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Formulation of the problem Space discretization Time discretization Higher order solution of ODEs arising from DG space semi-discretization of nonstationary convection-diffusion problems Miloslav Vlas ak V t Dolej s Charles

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Formulation of the problem Space discretization Time discretization

Higher order solution of ODEs arising from DG space semi-discretization of nonstationary convection-diffusion problems

Miloslav Vlas´ ak V´ ıt Dolejˇ s´ ı

Charles University Prague Faculty of Mathematics and Physics

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Introduction

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Introduction

Our aim: efficient numerical scheme for a simulation of unsteady compressible flows,

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Introduction

Our aim: efficient numerical scheme for a simulation of unsteady compressible flows, model problem: scalar nonstationary nonlinear convection-diffusion equation,

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Introduction

Our aim: efficient numerical scheme for a simulation of unsteady compressible flows, model problem: scalar nonstationary nonlinear convection-diffusion equation, space semi-discretization (e.g., DGFEM),

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Introduction

Our aim: efficient numerical scheme for a simulation of unsteady compressible flows, model problem: scalar nonstationary nonlinear convection-diffusion equation, space semi-discretization (e.g., DGFEM), suitable time discretization (e.g. BDF)

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Continuous problem

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Continuous problem

Let Ω ⊂ I Rd be convex, QT ≡ Ω × (0, T),

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Continuous problem

Let Ω ⊂ I Rd be convex, QT ≡ Ω × (0, T), we seek u : QT → I R such that

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Continuous problem

Let Ω ⊂ I Rd be convex, QT ≡ Ω × (0, T), we seek u : QT → I R such that ∂u ∂t +

d

∑

s=1

∂fs(u) ∂xs = 휀 Δu + g in QT,

Vlas´ ak, Dolejˇ s´ ı

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SLIDE 11

Formulation of the problem Space discretization Time discretization

Continuous problem

Let Ω ⊂ I Rd be convex, QT ≡ Ω × (0, T), we seek u : QT → I R such that ∂u ∂t +

d

∑

s=1

∂fs(u) ∂xs = 휀 Δu + g in QT,

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Continuous problem

Let Ω ⊂ I Rd be convex, QT ≡ Ω × (0, T), we seek u : QT → I R such that ∂u ∂t +

d

∑

s=1

∂fs(u) ∂xs = 휀 Δu + g in QT,

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Continuous problem

Let Ω ⊂ I Rd be convex, QT ≡ Ω × (0, T), we seek u : QT → I R such that ∂u ∂t +

d

∑

s=1

∂fs(u) ∂xs = 휀 Δu + g in QT, u

  • ∂Ω×(0,T) = uD,

u(x, 0) = u0(x), x ∈ Ω,

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Space settings

Let V ⊂ L2(Ω) be a space for exact solution

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Space settings

Let V ⊂ L2(Ω) be a space for exact solution Vh,p be a space for discrete solution

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Diffusive form Ah

Ah(v, w) be linear and symmetric

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Diffusive form Ah

Ah(v, w) be linear and symmetric ∣∣∣v∣∣∣2 = Ah(v, v) ∀v ∈ V

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Diffusive form Ah

Ah(v, w) be linear and symmetric ∣∣∣v∣∣∣2 = Ah(v, v) ∀v ∈ V Ah(v, w) ≤ C∣∣∣v∣∣∣ ∣∣∣w∣∣∣ ∀v, w ∈ Vh,p

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Diffusive form Ah

Ah(v, w) be linear and symmetric ∣∣∣v∣∣∣2 = Ah(v, v) ∀v ∈ V Ah(v, w) ≤ C∣∣∣v∣∣∣ ∣∣∣w∣∣∣ ∀v, w ∈ Vh,p Ah(v − Rhv, w) = 0 ∀w ∈ Vh,p

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Diffusive form Ah

Ah(v, w) be linear and symmetric ∣∣∣v∣∣∣2 = Ah(v, v) ∀v ∈ V Ah(v, w) ≤ C∣∣∣v∣∣∣ ∣∣∣w∣∣∣ ∀v, w ∈ Vh,p Ah(v − Rhv, w) = 0 ∀w ∈ Vh,p ∥Rhv − v∥ ≤ Chp+1

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Convective form bh

bh(v, w) be nonlinear in v and linear in w

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Convective form bh

bh(v, w) be nonlinear in v and linear in w bh(u, w) − bh(v, w) ≤ C∥u − v∥ ∣∣∣w∣∣∣ ∀u, v, w ∈ V

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Convective form bh

bh(v, w) be nonlinear in v and linear in w bh(u, w) − bh(v, w) ≤ C∥u − v∥ ∣∣∣w∣∣∣ ∀u, v, w ∈ V bh(v, w) − bh(Rhv, w) ≤ Chp+1∣∣∣w∣∣∣

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Source form ℓh

ℓh(v) be linear

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Semi-discrete problem

find uh ∈ C 1([0, T]; Vh,p) such that

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Semi-discrete problem

find uh ∈ C 1([0, T]; Vh,p) such that (∂uh ∂t (t), v ) + 휀Ah(uh(t), v) + bh(uh(t), v) = ℓh(v) (t) ∀ v ∈ Vh,p, ∀ t ∈ [0, T],

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Semi-discrete problem

find uh ∈ C 1([0, T]; Vh,p) such that (∂uh ∂t (t), v ) + 휀Ah(uh(t), v) + bh(uh(t), v) = ℓh(v) (t) ∀ v ∈ Vh,p, ∀ t ∈ [0, T], (uh(0), v) = (u0, v) ∀ v ∈ Vh,p

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Semi-discrete problem

find uh ∈ C 1([0, T]; Vh,p) such that (∂uh ∂t (t), v ) + 휀Ah(uh(t), v) + bh(uh(t), v) = ℓh(v) (t) ∀ v ∈ Vh,p, ∀ t ∈ [0, T], (uh(0), v) = (u0, v) ∀ v ∈ Vh,p

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Time discretization

Let ts = s휏 s = 0, . . . , r be a partition of [0, T] with a time step 휏 = T/r,

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Time discretization

Let ts = s휏 s = 0, . . . , r be a partition of [0, T] with a time step 휏 = T/r, let uh(ts) = us

h ≈ Us ∈ Vh,p for s = 0, . . . , r

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Euler method

Backward Euler method linearized by Forward Euler method

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Euler method

Backward Euler method linearized by Forward Euler method (Us+1 − Us, v) + 휏휀Ah(Us+1, v) + 휏bh(Us, v) = 휏ℓ(v)

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Euler method

Backward Euler method linearized by Forward Euler method (Us+1 − Us, v) + 휏휀Ah(Us+1, v) + 휏bh(Us, v) = 휏ℓ(v)

Vlas´ ak, Dolejˇ s´ ı

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SLIDE 34

Formulation of the problem Space discretization Time discretization

Euler method

Backward Euler method linearized by Forward Euler method (Us+1 − Us, v) + 휏휀Ah(Us+1, v) + 휏bh(Us, v) = 휏ℓ(v)

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Euler method

Backward Euler method linearized by Forward Euler method (Us+1 − Us, v) + 휏휀Ah(Us+1, v) + 휏bh(Us, v) = 휏ℓ(v) error estimates derived by Dolejˇ s´ ı, Feistauer, Hozman (2007)

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Euler method

Backward Euler method linearized by Forward Euler method (Us+1 − Us, v) + 휏휀Ah(Us+1, v) + 휏bh(Us, v) = 휏ℓ(v) error estimates derived by Dolejˇ s´ ı, Feistauer, Hozman (2007) ∥e∥2

h,휏,L∞(L2) = O(h2p + 휏 2),

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

BDF (one–leg)

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

BDF (one–leg)

BDF of order m = 1, . . . , 6: ∃훼j j = 0, . . . , m

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

BDF (one–leg)

BDF of order m = 1, . . . , 6: ∃훼j j = 0, . . . , m ⎛ ⎝

m

∑

j=0

훼jUs+j, v ⎞ ⎠ + 휏휀Ah(Us+m, v) + 휏bh(

m−1

∑

j=0

훽jUs+j, v) = 휏ℓ(v)

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

BDF (one–leg)

BDF of order m = 1, . . . , 6: ∃훼j j = 0, . . . , m ⎛ ⎝

m

∑

j=0

훼jUs+j, v ⎞ ⎠ + 휏휀Ah(Us+m, v) + 휏bh(

m−1

∑

j=0

훽jUs+j, v) = 휏ℓ(v)

Vlas´ ak, Dolejˇ s´ ı

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SLIDE 41

Formulation of the problem Space discretization Time discretization

BDF (one–leg)

BDF of order m = 1, . . . , 6: ∃훼j j = 0, . . . , m ⎛ ⎝

m

∑

j=0

훼jUs+j, v ⎞ ⎠ + 휏휀Ah(Us+m, v) + 휏bh(

m−1

∑

j=0

훽jUs+j, v) = 휏ℓ(v) 훽j j = 0, . . . , m − 1 chosen in such way as ∑m−1

j=0 훽jus+j ≈ us+m

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

BDF (one–leg)

BDF of order m = 1, . . . , 6: ∃훼j j = 0, . . . , m ⎛ ⎝

m

∑

j=0

훼jUs+j, v ⎞ ⎠ + 휏휀Ah(Us+m, v) + 휏bh(

m−1

∑

j=0

훽jUs+j, v) = 휏ℓ(v) 훽j j = 0, . . . , m − 1 chosen in such way as ∑m−1

j=0 훽jus+j ≈ us+m

∥e∥2

h,휏,L∞(L2) = O(h2p+2 + 휏 2m),

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

BDF coefficients-order conditions

m

∑

j=0

훼j =

m

∑

j=0

훼j(m − j) = −1

m

∑

j=0

훼j(m − j)s = s = 2, . . . , m

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

BDF coefficients

훼j = (−1)m−j (m

j

) m − j j = 0, . . . , m − 1 훼m =

m

∑

j=1

1 j 훽j = −훼j(m − j) = (−1)m−j−1 (m j )

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Technique of the proof

we divide the error es = Us − us = Us − Rhus

  • :=휉s

+ Rhus − us

  • :=휂s

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Technique of the proof

we divide the error es = Us − us = Us − Rhus

  • :=휉s

+ Rhus − us

  • :=휂s

for 휂 are standard estimates ∥휂s∥ ≤ Chp+1

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Technique of the proof

we divide the error es = Us − us = Us − Rhus

  • :=휉s

+ Rhus − us

  • :=휂s

for 휂 are standard estimates ∥휂s∥ ≤ Chp+1 it is sufficient to estimate ∥휉s∥

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Technique of the proof

substracting equation for weak solution from equation for discrete solution we obtain

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Technique of the proof

substracting equation for weak solution from equation for discrete solution we obtain ⎛ ⎝

m

∑

j=0

훼j휉s+j, v ⎞ ⎠ + 휏휀Ah(휉s+m, v) = ⎛ ⎝휏 ∂u ∂t (ts+m) −

m

∑

j=0

훼jus+j −

m

∑

j=0

훼j휂s+j, v ⎞ ⎠ +휏 ⎛ ⎝bh(us+m, v) − bh(

m−1

∑

j=0

훽jUs+j, v) ⎞ ⎠

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Technique of the proof

choice v = 훾n−m−s휉n

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Technique of the proof

choice v = 훾n−m−s휉n

  • perators 훾j such that

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Technique of the proof

choice v = 훾n−m−s휉n

  • perators 훾j such that

n−m

∑

s=0

⎛ ⎝

m

∑

j=0

훼j휉s+j, 훾n−m−s휉n ⎞ ⎠ + 휏휀Ah(휉s+m, 훾n−m−s휉n) = ∥휉n∥2 + ⎛ ⎝

m−1

∑

s=0 s

∑

j=0

훼j훾n−m−s+j휉s, 휉n ⎞ ⎠

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Technique of the proof

choice v = 훾n−m−s휉n

  • perators 훾j such that

n−m

∑

s=0

⎛ ⎝

m

∑

j=0

훼j휉s+j, 훾n−m−s휉n ⎞ ⎠ + 휏휀Ah(휉s+m, 훾n−m−s휉n) = ∥휉n∥2 + ⎛ ⎝

m−1

∑

s=0 s

∑

j=0

훼j훾n−m−s+j휉s, 휉n ⎞ ⎠ ∥훾j∥ ≤ C

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Technique of the proof

choice v = 훾n−m−s휉n

  • perators 훾j such that

n−m

∑

s=0

⎛ ⎝

m

∑

j=0

훼j휉s+j, 훾n−m−s휉n ⎞ ⎠ + 휏휀Ah(휉s+m, 훾n−m−s휉n) = ∥휉n∥2 + ⎛ ⎝

m−1

∑

s=0 s

∑

j=0

훼j훾n−m−s+j휉s, 휉n ⎞ ⎠ ∥훾j∥ ≤ C 휏휀 ∑∞

j=0 ∣∣∣훾jv∣∣∣2 ≤ C∥v∥2

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Technique of the proof

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Technique of the proof

∥휉n∥2 ≤ C(휏 m + hp+1 +

m−1

∑

j=0

∥휉j∥2) + C 휀

n−1

∑

j=0

∥휉j∥2

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Technique of the proof

∥휉n∥2 ≤ C(휏 m + hp+1 +

m−1

∑

j=0

∥휉j∥2) + C 휀

n−1

∑

j=0

∥휉j∥2 Gronwall lemma

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Conclusion

scalar nonstationary nonlinear convection-diffusion equation

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Conclusion

scalar nonstationary nonlinear convection-diffusion equation IMEX BDF-DGFE discretization discussed

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Conclusion

scalar nonstationary nonlinear convection-diffusion equation IMEX BDF-DGFE discretization discussed error estimates derived

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Thank you for your attention.

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Numerical example1

t ∈ [0, 1] y′(t) = −휀y(t) + y2(t) y(0) = 1

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Numerical example1

휏 =

1 20∗2m

BDF (one-leg) 2nd order BDF (IMEX) 2nd order m ∥e∥L∞ rate ∥e∥L∞ rate 1 3.2924

  • 4.6737
  • 2

1.2668 1.3780 2.0487 1.1898 3 0.3975 1.6723 0.7080 1.5329 4 0.1108 1.8425 0.2086 1.7629 5 0.0292 1.9259 0.0564 1.8866 6 0.0075 1.9647 0.0145 1.9461 7 0.0019 1.9829 0.0037 1.9740 8 0.0005 1.9916 0.0009 1.9873

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Numerical example1

휏 =

1 20∗2m

Runge–Kutta 2nd order Time DG 2nd order m ∥e∥L∞ rate ∥e∥L∞ rate 1 1.3770

  • 2.7966
  • 2

0.4270 1.6893 0.9946 1.4915 3 0.1173 1.8634 0.2912 1.7722 4 0.0306 1.9413 0.0774 1.9116 5 0.0078 1.9739 0.0198 1.9665 6 0.0020 1.9879 0.0050 1.9866 7 0.0005 1.9942 0.0013 1.9942 8 0.0001 1.9972 0.0003 1.9973

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Numerical example2

Ω = (0, 1) × (0, 1), T = 1, 휀 = 0.02 fs(u) = u2 2 , s = 1, 2 u = 16exp(10t) − 1 exp(10) − 1 xy(1 − x)(1 − y)

Vlas´ ak, Dolejˇ s´ ı

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Formulation of the problem Space discretization Time discretization

Numerical example2

휏 = 0.4

2m

BDF 2nd order Time DG 2nd order m ∥e∥L∞(L2) rate ∥e∥L∞(L2) rate 1 3.251E − 01

  • 3.650E − 02
  • 2

1.098E − 01 1.566 1.899E − 02 0.943 3 3.398E − 02 1.693 6.421E − 03 1.564 4 9.810E − 03 1.792 1.802E − 03 1.833 5 2.658E − 03 1.884 4.717E − 04 1.934 6 6.943E − 04 1.937 1.203E − 04 1.971

Vlas´ ak, Dolejˇ s´ ı