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Riemann Problems of the Shallow Water Equations
Nonlinear Systems of Conservation Laws
Brittany Boribong
Kathreen Yanit
PURE Math 2013
Interns Program
University of Hawai’i at Hilo
July 19, 2013
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Introduction
What happens if a dam breaks?
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Shallow Water Equations
One-Dimensional Shallow Water Equations h
hu
t
+
hu
hu2 + 1
2gh2
x
= 0
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Shallow Water Equations
One-Dimensional Shallow Water Equations h
hu
t
+
hu
hu2 + 1
2gh2
x
= 0
Using conserved quantaties,
q (x, t) =
h
hu
=
q 1q 2
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Shallow Water Equations
One-Dimensional Shallow Water Equations h
hu
t
+
hu
hu2 + 1
2gh2
x
= 0
Using conserved quantaties,
q (x, t) =
h
hu
=
q 1q 2
The Shallow Water Equations can be rewritten as,
q 1q 2
t
+
q 2q 2
2
q 1+
1
2gq 1
2
x= 0
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Riemann Problems
What is a Riemann Problem?
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Riemann Problems
What is a Riemann Problem?
Riemann ProblemAn initial boundary value problem for the conservation law with apiecewise constant initial condition.
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Riemann Problems
What is a Riemann Problem?
Riemann ProblemAn initial boundary value problem for the conservation law with apiecewise constant initial condition.
Riemann problems for a single nonlinear equation:
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Riemann Problems
What is a Riemann Problem?
Riemann ProblemAn initial boundary value problem for the conservation law with apiecewise constant initial condition.
Riemann problems for a single nonlinear equation:
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Riemann Problems
What is a Riemann Problem?
Riemann ProblemAn initial boundary value problem for the conservation law with apiecewise constant initial condition.
Riemann problems for a single nonlinear equation:
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Riemann Problems
What is a Riemann Problem?
Riemann ProblemAn initial boundary value problem for the conservation law with apiecewise constant initial condition.
Riemann problems for a nonlinear system of equations:
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Riemann Problems
What is a Riemann Problem?
Riemann ProblemAn initial boundary value problem for the conservation law with apiecewise constant initial condition.
Riemann problems for a nonlinear system of equations:
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Riemann Problems
What is a Riemann Problem?
Riemann ProblemAn initial boundary value problem for the conservation law with apiecewise constant initial condition.
Riemann problems for a nonlinear system of equations:
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Riemann Problems
What is a Riemann Problem?
Riemann ProblemAn initial boundary value problem for the conservation law with apiecewise constant initial condition.
Riemann problems for a nonlinear system of equations:
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Rarefaction Waves
What is a Rarefaction Wave?
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Rarefaction Waves
What is a Rarefaction Wave?
Rarefaction WaveIt is a continuous solution to a Riemann Problem.
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f
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Rarefaction Waves
What is a Rarefaction Wave?
Rarefaction WaveIt is a continuous solution to a Riemann Problem.
A rarefaction wave forms in a single nonlinear system,
Conditions of a Rarefaction Wave
F (ur) > F (ul)
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R f i W
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Rarefaction Waves
What is a Rarefaction Wave?
Rarefaction WaveIt is a continuous solution to a Riemann Problem.
A rarefaction wave forms in a single nonlinear system,
Conditions of a Rarefaction Wave
F (ur) > F (ul)
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R f ti W
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Rarefaction Waves
What is a Rarefaction Wave?
Rarefaction WaveIt is a continuous solution to a Riemann Problem.
A rarefaction wave forms in a single nonlinear system,
Conditions of a Rarefaction Wave
F (ur) > F (ul)
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R f ti W
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Rarefaction Waves
A rarefaction wave is solved in a single nonlinear equation by using asolution that is self-similar,
Self-Similar Solution for a Single Nonlinear Equation
u(x, t) = ux
t
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R f ti W
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Rarefaction Waves
A rarefaction wave is solved in a single nonlinear equation by using asolution that is self-similar,
Self-Similar Solution for a Single Nonlinear Equation
u(x, t) = ux
t
Consider Burgers’ equation as an example,Burgers’ Equation
F (u) = 1
2u2
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Ra fa ti Wa s
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Rarefaction Waves
A rarefaction wave is solved in a single nonlinear equation by using asolution that is self-similar,
Self-Similar Solution for a Single Nonlinear Equation
u(x, t) = ux
t
Consider Burgers’ equation as an example,Burgers’ Equation
F (u) = 1
2u2
Plugged into the conservation law,
∂
ux
t
∂t
+
∂
1
2ux
t
2
∂x
= 0
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Rarefaction Waves
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Rarefaction Waves
Rarefaction Wave Solution
u(x, t) = (F )−1
xt
where F is the speed of the characteristic paths u in the structure of the rarefaction wave.
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Rarefaction Waves
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Rarefaction Waves
Rarefaction Wave Solution
u(x, t) = (F )−1
xt
where F is the speed of the characteristic paths u in the structure of the rarefaction wave.
A similar process can be done to solve for the structure inside ararefaction wave in a nonlinear system of equations by choosing aself-similar solution,
Self-Similar Solution for a System of Nonlinear Equations
q (x, t) = q x
t
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Rarefaction Waves
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Rarefaction Waves
Plugged into the conservation law,
DF (q ) · q
xt
=
xt
· q
xt
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Rarefaction Waves
Plugged into the conservation law,
DF (q ) · q
xt
=
xt
· q
xt
where
DF (q ) =
0 1−
q 2
q 1
2+ gq 1
2q 2q 1
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Rarefaction Waves
Plugged into the conservation law,
DF (q ) · q
xt
=
xt
· q
xt
where
DF (q ) =
0 1−
q 2
q 1
2+ gq 1
2q 2q 1
Similar to the eigenvector equation,
Ax = λx
Therefore q x
t
is an eigenvector with corresponding eigenvalue λ.
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Rarefaction Waves
Finding the eigenvalues and eigenvectors of DF (q ),
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Rarefaction Waves
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Rarefaction Waves
Finding the eigenvalues and eigenvectors of DF (q ),
Eigenvalues
λ1 = u −
gh
λ2 = u + gh
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Rarefaction Waves
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Rarefaction Waves
Finding the eigenvalues and eigenvectors of DF (q ),
Eigenvalues
λ1 = u −
gh
λ2 = u +
gh
Eigenvectors
r1 = 1
u−√
gh r2 =
1
u +√
gh
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Rarefaction Waves
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Rarefaction Waves
Going back to the equation,
Self-Similar Equation
q x
t
= (λ p)
−1x
t
where q xt is the state on a curve corresponding to λ p.
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Rarefaction Waves
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Going back to the equation,
Self-Similar Equation
q x
t
= (λ p)
−1x
t
where q xt is the state on a curve corresponding to λ p.
What is an integral curve?
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Rarefaction Waves
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Going back to the equation,
Self-Similar Equation
q x
t
= (λ p)
−1x
t
where q xt is the state on a curve corresponding to λ p.
What is an integral curve?
Integral Curve
A curve of the vector field r p that has a tangent vector at each point q that is an eigenvector of DF (q ) corresponding to the eigenvalue λ p(q ).
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Integral Curves
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g
Finding the integral curves by plotting the points (h, u) on the vectorfield by solving for the system of differential equations obtained by theeigenvectors r1 and r2.
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Integral Curves
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g
Finding the integral curves by plotting the points (h, u) on the vectorfield by solving for the system of differential equations obtained by the
eigenvectors r1 and r2.
Differential Equations of Eigenvector r1
dq 1
dt
= 1
dq 2
dt =
q 2
q 1−√ g · q 1
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Integral Curves
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Finding the integral curves by plotting the points (h, u) on the vectorfield by solving for the system of differential equations obtained by the
eigenvectors r1 and r2.
Integral Curves of r1
h = h∗
hu = hu∗ + 2h
gh∗ −
gh
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Integral Curves
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Finding the integral curves by plotting the points (h, u) on the vectorfield by solving for the system of differential equations obtained by the
eigenvectors r1 and r2.
Differential Equations of Eigenvector r2
dq 1
dt
= 1
dq 2
dt =
q 2
q 1−√ g · q 1
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Integral Curves
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Finding the integral curves by plotting the points (h, u) on the vectorfield by solving for the system of differential equations obtained by the
eigenvectors r1 and r2.
Integral Curves of r2
h = h∗
hu = hu∗ + 2h
gh∗ −
gh
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Directional Derivatives
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Need to find the section of the integral curve that is increasing whendealing with rarefactions waves.
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Directional Derivatives
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Need to find the section of the integral curve that is increasing whendealing with rarefactions waves.
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Directional Derivatives
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Need to find the section of the integral curve that is increasing whendealing with rarefactions waves.
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Directional Derivatives
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Need to find the section of the integral curve that is increasing whendealing with rarefactions waves.
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Rarefaction Waves
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Recall the solution in the beginning,
Speed of Rarefaction Waves Equationq x
t
= (λ1)
−1x
t
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Rarefaction Waves
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Recall the solution in the beginning,
Speed of Rarefaction Waves Equationq x
t
= (λ1)
−1x
t
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Rarefaction Waves
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Once the speeds of the characteristic paths inside the rarefaction waveare found and the states corresponding to the speeds on the integral
curve, the structure inside the rarefaction wave can be displayed.
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All-Rarefaction Riemann Problem
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All-Rarefaction Riemann Problem
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All-Rarefaction Riemann Problem
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All-Rarefaction Riemann Problem
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All-Rarefaction Riemann Problem
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All-Rarefaction Riemann Problem
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Shockwaves
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What is a shockwave?
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Shockwaves
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What is a shockwave?
Shockwave
It is a discontinuous solution to a Riemann Problem.
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Shockwaves
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What is a shockwave?
Shockwave
It is a discontinuous solution to a Riemann Problem.
A shockwave forms in a single nonlinear equation:
Rankine-Hugoniot Jump Conditions
s(ul − ur) = F (ul) − F (ur)
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Shockwaves
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What is a shockwave?
Shockwave
It is a discontinuous solution to a Riemann Problem.
A shockwave forms in a single nonlinear equation:
Lax Entropy Condition
F (ul) > F (ur)
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Shockwaves
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A shockwave forms in a non-linear system of equations:
Jump Conditionss(q ∗ − q ) = f (q ∗) − f (q )
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Shockwaves
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A shockwave forms in a non-linear system of equations:
Jump Conditionss(q ∗ − q ) = f (q ∗) − f (q )
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Shockwaves
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A shockwave forms in a non-linear system of equations:
Entropy Conditionλ p(q L) > λ p(q R)
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Shockwaves
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A shockwave forms in a non-linear system of equations:
Entropy Conditionλ p(q L) > λ p(q R)
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State Space
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Note: Integral curves of the state space for an arbitrary state q , sharethe same tangent vector at the base state but are not equal curves.
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State Space
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Note: Integral curves of the state space for an arbitrary state q , sharethe same tangent vector at the base state but are not equal curves.
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State Space
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Note: Integral curves of the state space for an arbitrary state q , sharethe same tangent vector at the base state but are not equal curves.
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State Space
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Note: Integral curves of the state space for an arbitrary state q , sharethe same tangent vector at the base state but are not equal curves.
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State Space
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Note: Integral curves of the state space for an arbitrary state q , sharethe same tangent vector at the base state but are not equal curves.
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Shockwaves
P d t f l b i ti h l d i t th j
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Produces a system of algebraic equations when plugged into the jumpconditions,
Shallow Water Equations
s(h∗ − h) = h∗u∗ − hu
s(h∗q ∗−
hu) = h∗u2∗
−hu2 +
1
2
g(h2∗
−h2)
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Shockwaves
P d t f l b i ti h l d i t th j
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Produces a system of algebraic equations when plugged into the jumpconditions,
Shallow Water Equations
s(h∗ − h) = h∗u∗ − hu
s(h∗q ∗−
hu) = h∗u2∗
−hu2 +
1
2
g(h2∗
−h2)
After solving for the 3 unknowns,
Equation of the Shockwave Solution
u(h) = u∗ ± (h∗ − h)
g
2
1
h +
1
h∗
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Shockwaves
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Hugoniot Loci
“–” corresponds to the 1-Shockwave
“+” corresponds to the 2-Shockwave
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Shockwaves
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Hugoniot Loci
“–” corresponds to the 1-Shockwave
“+” corresponds to the 2-Shockwave
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All-Shockwave
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All-Shockwave
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All-Shockwave
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All-Shockwave
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All-Shockwave
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All-Shockwave
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Brittany Boribong Kathreen Yanit Riemann Problems of the Shallow Water Equations July 19, 2013 22 / 28
Shallow Water Equations
Ge e al Rie a Sol e fo Shallo Wate E atio s
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General Riemann Solver for Shallow Water Equations
φl(h) =
ul + 2
ghl − gh , if h < hl,ul − (h − hl)
g
2
1
h +
1
hl
, if h > hl
φr(h) =
ur − 2 ghr − gh , if h < hr,ur + (h − hr)
g
2
1
h +
1
hr
, if h > hr
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Dam-Break Solution
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Dam-Break Solution
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Dam-Break Solution
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Brittany Boribong Kathreen Yanit Riemann Problems of the Shallow Water Equations July 19, 2013 24 / 28
Dam-Break Solution
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Brittany Boribong Kathreen Yanit Riemann Problems of the Shallow Water Equations July 19, 2013 24 / 28
Dam-Break Solution
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Brittany Boribong Kathreen Yanit Riemann Problems of the Shallow Water Equations July 19, 2013 24 / 28
Dam-Break Solution
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Shallow Water Equations
Theorem
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Theorem
Given q l = q 1l
q 2l
and q r = q 1r
q 2r
, where q 1l , q 1r > 0, there exists a
solution to determine the intermediate state q m to the Riemann
Problem of the Shallow Water Equations.
Brittany Boribong Kathreen Yanit Riemann Problems of the Shallow Water Equations July 19, 2013 26 / 28
References
Randall J. LeVeque Finite Volume Methods for Hyperbolic
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q f ypProblems. Pages 253-283. 2002.
Randall J. LeVeque Nonlinear Conservation Laws and Finite Volume Methods for Astrophysical Fluid Flow. Pages 35-43. 1998.
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