large-time behavior and numerics for …...large-time behavior and numerics for scalar conservation...

LARGE-TIME BEHAVIOR AND NUMERICSFOR SCALAR CONSERVATION LAWS

Alejandro PozoJoint work with L. I. Ignat (IMAR) and E. Zuazua (BCAM&Ikerbasque)

BCAM – Basque Center for Applied Mathematics

The Second BCAM Workshop on Computational MathematicsOctober 18, 2013

MotivationInviscid vs. Viscous

Numerics for 1-D scalar conservation lawsSimilarity variables

Outline

1 Motivation

2 Inviscid vs. Viscous

3 Numerics for 1-D scalar conservation laws

4 Similarity variables

Alejandro Pozo Large-time behavior and numerics



Motivation

Source: J.J.Alonso-M.R.Colonno (2012) Source: National Oceanic and Atmospheric Administration

Source: European Space Agency




Burgers equation

Given u0 ∈ L1(R), consider the 1-D conservation law with or without viscosity:ut +

(u2

2

)x

= εuxx , x ∈ R, t > 0,

u(x , 0) = u0(x), x ∈ R.

Then:If ε = 0, u(·, t) ∼ Np,q(·, t) as t →∞.If ε > 0, u(·, t) ∼ uM (·, t) as t →∞.




N-wave vs. diffusive wave

N-wave: Np,q is the so-called N-wave, defined as:

Np,q(x , t) :=

{xt , if √pt < x < √qt,0 otherwise,

where

p := −2 miny∈R

∫ y

−∞u0(x)dx , q := 2 max

y∈R

∫ ∞y

u0(x)dx .

Diffusive wave: uM is the solution of the Burgers equation with initial data Mδ0:

uM (x , t) :=

∫R

( x−yt

)e−

FM (x,y,t)

2ε dy∫R e−

−FM (x,y,t)

2ε dy,

where

FM (x , y , t) := M∫ y

−∞δ0(r)dr +

(x − y)2

2t, M :=

∫R

u0(x)dx .




Intermediate asymptotics




Conservative schemes

Let us consider now numerical approximation schemes (ε = 0):un+1j = un

j −∆t∆x(

gnj+1/2 − gn

j−1/2), j ∈ Z, n > 0,

u0j = 1

∆x

∫ xj+1/2xj−1/2

u0(x)dx , j ∈ Z.

The approximated solution u∆ is given by

u∆(t, x) = unj , xj−1/2 < x < xj+1/2, tn ≤ t < tn+1,

where tn = n∆t and xj+1/2 = (j + 1/2)∆x .

Which is the behavior of u∆?Is the N-wave preserved?




Examples

Three-point conservative schemes for the Burgers equation:1 Lax-Friedrichs

gLF (u, v) =u2 + v2

4−

∆x∆t

( v − u2

),

2 Engquist-Osher

gEO(u, v) =u(u + |u|)

4+

v(v − |v |)4

,

3 Godunov

gG (u, v) =

minw∈[u,v ]

w2

2 , if u ≤ v ,

maxw∈[v,u]

w2

2 , if v ≤ u.

We denote gnj+1/2 = g(un

j , unj+1).




Numerical viscosity

We can rewrite three-point monotone schemes in the form

un+1j − un

j

∆t+

(unj+1)2 − (un

j−1)2

4∆x= R(un

j , unj+1)− R(un

j−1, unj ).

R is the numerical viscosity:

R(u, v) =1

2∆x

(u2

2+

v2

2− 2g(u, v)

).

For instance:1 Lax-Friedrichs

RLF (u, v) =v − u

2,

2 Engquist-OsherREO(u, v) =

14∆x

(v |v | − u|u|),

3 Godunov

RG (u, v) =

{ 14∆x sign(|u| − |v |)(v2 − u2), v ≤ 0 ≤ u,

14∆x (v |v | − u|u|), elsewhere.




Numerical viscosity


un+1j − un

j

∆t︸︷︷︸ut

+(un

j+1)2 − (unj−1)2

4∆x︸︷︷︸u2/2

= R(unj , un

j+1)− R(unj−1, un

j ).


R(u, v) =1

2∆x

(u2

2+

v2

2− 2g(u, v)

).


RLF (u, v) =v − u

2,


14∆x

(v |v | − u|u|),3 Godunov

RG (u, v) =

{ 14∆x sign(|u| − |v |)(v2 − u2), v ≤ 0 ≤ u,





Numerical viscosity

We can rewrite three-point monotone schemes in the formun+1

j − unj

∆t+

(unj+1)2 − (un

j−1)2

4∆x= R(un

j , unj+1)− R(un

j−1, unj )︸︷︷︸

?.


R(u, v) =1

2∆x

(u2

2+

v2

2− 2g(u, v)

).


RLF (u, v) =v − u

2,


14∆x

(v |v | − u|u|),3 Godunov

RG (u, v) =

{ 14∆x sign(|u| − |v |)(v2 − u2), v ≤ 0 ≤ u,





Numerical viscosity


un+1j − un

j

∆t+

(unj+1)2 − (un

j−1)2

4∆x= R(un

j , unj+1)− R(un

j−1, unj ).


R(u, v) =1

2∆x

(u2

2+

v2

2− 2g(u, v)

).


RLF (u, v) =v − u

2,


14∆x

(v |v | − u|u|),

3 Godunov

RG (u, v) =

{ 14∆x sign(|u| − |v |)(v2 − u2), v ≤ 0 ≤ u,





Properties

The three schemes ensure:

Monotonicity:

unj ≥ vn

j , ∀j ∈ Z =⇒ un+1j ≥ vn+1

j , ∀j ∈ Z.

Conservation of massEntropyConsistency with OSLCL1 → L∞ decay with a rate O(t−1/2)

BV estimates




Properties


MonotonicityConservation of mass: ∑

j∈Z

unj =∑j∈Z

u0j , ∀n ∈ N.

EntropyConsistency with OSLCL1 → L∞ decay with a rate O(t−1/2)

BV estimates




Properties


MonotonicityConservation of massEntropy

Consistency with OSLCL1 → L∞ decay with a rate O(t−1/2)

BV estimates




Properties


MonotonicityConservation of massEntropyConsistency with OSLC:

unj−1 − un

j+1

2∆x≤

2n∆t

, ∀j ∈ Z.

L1 → L∞ decay with a rate O(t−1/2)

BV estimates




Properties


MonotonicityConservation of massEntropyConsistency with OSLCL1 → L∞ decay with a rate O(t−1/2):

‖un‖∞,∆ := maxj∈Z|un

j | ≤ C(n∆t)−1/2‖u0‖1.

BV estimates




Properties


MonotonicityConservation of massEntropyConsistency with OSLCL1 → L∞ decay with a rate O(t−1/2)

BV estimates: ∑j∈Z

|unj+k − un

j | =∑j∈Z

|u0j+k − u0

j |, ∀n, k ∈ N.




Properties


MonotonicityConservation of massEntropyConsistency with OSLCL1 → L∞ decay with a rate O(t−1/2)

BV estimates




Conservation of positive and negative masses

Theorem (L. I. Ignat, A.P, E. Zuazua ’13)

Assume that u0 ∈ L1(R) and that the CFL condition λ‖un‖∞,∆ ≤ 1 is fulfilled (whereλ = ∆t/∆x). If the numerical flux of a 3-point monotone conservative schemesatisfies

g(η, ξ) = 0, when − 1/λ ≤ η ≤ 0 ≤ ξ ≤ 1/λ,

andξ − λg(ξ,−ξ) ≥ 0, when 0 ≤ ξ ≤ 1/λ,

then, for any n ∈ N the following holds:

mink∈Z

k∑j=−∞

unj = min

k∈Z

k∑j=−∞

u0j and max

k∈Z

∞∑j=k

unj = max

k∈Z

∞∑j=k

u0j .




Examples

Engquist-Osher and Godunov schemes satisfy the theorem:

gEO(η, ξ) = gG (η, ξ) = 0, ∀η, ξ s.t. − 1/λ ≤ η ≤ 0 ≤ ξ ≤ 1/λ

andξ − λgEO(ξ,−ξ) = ξ − λξ2 = ξ(1− λξ) ≥ 0, ∀ξ s.t. 0 ≤ ξ ≤ 1/λ

ξ − λgG (ξ,−ξ) = ξ − λξ2

2= ξ(1− λ

ξ

2) ≥ 0, ∀ξ s.t. 0 ≤ ξ ≤ 1/λ

BUT Lax-Friedrichs scheme does not!

gLF (η, ξ) = 0 ⇐⇒ ξ = η = 0, ∀η, ξ s.t. − 1/λ ≤ η ≤ 0 ≤ ξ ≤ 1/λ,




Main result


Consider u0 ∈ L1(R) and ∆x and ∆t such that λ‖un‖∞,∆ ≤ 1, λ = ∆t/∆x. Then,for any p ∈ [1,∞), the numerical solution u∆ given by the Lax-Friedrichs schemesatisfies

limt→∞

t12 (1− 1

p )‖u∆(t)− w(t)‖Lp (R) = 0,

where the profile w = wM∆is the unique solution ofwt +

(w2

2

)x

=(∆x)2

2 wxx , x ∈ R, t > 0,

w(0) = M∆δ0,

with M∆ =∫R u0

∆.




Main result


Consider u0 ∈ L1(R) and ∆x and ∆t such that λ‖un‖∞,∆ ≤ 1, λ = ∆t/∆x. Then,for any p ∈ [1,∞), the numerical solutions u∆ given by Engquist-Osher and Godunovschemes satisfy

limt→∞

t12 (1− 1

p )‖u∆(t)− w(t)‖Lp (R) = 0,

where the profile w = wp∆,q∆ is the unique solution ofwt +

(w2

2

)x

= 0, x ∈ R, t > 0,

w(0) = M∆δ0, limt→0

∫ x

0w(t, z)dz =

0, x < 0,−p∆, x = 0,q∆ − p∆, x > 0,

with M∆ =∫R u0

∆ and

p∆ = −minx∈R

∫ x

−∞u0

∆(z)dz and q∆ = maxx∈R

∫ ∞x

u0∆(z)dz.




Example

Let us consider the initial data

u0(x) =

−0.05, x ∈ [−1, 0],

0.15, x ∈ [0, 2],

0, elsewhere.

The parameters that describe the asymptotic N-wave profile are:

M = 0.25 , p = 0.05 and q = 0.3.

We take ∆x = 0.1 as the mesh size for the interval [−350, 800] and ∆t = 0.5.




Comparison




Similarity variables

Consider the change of variables:

s = ln(t + 1), ξ = x/√

t + 1, w(ξ, s) =√

t + 1 u(x , t).

This transforms Burgers equation into

ws +

(12

w2 −12ξw)ξ

= εwξξ, ξ ∈ R, s > 0.

The asymptotic profile is a N-wave too:

Np,q(ξ) =

{ξ, −

√2p < ξ <

√2q,

0, elsewhere,




Examples

Engquist-Osher scheme (with ∆ξ = 0.01 and ∆s = 0.0005)




Examples

Lax-Friedrichs scheme (with ∆ξ = 0.01 and ∆s = 0.0005)




Physical vs. Similarity variables

Comparison of numerical and exact solutions at t = 1000. We choose ∆ξ such thatthe ‖ · ‖1,∆ error is similar. The time-steps are ∆t = ∆x/2 and ∆s = ∆ξ/20,respectively, enough to satisfy the CFL condition.For ∆x = 0.1:

Nodes Time-steps ‖ · ‖1,∆ ‖ · ‖2,∆ ‖ · ‖∞,∆Physical 1501 19987 0.0867 0.0482 0.0893Similarity 215 4225 0.0897 0.0332 0.0367

For ∆x = 0.01:

Nodes Time-steps ‖ · ‖1,∆ ‖ · ‖2,∆ ‖ · ‖∞,∆Physical 15001 199867 0.0093 0.0118 0.0816Similarity 2000 39459 0.0094 0.0106 0.0233




Conclusions

Be careful!Numerical viscosity is necessary.BUT too much viscosity can produce undesired large-time dynamics.

To-Do list:Develop numerical schemes preserving large-time asymptotics for nonlinear PDEs.Study the use of self-similarity variables to improve computational cost.Analyze influence on optimal control problems.

Thank you very much! Eskerrik asko!

Acknowledgments:


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