[PPT] - Schemes with Well-Controlled Dissipation (WCD) Jan Ernest Seminar PowerPoint Presentation

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Schemes with Well-Controlled Dissipation (WCD)

Jan Ernest

Seminar of Applied Mathematics (SAM), ETH Z¨ urich, Switzerland

Jan Ernest WCD Schemes

SLIDE 2

Joint work with

◮ Siddhartha Mishra

◮ Center of Mathematics for Applications (CMA),

University of Oslo, Norway.

◮ Philippe G. LeFloch

◮ Laboratoire Jacques-Louis Lions, Centre National de la

Recherche Scientifique, Universit´ e Pierre et Marie Curie, Paris, France.

Jan Ernest WCD Schemes

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Scope of the Talk

Hyperbolic conservation law ut + f (u)x = 0, u = u(x, t), (x, t) ∈ R × R+.

Jan Ernest WCD Schemes

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Scope of the Talk

Hyperbolic conservation law with small-scale effects uε

t + f (uε)x = Rε(uε),

uε = uε(x, t), (x, t) ∈ R × R+. Interested in limit as ε → 0.

Jan Ernest WCD Schemes

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Scope of the Talk

Two cases:

1. diffusive-dispersive regularization

Rε(u) = εuxx + δε2uxxx

Jan Ernest WCD Schemes

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Scope of the Talk

Two cases:

1. diffusive-dispersive regularization

Rε(u) = εuxx + δε2uxxx

2. pseudo-parabolic regularization

Rε(u) = εuxx + δε2uxxt

Jan Ernest WCD Schemes

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Scope of the Talk

Two cases:

1. diffusive-dispersive regularization

Rε(u) = εuxx + δε2uxxx

2. pseudo-parabolic regularization

Rε(u) = εuxx + δε2uxxt Goal: Approximate solutions for fixed δ and ε → 0 numerically.

Jan Ernest WCD Schemes

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Classical vs. Nonclassical Solutions

Why interesting?

◮ Classical and Nonclassical shock waves may arise depending

n value of δ

Jan Ernest WCD Schemes

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Classical vs. Nonclassical Solutions

Why interesting?

◮ Classical and Nonclassical shock waves may arise depending

n value of δ

◮ Classical solutions

◮ satisfy Lax / Oleinik entropy inequalities ◮ S(u)t + Q(u)x ≤ 0 for all entropy pairs (S, Q) ◮ are TVD Jan Ernest WCD Schemes

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Classical vs. Nonclassical Solutions

Why interesting?

◮ Classical and Nonclassical shock waves may arise depending

n value of δ

◮ Classical solutions

◮ satisfy Lax / Oleinik entropy inequalities ◮ S(u)t + Q(u)x ≤ 0 for all entropy pairs (S, Q) ◮ are TVD

◮ Nonclassical solutions

◮ do not satisfy Lax / Oleinik entropy conditions ◮ satisfy S(u)t + Q(u)x ≤ 0 for a single entropy pair (S, Q) ◮ not TVD ◮ additional criterion for uniqueness: Kinetic relation (Hayes &

LeFloch (1997))

◮ Controls entropy dissipation at a shock as a function of the

shock speed.

Jan Ernest WCD Schemes

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Example: Cubic Conservation Law

Regularized cubic conservation law: ut +

u3

x = εuxx + δε2uxxx

Jacobs, McKinney, Shearer (1995):

◮ Systematic study of traveling wave solutions ◮ Analytic expressions for nonclassical solutions

Jan Ernest WCD Schemes

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Cubic CL: Classical Solution

Classical solution (δ = 0): ut +

u3

x = εuxx

Jan Ernest WCD Schemes

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Cubic CL: Classical Solution

Classical solution (δ = 0): ut +

u3

x = εuxx

−0.5 0.5 1 1.5 2 2.5 −5 5

(a) t = 0

Jan Ernest WCD Schemes

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Cubic CL: Classical Solution

Classical solution (δ = 0): ut +

u3

x = εuxx

−0.5 0.5 1 1.5 2 2.5 −5 5

(a) t = 0

−0.5 0.5 1 1.5 2 2.5 −5 5

(b) t = 0.05

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Cubic CL: Nonclassical Solution

Nonclassical solution (δ = 1): ut +

u3

x = εuxx + ε2uxxx

Jan Ernest WCD Schemes

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Cubic CL: Nonclassical Solution

Nonclassical solution (δ = 1): ut +

u3

x = εuxx + ε2uxxx

−0.5 0.5 1 1.5 2 2.5 −5 5

(a) t = 0

Jan Ernest WCD Schemes

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Cubic CL: Nonclassical Solution

Nonclassical solution (δ = 1): ut +

u3

x = εuxx + ε2uxxx

−0.5 0.5 1 1.5 2 2.5 −5 5

(a) t = 0

−0.5 0.5 1 1.5 2 2.5 −5 5

(b) t = 0.05

Jan Ernest WCD Schemes

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Numerical Approximation

Standard schemes fail to approximate nonclassical solutions

completely. They always converge to the classical solution (δ = 0).

0.5 1 1.5 2 −5 5 Lax−Friedrichs Rusanov Exact

⇒ Design schemes that approximate correct solution for δ = 0.

Jan Ernest WCD Schemes

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Approaches to Resolve Nonclassical Shocks

Numerical solvers for regularized cubic conservation law:

◮ Kissling & Rohde (2010): Heterogeneous multiscale approach ◮ LeFloch & Mohammadian (2008): High-order finite difference

schemes with controlled dissipation

dui dt = − 1 ∆x j=p

j=−p

αj(u3)i+j

+

ε (∆x)2 j=p

j=−p

βjui+j

+ δε2

(∆x)3 j=p

j=−p

γjui+j

◮ αj, βj and γj chosen such that scheme is of order 2p and

leading order terms in equivalent equation match the small-scale effects.

Jan Ernest WCD Schemes

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Failure at Strong Shocks

Observation: Approximations for ε = 5 · 10−3dx.

0.8 1 1.2 1.4 −4 −2 2 4

(a) uL = 4

Jan Ernest WCD Schemes

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Failure at Strong Shocks

Observation: Approximations for ε = 5 · 10−3dx.

0.8 1 1.2 1.4 −4 −2 2 4

(a) uL = 4

0.95 1 1.05 1.1 −15 −10 −5 5 10 15 20

(b) uL = 14

Jan Ernest WCD Schemes

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Failure at Strong Shocks

Observation: Approximations for ε = 5 · 10−3dx.

0.8 1 1.2 1.4 −4 −2 2 4

(a) uL = 4

0.95 1 1.05 1.1 −15 −10 −5 5 10 15 20

(b) uL = 14

0.9 0.95 1 1.05 1.1 −15 −10 −5 5 10 15 20

(c) uL = 18

Jan Ernest WCD Schemes

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Failure at Strong Shocks

Observation: Approximations for ε = 5 · 10−3dx.

0.8 1 1.2 1.4 −4 −2 2 4

(a) uL = 4

0.95 1 1.05 1.1 −15 −10 −5 5 10 15 20

(b) uL = 14

0.9 0.95 1 1.05 1.1 −15 −10 −5 5 10 15 20

(c) uL = 18

◮ Scheme performs well for small shocks but fails for large

shocks.

Jan Ernest WCD Schemes

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Construction of Schemes with Well-Controlled Dissipation

Cubic conservation law: ut +

u3

x = εuxx + δε2uxxx

Idea 1 (Controlled Dissipation): Develop schemes such that leading-order terms of equivalent equation match the regularizing terms exactly.

dui dt = − 1 ∆x  

j=p

j=−p

αj(u3)i+j  + c ∆x  

j=p

j=−p

βjui+j  +δc2 ∆x  

j=p

j=−p

γjui+j  

Jan Ernest WCD Schemes

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Construction of Schemes with Well-Controlled Dissipation

Cubic conservation law: ut +

u3

x = εuxx + δε2uxxx

Idea 1 (Controlled Dissipation): Develop schemes such that leading-order terms of equivalent equation match the regularizing terms exactly.

dui dt = − 1 ∆x  

j=p

j=−p

αj(u3)i+j  + c ∆x  

j=p

j=−p

βjui+j  +δc2 ∆x  

j=p

j=−p

γjui+j  

Jan Ernest WCD Schemes

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Equivalent Equation

Cubic conservation law: ut +

u3

x = εuxx + δε2uxxx

Equivalent equation: ut = −(u3)x + c∆xuxx + δc2∆x2uxxx

leading order terms (l.o.t.)

+ higher order terms

(h.o.t.)

Jan Ernest WCD Schemes

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Equivalent Equation

Cubic conservation law: ut +

u3

x = εuxx + δε2uxxx

Equivalent equation: ut = −(u3)x +

εuxx

c∆xuxx +

δε2uxxx

δc2∆x2uxxx
leading order terms (l.o.t.)

+ higher order terms

(h.o.t.)

Jan Ernest WCD Schemes

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Recall: Failure at Strong Shocks

0.8 1 1.2 1.4 −4 −2 2 4

(a) uL = 4

0.95 1 1.05 1.1 −15 −10 −5 5 10 15 20

(b) uL = 14

0.9 0.95 1 1.05 1.1 −15 −10 −5 5 10 15 20

(c) uL = 18

◮ Scheme performs well for small shocks but fails for large

shocks.

Jan Ernest WCD Schemes

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Equivalent Equation

Equivalent equation: ut = −(u3)x + c∆xuxx + δc2∆x2uxxx

leading order terms (l.o.t.)

+ higher order terms

(h.o.t.)

◮ For large shocks the higher-order terms (h.o.t.) in the

equivalent equation start to influence the approximation.

Jan Ernest WCD Schemes

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Schemes with Well-Controlled Dissipation

Idea 2: Impose a balance between leading-order and higher-order terms of the equivalent equation. Want: |h.o.t.| < τ |l.o.t.| for τ << 1.

◮ Ensures that the nonclassical behaviour in approximation

mostly comes from the correct small-scale mechanisms

Jan Ernest WCD Schemes

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Derivation of the WCD Condition

Scheme:

dui dt = − 1 ∆x j=p

j=−p

αjfi+j

+

c ∆x j=p

j=−p

βjui+j

+ δc2

∆x j=p

j=−p

γjui+j

Equivalent equation:

ut = − f (u)x + c∆xuxx + δc2∆x2uxxx −

∞

k=2p+1

∆xk−1 k! Ap

kf [k] + c ∞

k=2p+1

∆xk−1 k! Bp

k u[k] + δc2 ∞

k=2p+1

∆xk−1 k! C p

k u[k]

with Ap

k = p

j=−p

αjjk, Bp

k = p

j=−p

βjjk, C p

k = p

j=−p

γjjk

Jan Ernest WCD Schemes

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Derivation of the WCD Condition

Scheme:

dui dt = − 1 ∆x j=p

j=−p

αjfi+j

+

c ∆x j=p

j=−p

βjui+j

+ δc2

∆x j=p

j=−p

γjui+j

Equivalent equation:

ut =−f (u)x + c∆xuxx + δc2∆x2uxxx −

∞

k=2p+1

∆xk−1 k! Ap

kf [k] + c ∞

k=2p+1

∆xk−1 k! Bp

k u[k] + δc2 ∞

k=2p+1

∆xk−1 k! C p

k u[k]

with Ap

k = p

j=−p

αjjk, Bp

k = p

j=−p

βjjk, C p

k = p

j=−p

γjjk

Jan Ernest WCD Schemes

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Derivation of the WCD Condition

Scheme:

dui dt = − 1 ∆x j=p

j=−p

αjfi+j

+

c ∆x j=p

j=−p

βjui+j

+ δc2

∆x j=p

j=−p

γjui+j

Equivalent equation:

ut = − f (u)x + c∆xuxx + δc2∆x2uxxx −

∞

k=2p+1

∆xk−1 k! Ap

kf [k] + c ∞

k=2p+1

∆xk−1 k! Bp

k u[k] + δc2 ∞

k=2p+1

∆xk−1 k! C p

k u[k]

with Ap

k = p

j=−p

αjjk, Bp

k = p

j=−p

βjjk, C p

k = p

j=−p

γjjk

Jan Ernest WCD Schemes

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Derivation of the WCD Condition

Analysis at an isolated discontinuity connecting uL and uR. We assume [ [u] ] := uL − uR > 0. Formally, u[k] = [ [u] ] ∆xk , f [k] = [ [f ] ] ∆xk = [ [u3] ] ∆xk .

Jan Ernest WCD Schemes

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Derivation of the WCD Condition

Analysis at an isolated discontinuity connecting uL and uR. We assume [ [u] ] := uL − uR > 0. Formally, u[k] = [ [u] ] ∆xk , f [k] = [ [f ] ] ∆xk = [ [u3] ] ∆xk . Equivalent equation at isolated discontinuity du dt + [ [u3] ] ∆x − c[ [u] ] ∆x − δc2[ [u] ] ∆x

l.o.t

= SD

p c[

[u] ] ∆x + SC

p δc2[

[u] ] ∆x − Sf

p[

[u3] ] ∆x

h.o.t

with Sf

p = ∞

k=2p+1

Ap

k

k! SD

p = ∞

k=2p+1

Bp

k

k! SC

p = ∞

k=2p+1

C p

k

k! .

Jan Ernest WCD Schemes

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Derivation of the WCD Condition

Bounds on the h.o.t. and l.o.t. from above and below respectively.

|l.o.t| ≥

|δ|c2 + c − σ

|[ [u] ]| ∆x . and |h.o.t| ≤

ˆ

SD

p c + ˆ

SC

p |δ|c2 + ˆ

Sf

pσ

|[ [u] ]| ∆x , ˆ SD

p , ˆ

SC

p and ˆ

Sf

p only depend on the order p and can be precomputed

before any simulations.

Jan Ernest WCD Schemes

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WCD Condition

Want: |h.o.t| < τ |l.o.t| for τ << 1 → WCD condition

|δ| −

ˆ SC

p |δ|

τ

c2 +
1 −

ˆ SD

p

τ

c −
1 +

ˆ Sf

p

τ

σ > 0

◮ Quadratic inequality for scheme parameter c ◮ The smaller we choose τ the larger the order p

(ˆ SC

p decreases in p)

Jan Ernest WCD Schemes

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Numerical Experiments

Cubic Conservation Law ut + (u3)x = εuxx + δε2uxxx Riemann problems u(x, 0) =

uL

x ≤ 0.4 −2 x > 0.4

Jan Ernest WCD Schemes

SLIDE 39

Recall: Failure at Strong Shocks

0.8 1 1.2 1.4 −4 −2 2 4

(a) uL = 4

0.95 1 1.05 1.1 −15 −10 −5 5 10 15 20

(b) uL = 14

0.9 0.95 1 1.05 1.1 −15 −10 −5 5 10 15 20

(c) uL = 18

◮ Scheme with controlled dissipation performs well for small

shocks but fails for large shocks.

Jan Ernest WCD Schemes

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Numerical Experiments: Small Shocks

0.9 1 1.1 1.2 1.3 1.4 1.5 −3 −2 −1 1 2 3 4 200 mesh points 500 mesh points 2000 mesh points exact

(a) uL = 4: Double shock

1 1.2 1.4 1.6 −1.5 −1 −0.5 0.5 1 1.5 2 200 mesh points 500 mesh points 2000 mesh points exact

(b) uL = 2: Shock-rarefaction

Jan Ernest WCD Schemes

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Numerical Experiments: Moderate Shocks

1 1.05 1.1 1.15 −30 −20 −10 10 20 30 τ = 0.1, order 8 τ = 0.3, order 4 exact

(a) uL = 30

1.06 1.08 1.1 1.12 1.14 −30 −29 −28 −27 −26 −25 −24 −23 τ = 0.1, order 8 τ = 0.3, order 4 exact

(b) Zoom at nonclassical state

Jan Ernest WCD Schemes

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Numerical Experiments: Large Shocks

1.05 1.055 1.06 1.065 1.07 1.075 1.08 −40 −20 20 40 60 5000 mesh points 20000 mesh points exact

(a) uL = 55

1.06 1.065 1.07 1.075 −56 −55 −54 −53 −52 −51 −50 5000 mesh points 20000 mesh points exact

(b) Zoom at nonclassical state

Jan Ernest WCD Schemes

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Violating WCD condition

Scheme parameter c is chosen such that it does satisfy (red) and it does not satisfy (blue) the WCD condition.

0.9 1 1.1 1.2 −15 −10 −5 5 10 15 c = 0.25 cWCD c = cWCD exact Jan Ernest WCD Schemes

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Kinetic Relation

Hayes & LeFloch (1997): Concept of kinetic relations as selection principle for physically meaningful nonclassical solution.

5 10 15 −14 −12 −10 −8 −6 −4 −2 2nd order 6th order 8th order exact

(a) Controlled Dissipation

20 40 60 80 100 −100 −80 −60 −40 −20 exact 8th order WCD

(b) Well-Controlled Dissipation

Jan Ernest WCD Schemes

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Extension to Systems of Conservation Laws

Extension to systems by applying the concepts componentwise. Example: Van der Waals fluids with viscosity and capillarity τt − ux = 0 ut + p(τ)x = εuxx − δ ε2τxxx. u: velocity of the fluid τ: volume of the fluid p(τ) = R T (τ − 1/3) − 3 τ 2 where R = 8/3 and T = 1.005.

Jan Ernest WCD Schemes

SLIDE 46

0.2 0.4 0.6 0.8 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 200 mesh points 1000 mesh points reference

(a) δ = 0

0.2 0.4 0.6 0.8 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 200 mesh points 1000 mesh points reference

(b) δ = 1

Jan Ernest WCD Schemes

SLIDE 47

0.1 0.2 0.3 0.4 0.5 0.6 0.5 1 1.5 2 2.5 3

(a) τR = 25

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.36 0.365 0.37 0.375 0.38 0.385 0.39

(b) Zoom at nonclassical state

Jan Ernest WCD Schemes

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Extension to the Pseudo-Parabolic Regularization

Example: Regularized Buckley-Leverett equation ut + f (u)x = εuxx + δε2uxxt with f (u) = u2 u2 + κ (1 − u)2 , 0 ≤ u ≤ 1, κ > 0

Jan Ernest WCD Schemes

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Extension to the Pseudo-Parabolic Regularization

Example: Regularized Buckley-Leverett equation ut + f (u)x = εuxx + δε2uxxt with f (u) = u2 u2 + κ (1 − u)2 , 0 ≤ u ≤ 1, κ > 0

1.1 1.2 1.3 1.4 1.5 0.2 0.4 0.6 0.8

(a) δ = 0

0.8 0.9 1 1.1 1.2 1.3 1.4 0.2 0.4 0.6 0.8 1

(b) δ = 5

Jan Ernest WCD Schemes

SLIDE 50

References

◮ Jacobs, McKinney, Shearer (1995)

Traveling wave solutions of the modified Korteweg-deVries-Burgers

equation. Journal of Differential Equations, 116(2):448-467, 1995.

◮ Kissling, Rohde (2010)

The computation of nonclassical shock waves with a heterogeneous multiscale method. NHM, 5(3):661-674, 2010.

◮ LeFloch, Mohammadian (2008)

Why many theories of shock waves are necessary: Kinetic functions, equivalent equations, and fourth-order models. Journal of Computational Physics, 227(8):4162-4189, 2008.

Jan Ernest WCD Schemes