Numerical schemes for Hyperbolic CL over non-uniform ... · upd values next values. FD schemes over non-uniform grids for HCL Nikolaos Sfakianakis Introduction Uniform mesh Non-uniform

FD schemes overnon-uniform grids

for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps

Some numericalschemes

Remarks

Consistency

EntropyConservativeSchemes

More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Numerical schemes for Hyperbolic CL overnon-uniform adaptively redistributed meshes

Nikolaos Sfakianakis

RICAM, Vienna

Paris / January 2011


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Scalar Conservation Laws

u(x , t)t + f (u(x , t))x = 0, x ∈ [a,b], t ∈ [0,T ]

f smooth, convex flux function.

• Loss of smoothness =⇒Weak solutions• Loss of uniqueness =⇒ Entropy conditions

x

t


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


u(x , t)t + f (u(x , t))x = 0, x ∈ [a,b], t ∈ [0,T ]


• Loss of smoothness =⇒Weak solutions

• Loss of uniqueness =⇒ Entropy conditions

x

t


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


u(x , t)t + f (u(x , t))x = 0, x ∈ [a,b], t ∈ [0,T ]


• Loss of smoothness =⇒Weak solutions• Loss of uniqueness =⇒ Entropy conditions

x

t


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Uniform mesh

{(xi , tn), xi+1 − xi = ∆x, tn+1 − tn = ∆t

}Derivative approximations

∂∂t u(xi , tn) ≈ u(xi ,tn+1)−u(xi ,tn)

∆t∂∂x u(xi , tn) ≈ u(xi+1,tn)−u(xi−1,tn)

2∆x

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

t


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Non-uniform mesh

{(xn

i , tn), xn

i+1 − xni = ∆xn

i+1/2, tn+1 − tn = ∆tn+1/2}

Derivative approximations

∂∂t u(xn

i , tn) ≈ u(xn

i ,tn+1)−u(xn

i ,tn)

∆tn+1/2

∂∂x u(xn

i , tn) ≈ u(xn

i+1,tn)−u(xn

i−1,tn)

∆xni−1/2+∆xn

i+1/2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

t


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Non-uniform mesh

• How do we create/update such grids?

• Do we use the same numerical schemes?• Do they have the same properties?• How does the mesh adaptation affect the numerical results?


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Non-uniform mesh

• How do we create/update such grids?• Do we use the same numerical schemes?

• Do they have the same properties?• How does the mesh adaptation affect the numerical results?


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Non-uniform mesh

• How do we create/update such grids?• Do we use the same numerical schemes?• Do they have the same properties?

• How does the mesh adaptation affect the numerical results?


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Non-uniform mesh

• How do we create/update such grids?• Do we use the same numerical schemes?• Do they have the same properties?• How does the mesh adaptation affect the numerical results?


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Main Adaptive Scheme (MAS)

Mnx = {xn

1 , . . . , xnN}, Un = {un

1 , . . . ,unN}

Step 1 Mesh reconstruction Mn+1x = {xn+1

1 , . . . , xn+1N }

Step 2 Solution update Un = {un1 , . . . , u

nN}

Step 3 Time evolution Un+1 ={

un+11 , . . . , un+1

N

}

valuesnodes

valuesnodes

new nodes

valuesnodes

new nodesupd values

valuesnodes

new nodesupd valuesnext values


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Mnx = {xn

1 , . . . , xnN}, Un = {un

1 , . . . ,unN}


1 , . . . , xn+1N }


nN}


un+11 , . . . , un+1

N

}valuesnodes

valuesnodes

new nodes

valuesnodes

new nodesupd values

valuesnodes



for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Mnx = {xn

1 , . . . , xnN}, Un = {un

1 , . . . ,unN}


1 , . . . , xn+1N }


nN}


un+11 , . . . , un+1

N

}valuesnodes

valuesnodes

new nodes

valuesnodes

new nodesupd values

valuesnodes



for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Mnx = {xn

1 , . . . , xnN}, Un = {un

1 , . . . ,unN}


1 , . . . , xn+1N }


nN}


un+11 , . . . , un+1

N

}

valuesnodes

valuesnodes

new nodes

valuesnodes

new nodesupd values

valuesnodes



for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Main Adaptive Scheme (MAS)Step 1 (Mesh reconstruction)

Mnx = {xn

1 , . . . , xnN}, Un = {un

1 , . . . ,unN}

Comput. cost O(N)


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Mnx = {xn

1 , . . . , xnN}, Un = {un

1 , . . . ,unN}

• Estimator function KUn (·)

• Curvature, continuous version for smooth u

Ku =|u′′(x)|

(1+(u′(x))2)32

• Curvature, discrete version

K dscrUn,i =

2xi+1−xi−1

∣∣∣∣∣ uni −un

i−1xni −xn

i−1−

uni+1−un

ixni+1−xn

i

∣∣∣∣∣1+

(un

i −uni−1

xni −xn

i−1

)21+

(uni+1−un

ixni+1−xn

i

)21+

(uni+1−un

i−1xni+1−xn

i−1

)21/2

• Interpolate to KUn (x)

Comput. cost O(N)


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Mnx = {xn

1 , . . . , xnN}, Un = {un

1 , . . . ,unN}

• Estimator function KUn (·)• Curvature, continuous version for smooth u

Ku =|u′′(x)|

(1+(u′(x))2)32


K dscrUn,i =

2xi+1−xi−1

∣∣∣∣∣ uni −un

i−1xni −xn

i−1−

uni+1−un

ixni+1−xn

i

∣∣∣∣∣1+

(un

i −uni−1

xni −xn

i−1

)21+

(uni+1−un

ixni+1−xn

i

)21+

(uni+1−un

i−1xni+1−xn

i−1

)21/2


Comput. cost O(N)


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Mnx = {xn

1 , . . . , xnN}, Un = {un

1 , . . . ,unN}


Ku =|u′′(x)|

(1+(u′(x))2)32


K dscrUn,i =

2xi+1−xi−1

∣∣∣∣∣ uni −un

i−1xni −xn

i−1−

uni+1−un

ixni+1−xn

i

∣∣∣∣∣1+

(un

i −uni−1

xni −xn

i−1

)21+

(uni+1−un

ixni+1−xn

i

)21+

(uni+1−un

i−1xni+1−xn

i−1

)21/2


Comput. cost O(N)


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Mnx = {xn

1 , . . . , xnN}, Un = {un

1 , . . . ,unN}


Ku =|u′′(x)|

(1+(u′(x))2)32


K dscrUn,i =

2xi+1−xi−1

∣∣∣∣∣ uni −un

i−1xni −xn

i−1−

uni+1−un

ixni+1−xn

i

∣∣∣∣∣1+

(un

i −uni−1

xni −xn

i−1

)21+

(uni+1−un

ixni+1−xn

i

)21+

(uni+1−un

i−1xni+1−xn

i−1

)21/2


Comput. cost O(N)


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Mnx = {xn

1 , . . . , xnN}, Un = {un

1 , . . . ,unN}

• Estimator function KUn (·)• Monitor function MUn (·)

• Discrete monitor MdscrUn,i =

∫ xni

a KUn (x)dx• Interpolate to MUn (x)

Comput. cost O(N)


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Mnx = {xn

1 , . . . , xnN}, Un = {un

1 , . . . ,unN}



∫ xni

a KUn (x)dx

• Interpolate to MUn (x)

Comput. cost O(N)


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Mnx = {xn

1 , . . . , xnN}, Un = {un

1 , . . . ,unN}



∫ xni

a KUn (x)dx• Interpolate to MUn (x)

Comput. cost O(N)


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Mnx = {xn

1 , . . . , xnN}, Un = {un

1 , . . . ,unN}


• New nodes

{xn+1

1 = xn1 ,

MUn (xn+1i+1 )−MUn (xn+1

i ) = 1N MUn (xn

N), i

Comput. cost O(N)


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Mnx = {xn

1 , . . . , xnN}, Un = {un

1 , . . . ,unN}


• New nodes

{xn+1

1 = xn1 ,


i ) = 1N MUn (xn

N), iComput. cost O(N)


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Mnx = {xn

1 , . . . , xnN}, Un = {un

1 , . . . ,unN}


• New nodes

{xn+1

1 = xn1 ,


i ) = 1N MUn (xn


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

adamon

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

adamon


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Mnx = {xn

1 , . . . , xnN}, Un = {un

1 , . . . ,unN}


• New nodes

{xn+1

1 = xn1 ,


i ) = 1N MUn (xn


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

adamon

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

adamon


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Main Adaptive Scheme (MAS)Step 2 (Solution update)

Mnx = {xn

1 , . . . , xnN}, Un = {un

1 , . . . ,unN}, Mn+1

x ={

xn+11 , . . . , xn+1

N

}

• Grids• Vertex centered Cn

i = (xni−1/2, x

ni+1/2), xn

i−1/2 =xn

i−1+xni

2• Finite element cells Cn

i = (xni−1, x

ni )

• Approximation function Un(x)• Piecewise constant• Piecewise linear

• Update to Un = {un1 , . . . , u

nN}

• Conservative reconstruction• Interpolation

graph


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Mnx = {xn

1 , . . . , xnN}, Un = {un

1 , . . . ,unN}, Mn+1

x ={

xn+11 , . . . , xn+1

N

}• Grids

• Vertex centered Cni = (xn

i−1/2, xni+1/2), xn

i−1/2 =xn

i−1+xni


i = (xni−1, x

ni )


• Update to Un = {un1 , . . . , u

nN}


graph


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Mnx = {xn

1 , . . . , xnN}, Un = {un

1 , . . . ,unN}, Mn+1

x ={

xn+11 , . . . , xn+1

N

}• Grids


i−1/2, xni+1/2), xn

i−1/2 =xn

i−1+xni


i = (xni−1, x

ni )


• Update to Un = {un1 , . . . , u

nN}


graph


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Mnx = {xn

1 , . . . , xnN}, Un = {un

1 , . . . ,unN}, Mn+1

x ={

xn+11 , . . . , xn+1

N

}• Grids


i−1/2, xni+1/2), xn

i−1/2 =xn

i−1+xni


i = (xni−1, x

ni )


• Update to Un = {un1 , . . . , u

nN}


graph


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Main Adaptive Scheme (MAS)Step 3 (Time evolution)

Mn+1x =

{xn+1

1 , . . . , xn+1N

}, Un = {un

1 , . . . , unN},

Progress in time, with numerical scheme:

Un+1 = N(

Mn+1x , Un

)

Analysis tool - Hirt, Warming & HyettModified Equation(ME): PDE the numerical scheme resolves

MAS applications can be found in...

• Ch. Arvanitis, Th. Katsaounis, Ch. Makridakis, 2001• Ch. Arvanitis, Ch. Makridakis, A. Tzavaras, 2004• Ch. Arvanitis, A. I. Delis, 2006• Ch. Arvanitis, Ch. Makridakis, N. Sfakianakis, 2010• N. Sfakianakis, 2010


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Mn+1x =

{xn+1

1 , . . . , xn+1N

}, Un = {un

1 , . . . , unN},


Un+1 = N(

Mn+1x , Un

)Analysis tool - Hirt, Warming & HyettModified Equation(ME): PDE the numerical scheme resolves




for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Mn+1x =

{xn+1

1 , . . . , xn+1N

}, Un = {un

1 , . . . , unN},


Un+1 = N(

Mn+1x , Un

)Analysis tool - Hirt, Warming & HyettModified Equation(ME): PDE the numerical scheme resolves




for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Some schemes:Generalised LxF

un+1i = un

i −∆t|Ci |

(Fi+1/2 − Fi−1/2

)Fi+1/2 = −hi+1

2∆t(un

i+1 − uni ) +

12(f (un

i ) + f (uni+1)

)((hi = xi − xi−1, xi− 1

2=

xi−1+xi2 , Ci = (xi− 1

2, xi+ 1

2), |Ci | =

hi +hi+12 ))

ME : ut + ux = hi+1−hi∆t ux + (hi+1−hi )(∆t−hi+1)+(hi−∆t)(hi +∆t)

2∆t uxx

remarks


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


un+1i = un

i −∆t|Ci |

(Fi+1/2 − Fi−1/2

)Fi+1/2 = −hi+1

2∆t(un

i+1 − uni ) +

12(f (un

i ) + f (uni+1)

)((hi = xi − xi−1, xi− 1

2=

xi−1+xi2 , Ci = (xi− 1

2, xi+ 1

2), |Ci | =

hi +hi+12 ))


2∆t uxx

remarks


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


un+1i = un

i −∆t|Ci |

(Fi+1/2 − Fi−1/2

)Fi+1/2 = −hi+1

2∆t(un

i+1 − uni ) +

12(f (un

i ) + f (uni+1)

)((hi = xi − xi−1, xi− 1

2=

xi−1+xi2 , Ci = (xi− 1

2, xi+ 1

2), |Ci | =

hi +hi+12 ))


2∆t uxx

remarks

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

uniada

exact

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

uniada

exact

Burgers single shock


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


un+1i = un

i −∆t|Ci |

(Fi+1/2 − Fi−1/2

)Fi+1/2 = −hi+1

2∆t(un

i+1 − uni ) +

12(f (un

i ) + f (uni+1)

)((hi = xi − xi−1, xi− 1

2=

xi−1+xi2 , Ci = (xi− 1

2, xi+ 1

2), |Ci | =

hi +hi+12 ))


2∆t uxx

remarks

-0.01

-0.005

0

0.005

0.01

0 0.2 0.4 0.6 0.8 1

uniada

exact

-0.01

-0.005

0

0.005

0.01

0 0.2 0.4 0.6 0.8 1

uniada

exact

Transport step slope step (inconsistency)


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Some schemes:A first order

un+1i = un

i −∆t|Ci |

(Fi+1/2 − Fi−1/2

)Fi+1/2 = − ∆t

2hi+1(un

i+1 − uni ) +

12(f (un

i ) + f (uni+1)

)((hi = xi − xi−1, xi− 1

2=

xi−1+xi2 , Ci = (xi− 1

2, xi+ 1

2), |Ci | =

hi +hi+12 ))

ME : ut + αux = α(hi+1−hi )+(1−α2)∆t2 uxx


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


un+1i = un

i −∆t|Ci |

(Fi+1/2 − Fi−1/2

)Fi+1/2 = − ∆t

2hi+1(un

i+1 − uni ) +

12(f (un

i ) + f (uni+1)

)((hi = xi − xi−1, xi− 1

2=

xi−1+xi2 , Ci = (xi− 1

2, xi+ 1

2), |Ci | =

hi +hi+12 ))



for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


un+1i = un

i −∆t|Ci |

(Fi+1/2 − Fi−1/2

)Fi+1/2 = − ∆t

2hi+1(un

i+1 − uni ) +

12(f (un

i ) + f (uni+1)

)((hi = xi − xi−1, xi− 1

2=

xi−1+xi2 , Ci = (xi− 1

2, xi+ 1

2), |Ci | =

hi +hi+12 ))


-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

uniuniLxF

adaexact

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

uniuniLxF

adaexact

Burgers single shock (anti-diffusion instability)


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Some schemes:Unstable centered, FTCS

un+1i = un

i −∆t|Ci |

(Fi+1/2 − Fi−1/2)

Fi+1/2 =f (ui ) + f (ui+1)

2

((hi = xi − xi−1, xi− 12

=xi−1+xi

2 , Ci = (xi− 12, xi+ 1

2), |Ci | =

hi +hi+12 ))

ut + αux = α hi−hi+1−α∆t2 uxx +

α(hi−6α∆t)(hi+1−hi )−2α3∆t2−αh2i+1

6 uxxx


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


un+1i = un

i −∆t|Ci |

(Fi+1/2 − Fi−1/2)

Fi+1/2 =f (ui ) + f (ui+1)

2

((hi = xi − xi−1, xi− 12

=xi−1+xi

2 , Ci = (xi− 12, xi+ 1

2), |Ci | =

hi +hi+12 ))



6 uxxx


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


un+1i = un

i −∆t|Ci |

(Fi+1/2 − Fi−1/2)

Fi+1/2 =f (ui ) + f (ui+1)

2

((hi = xi − xi−1, xi− 12

=xi−1+xi

2 , Ci = (xi− 12, xi+ 1

2), |Ci | =

hi +hi+12 ))



6 uxxx

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5

uniada

exact

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5

uniada

exact

Burgers double shock & rarefaction (anti-diffusion instability)


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Some schemes:MacCormack

u∗i = uni −

∆t|Ci |

(f (ui+1)− f (ui ))

u∗∗i = u∗i −∆t|Ci−1|

(f (u∗i )− f (u∗i−1))

un+1i =

uni + u∗∗i

2((hi = xi − xi−1, xi− 1

2=

xi−1+xi2 , Ci = (xi− 1

2, xi+ 1

2), |Ci | =

hi +hi+12 ))

ME : ut + αux = α(hi−1−hi+1)(hi−1+hi−2α∆t)

8(hi−1+hi )uxx


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


u∗i = uni −

∆t|Ci |

(f (ui+1)− f (ui ))

u∗∗i = u∗i −∆t|Ci−1|

(f (u∗i )− f (u∗i−1))

un+1i =

uni + u∗∗i

2((hi = xi − xi−1, xi− 1

2=

xi−1+xi2 , Ci = (xi− 1

2, xi+ 1

2), |Ci | =

hi +hi+12 ))


8(hi−1+hi )uxx


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


u∗i = uni −

∆t|Ci |

(f (ui+1)− f (ui ))

u∗∗i = u∗i −∆t|Ci−1|

(f (u∗i )− f (u∗i−1))

un+1i =

uni + u∗∗i

2((hi = xi − xi−1, xi− 1

2=

xi−1+xi2 , Ci = (xi− 1

2, xi+ 1

2), |Ci | =

hi +hi+12 ))


8(hi−1+hi )uxx

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5

uniada

exact

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5

uniada

exact

Burgers double shock & rarefaction (entropy instability)


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


u∗i = uni −

∆t|Ci |

(f (ui+1)− f (ui ))

u∗∗i = u∗i −∆t|Ci−1|

(f (u∗i )− f (u∗i−1))

un+1i =

uni + u∗∗i

2((hi = xi − xi−1, xi− 1

2=

xi−1+xi2 , Ci = (xi− 1

2, xi+ 1

2), |Ci | =

hi +hi+12 ))


8(hi−1+hi )uxx

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

uniada

exact

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

uniada

exact

Burgers single shock (conservation)


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Some schemes:Non-uniform 2-nd order (S. Noelle & N. Sfakianakis)

un+1i =

α∆t(α∆t + hi+1)

hi (hi + hi+1)un

i−1

+(−α∆t + hi )(α∆t + hi+1)

hihi+1un

i +α∆t(α∆t − hi )

(hi + hi+1)hi+1un

i+1

((hi = xi − xi−1))

ME : ut + αux = α(α∆t−hi )(α∆t+hi+1)6 uxxx


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


un+1i =


hi (hi + hi+1)un

i−1

+(−α∆t + hi )(α∆t + hi+1)

hihi+1un


(hi + hi+1)hi+1un

i+1

((hi = xi − xi−1))



for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


un+1i =


hi (hi + hi+1)un

i−1

+(−α∆t + hi )(α∆t + hi+1)

hihi+1un


(hi + hi+1)hi+1un

i+1

((hi = xi − xi−1))


0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

uniada

exact

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

uniada

exact

Transport single shock (dispersion instability)


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Some remarks

Remark 1The MAS cannot treat consistency or conservation problems

Remark 2The MAS can treat stability problems (anti-diffusion, dispersion,entropy)

Remark 3Inconsistency of the flux consistency criterion & the modifiedequation analysis

example


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Some remarks




example


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Some remarks




example


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Extended consistency criterion (N. Sfakianakis)

Uniform mesh

un+1i = un

i −∆t∆x

(Fi+1/2 − Fi−1/2

)Fi+1/2 = F (ui ,ui+1)

criterion:

F (u, u) = f (u), ∀u ∈ R

Non-uniform mesh

un+1i = un

i −∆t|Ci |

(F R

i+1/2 − F Li−1/2

)F R

i+1/2 = F R(ui ,ui+1) = F (ui ,ui+1,hi ,hi+1)

((hi = xi − xi−1, xi− 12

=xi−1+xi

2 , Ci = (xi− 12, xi+ 1

2), |Ci | =

hi +hi+12 ))

extended criterion:

F R(u, u) = F L(u, u), ∀u ∈ R2hi

hi +hi+1F L

1 (uni ,u

ni ) + 2hi+1

hi +hi+1F R

2 (uni ,u

ni ) = f ′(un

i )


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Extended consistency criterion (N. Sfakianakis)

Uniform mesh

un+1i = un

i −∆t∆x

(Fi+1/2 − Fi−1/2

)Fi+1/2 = F (ui ,ui+1)

criterion:

F (u, u) = f (u), ∀u ∈ R

Non-uniform mesh

un+1i = un

i −∆t|Ci |

(F R

i+1/2 − F Li−1/2

)F R

i+1/2 = F R(ui ,ui+1) = F (ui ,ui+1,hi ,hi+1)

((hi = xi − xi−1, xi− 12

=xi−1+xi

2 , Ci = (xi− 12, xi+ 1

2), |Ci | =

hi +hi+12 ))

extended criterion:

F R(u, u) = F L(u, u), ∀u ∈ R2hi

hi +hi+1F L

1 (uni ,u

ni ) + 2hi+1

hi +hi+1F R

2 (uni ,u

ni ) = f ′(un

i )


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Entropy conservative schemes (Ch. Arvanitis, Ch. Makridakis, N. Sfakianakis)

PDE: ut + f (u)x = 0Entropy Pair: (U,F ), U ′′ > 0 and F ′(u) = U ′(u)f ′(u)

Symmetrisation “on the right” - MockEntropy variables: v = U ′(u)

PDE: u(v)t + g(v)x = 0, g(v) = f (u(v))Entropy potential: φ(v) = vu(v)− U(u(v)), u(v) = φ′(v)

Entropy flux potential: ψ(v) = vg(v)− F (u(v)), g(v) = ψ′(v)

Semi-discrete entropy conservative scheme - Tadmorddt ui (t) = − 1

∆xi

(gi+1/2 − gi−1/2

),

gi+1/2 =∫ 1ξ=0 g (vi + ξ(vi+1 − vi )) dξ

= ψ(vi+1)−ψ(vi )vi+1−vi

Remarks - Tadmor

• More numerical viscosity⇒ “More” entropy stability• Implicit time discretisation⇒ Entropy dissipation• Explicit time discretisation⇒ Entropy production


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary







∆xi

(gi+1/2 − gi−1/2

),

gi+1/2 =∫ 1ξ=0 g (vi + ξ(vi+1 − vi )) dξ


Remarks - Tadmor



for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary







∆xi

(gi+1/2 − gi−1/2

),

gi+1/2 =∫ 1ξ=0 g (vi + ξ(vi+1 − vi )) dξ


Remarks - Tadmor



for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary







∆xi

(gi+1/2 − gi−1/2

),

gi+1/2 =∫ 1ξ=0 g (vi + ξ(vi+1 − vi )) dξ


Remarks - Tadmor



for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Entropy conservative schemes (Ch. Arvanitis, Ch. Makridakis, N. Sfakianakis)Example 1 - Tadmor

PDE: ut + (eu)x = 0Ent. pair: (U(u),F (u)) = (eu, 1

2 e2u)Entr. var.: v(u) = U ′(u) = eu ⇒ u = ln v

Entr. var.f.: g(v) = f (u(v)) = vEntr. f.pot.: ψ(v) = vg(v)− F (u(v)) = 1

2 v2

Semi: ddt ui(t) = − eun

i+1(t)−euni−1(t)

xni+1−xn

i−1

Fully: Explicit

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5

uniada

exact

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5

uniada

exact


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Entropy conservative schemes (Ch. Arvanitis, Ch. Makridakis, N. Sfakianakis)Example 2 - LeFloch & Rohde

PDE: ut +( 1

2 u2)x = 0

Ent. pair: (U(u),F (u)) = (∫ u f (s)ds = 1

6 u3, 18 u4), u ≥ 0

Entr. var.: v(u) = U ′(u) = 12 u2 ⇒ u =

√2v

Entr. var.f.: g(v) = f (u(v)) = vEntr. f.pot.: ψ(v) = vg(v)− F (u(v)) = 1

2 v2

Semi: ddt ui(t) = − 1

4un

i+1(t)2−uni−1(t)2

xni+1−xn

i−1

Fully: Uniform: Runge-Kutta 4th orderNon-uniform: Explicit

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

adaptiveuniform

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

adaptiveuniform


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Entropy conservative schemes (Ch. Arvanitis, Ch. Makridakis, N. Sfakianakis)Example 3 - LeFloch & Rohde

PDE: ut +(u3)

x = 0Ent. pair: (U(u),F (u)) = ( 1

4 u4, 12 u6)

Entr. var.: v(u) = U ′(u) = u3 ⇒ u = 3√

vEntr. var.f.: g(v) = f (u(v)) = vEntr. f.pot.: ψ(v) = vg(v)− F (u(v)) = 1

2 v2

Semi: ddt ui(t) = − 1

2∆xi

(g(vi+1)− g(vi−1)

)− 5

12∆xi

(g(vi+2) + 2g(vi+1)− 2g(vi−1) + g(vi−2)

)Fully: Runge-Kutta 4th order

-6

-4

-2

0

2

4

0 2 4 6 8 10 12 14 16

adaptiveuniform

-6

-4

-2

0

2

4

0 2 4 6 8 10 12 14 16

adaptiveuniform


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Entropy conservative schemes (Ch. Arvanitis, Ch. Makridakis, N. Sfakianakis)Some more remarks

Remark 4The MAS provides enough entropy dissipation to stabilize explicitin time Entropy Conservative schemes

Remark 5On computing non-classical shocks they converge to theclassical shock solution


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Entropy conservative schemes (Ch. Arvanitis, Ch. Makridakis, N. Sfakianakis)Some more remarks

Remark 4The MAS provides enough entropy dissipation to stabilize explicitin time Entropy Conservative schemes

Remark 5On computing non-classical shocks they converge to theclassical shock solution


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Total variation bound (N. Sfakianakis)

Main Adaptive Scheme (MAS)Mn

x = {xn1 , . . . , x

nN}, Un = {un

1 , · · · ,unN}

Step 1 Mesh reconstructionMn+1

x = {xn+11 , . . . , xn+1

N }Step 2 Solution update

Un = {un1 , . . . , u

nN}

Step 3 Time evolutionUn+1 =

{un+1

1 , . . . , un+1N

}

RemarkSteps 1 & 2 are not depicted with usual analysis


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Main Adaptive Scheme (MAS)Mn

x = {xn1 , . . . , x

nN}, Un = {un

1 , · · · ,unN}

Step 1 Mesh reconstructionMn+1

x = {xn+11 , . . . , xn+1

N }Step 2 Solution update

Un = {un1 , . . . , u

nN}

Step 3 Time evolutionUn+1 =

{un+1

1 , . . . , un+1N

}RemarkSteps 1 & 2 are not depicted with usual analysis


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


0.999

0.9992

0.9994

0.9996

0.9998

1

1.0002

1.0004

1.0006

1.0008

1.001

0.495 0.4955 0.496 0.4965 0.999

0.9992

0.9994

0.9996

0.9998

1

1.0002

1.0004

1.0006

1.0008

1.001

0.495 0.4955 0.496 0.4965

0.999

0.9992

0.9994

0.9996

0.9998

1

1.0002

1.0004

1.0006

1.0008

1.001

0.495 0.4955 0.496 0.4965 0.999

0.9992

0.9994

0.9996

0.9998

1

1.0002

1.0004

1.0006

1.0008

1.001

0.495 0.4955 0.496 0.4965

The 3 steps of MAS: Before, After mesh reconstruction and solution update & time update


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Total variation bound (N. Sfakianakis)Requirements (interpolation over piecewise linears)

Evolution Requirement|un+1

i − uni | ≤ C max

{|un

i − uni−1|, |un

i+1 − uni |}

λ-rule RequirementAvoid the extremes by a factor of λ details

Coupling of the Requirementsλ+ 3λC < 1


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary




{|un

i − uni−1|, |un

i+1 − uni |}




for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary




{|un

i − uni−1|, |un

i+1 − uni |}




for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Recursive relationsEk

m = λ(

Ek−1m + C ·

(Ek−1

m + Ek−1m−1

)), m > 1

Ek1 = λ

(Ek−1

1 + 2C · Ek−11 + αk−1

), m = 1

αk = C(|uk

i − uki+1| − 2Ek

1

)+

E01 E1

1

E12 E2

2 E11 E2

1

E23 E3

3

E22 E3

2 E21

E31


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Recursive relationsEk

m = λ(

Ek−1m + C ·

(Ek−1

m + Ek−1m−1

)), m > 1

Ek1 = λ

(Ek−1

1 + 2C · Ek−11 + αk−1

), m = 1

αk = C(|uk

i − uki+1| − 2Ek

1

)+

E01 E1

1

E12 E2

2 E11 E2

1

E23 E3

3

E22 E3

2 E21

E31


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Total variation bound (N. Sfakianakis)TV - Approach 1

Lemma (Magnitude of the extremes)Ek

m = λmCm−1∑m−1l=k−1

( ll−m+1

)λl−m+1(1 + 2C)l−m+1αk−l

Lemma (Uniform time bound)If αi ≤ CM & λ+ 2λC < 1, we bound Ek

m uniform in k,

Ekm ≤ M

(λC

1−λ−2λC

)m

proof

Theorem (Total Variation Bound)If moreover λ+ 3λC < 1

TVI ≤ 2MC

1−λ−2λC1−λ−3λC

proof


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary




( ll−m+1

)λl−m+1(1 + 2C)l−m+1αk−l


m uniform in k,

Ekm ≤ M

(λC

1−λ−2λC

)m

proof


TVI ≤ 2MC


proof


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary




( ll−m+1

)λl−m+1(1 + 2C)l−m+1αk−l


m uniform in k,

Ekm ≤ M

(λC

1−λ−2λC

)m

proof


TVI ≤ 2MC


proof


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary


Lemma (Contribution of αm)Ikαm

= λ (λ+ 3λC)k−mαm

Lemma (Total contribution)Iktot ≤ λM 1−(λ+3λC)k

1−(λ+3λC)

k→∞→ λ1−(λ+3λC) M

Theorem (Total Variation Bound)If αi < CM and λ+ 3λC < 1

TVI ≤ 2λ1−(λ+3λC) M

Theorem (Total Variation Increase Diminishing)If moreover

∑αi <∞,

Iktot → 0

proof


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary





1−(λ+3λC)

k→∞→ λ1−(λ+3λC) M




∑αi <∞,

Iktot → 0

proof


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary





1−(λ+3λC)

k→∞→ λ1−(λ+3λC) M




∑αi <∞,

Iktot → 0

proof


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary





1−(λ+3λC)

k→∞→ λ1−(λ+3λC) M




∑αi <∞,

Iktot → 0

proof


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Summary

• Main Adaptive Scheme• Non-uniform numerical schemes• Non-uniform consistency criterion• MAS & entropy conservative schemes• Mesh adaptation & Total variation

Ongoing work

• Entropy dissipation via mesh reconstruction -(with M. Lukacova)

• Consistency criterion• Total variation for conservative mesh reconstruction• Incorporate numerical schemes in mesh reconstruction


for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Summary


Ongoing work




for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Summary


Ongoing work




for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Summary


Ongoing work




for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Summary


Ongoing work




for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Summary


Ongoing work




for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Summary


Ongoing work




for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Summary


Ongoing work




for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Summary


Ongoing work




for HCL

NikolaosSfakianakis

Introduction

Uniform mesh

Non-uniform mesh

Main AdaptiveScheme

Steps


Remarks

Consistency


More remarks

TVB

Extremes

TV - Approach 1

TV - Approach 2

Summary

Summary


Ongoing work




for HCL

NikolaosSfakianakis

Appendix

Solution update

Consistency

λ-Rule

TV

Solution update, vertex centered, pw constant

uni = 1

|Cn+1i |

∫Cn+1

iUn(x)dx

Cn+1i = (xn+1

i−1/2, xn+1i+1/2), xn+1

i−1/2 =xn+1

i−1 +xn+1i

2return

oldnew

xn+1j+1xn+1

jxn+1j−1

xni+1xn

ixni−1


for HCL

NikolaosSfakianakis

Appendix

Solution update

Consistency

λ-Rule

TV

Uniform mesh

Discretise ut + f (u)x = 0︸︷︷︸pde

, forward in time: Un+1 = N (Un)︸︷︷︸num.scheme

Local Truncation Errorτn(u) = 1

∆t

(un+1 −N (un)

), un = u(·, tn) smooth pde solution

Consistent schemelim∆t→0 τ

n(u) = 0, for u smooth pde solution


for HCL

NikolaosSfakianakis

Appendix

Solution update

Consistency

λ-Rule

TV

Uniform mesh




∆t

(un+1 −N (un)





for HCL

NikolaosSfakianakis

Appendix

Solution update

Consistency

λ-Rule

TV

Uniform mesh




∆t

(un+1 −N (un)





for HCL

NikolaosSfakianakis

Appendix

Solution update

Consistency

λ-Rule

TV

Extended consistency criterion

Proof.For tr = τn

i ∆t

tr = un+1i − un

i + ∆thi

(F R(un

i , uni+1)− F L(un

i−1, uni ))

after Taylor expansions and F R(uni , u

ni ) = F L(un

i , uni ).

tr = ∆t(

(ut )ni +

(F R

2 (uni , u

ni )

hi +hi+12hi

+ F L1 (un

i , uni )

hi−1+hi2hi

)(ux )n

i

)+O(∆t2) +O

(∆thi

h2)

We use hi−1+hi2hi

F L1 (un

i , uni ) +

hi +hi+12hi

F R2 (un

i , uni ) = f ′(un

i ) so

tr =

((ut )

ni + f ′(un

i )(ux )ni

)∆t +O(∆t2) +O( ∆t

hih2),

finally

τni = tr

∆t = O(∆t) +O( h2

hi)

return


for HCL

NikolaosSfakianakis

Appendix

Solution update

Consistency

λ-Rule

TV

λ-rule for interpolation on piecewise linears

There exist a global constant λ such that for every pair xoldi0

- xnewj0

• If xoldi0−1 ≤ xnew

j0≤ xold

i0then

xoldi0− xnew

j0≥ (1− λ)(xold

i0− xold

i0−1)

hence

xnewj0− xold

i0−1 ≤ λ(xoldi0− xold

i0−1)

• If xoldi0≤ xnew

j0≤ xold

i0+1 then

xnewj0− xold

i0≥ (1− λ)(xold

i0+1 − xoldi0

)

hence

xoldi0+1 − xnew

j0≤ λ(xold

i0+1 − xoldi0

)

return

xoldi0−1 xold

i0xnew

j0xold

i0+1


for HCL

NikolaosSfakianakis

Appendix

Solution update

Consistency

λ-Rule

TV

Uniform time bound

Proof.

Ekm ≤ λmCm−1∑k−1

l=m−1

( ll−m+1

)λl−m+1(1 + 2C)l−m+1ak−l

Since ai ≤ CM we get:

Ekm ≤ λmCmM

∑k−1l=m−1

( ll−m+1

)λl−m+1(1 + 2C)l−m+1

or by setting ν = l −m + 1:

Ekm ≤ λmCmM

∑k−mν=0

(ν+m−1ν

)λν(1 + 2C)ν

or

Ekm ≤ λmCmM

∑∞ν=0

(ν+m−1ν

)(λ+ 2λC)ν

we recall that:∑∞ν=0

(ν+m−1ν

)tν = 1

(1−t)m , for |t | < 1, and

Ekm ≤ λmCmM 1

(1−λ−2λC)m = M(

λC1−λ−2λC

)m

return


for HCL

NikolaosSfakianakis

Appendix

Solution update

Consistency

λ-Rule

TV

TV Bound

Proof.

∑km=1 Ek

m ≤ M∑k

m=1

(λC

1−λ−2λC

)m≤ M

∑∞m=1

(λC

1−λ−2λC

)m

= M 11− λC

1−λ−2λC≤ M 1−λ−2λC

1−λ−3λC

since λ+ 3λC < 1. The variation of the oscillatory part is bounded by

TVI ≤ 2M∑∞

m=1

(λC

1−λ−2λC

)m≤ 2M 1−λ−2λC

1−λ−3λC

return


for HCL

NikolaosSfakianakis

Appendix

Solution update

Consistency

λ-Rule

TV

TV increase diminishing

Proof.The sum ∑∞

k=1 Iktot =

∑∞k=1

(λ∑k

j=1(λ+ 3λC)k−jαj

)= λ

∑∞k=1

∑kj=1(λ+ 3λC)k−jαj

is the Cauchy product of∑∞i=0(λ+ 3λC)i = 1

1−(λ+3λC)<∞

and ∑∞i=0 αi <∞

So ∑∞k=1 Ik

tot <∞

hencelimk→∞ Ik

tot = 0

return

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Numerical schemes for Hyperbolic CL over non-uniform ... · upd values next values. FD schemes over non-uniform grids for HCL Nikolaos Sfakianakis Introduction Uniform mesh Non-uniform