume methods, by RJL, D. L. George, M. J. Berger, Acta Numerica 2011 The GeoClaw software for depth-averaged flows with adaptive refinement, by M. J. Berger, D. L. George, RJL, and K. M. Mandli, 2011 Advances in Water Resources GeoClaw results for the NTHMP tsunami benchmark problems, with Chamberlain, González, Hirai, Varkovitzky, 2011.

R.J. LeVeque, University of Washington Gene Golub SIAM Summer School 2012