[PPT] - SLDG schemes for 1st and 2nd order PDEs Olivier Bokanowski PowerPoint Presentation

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SLDG schemes for 1st and 2nd order PDEs

Olivier Bokanowski

Laboratory Jacques Louis Lions (Paris 6) University Paris-Diderot (Paris 7) Commands (INRIA Saclay / Ensta ParisTech) Joint work with

G. Simarmata

(Rabobank, Netherlands) HYP 2012, Padova, 25-29 June 2012

Olivier Bokanowski SLDG for 2nd order PDEs

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ADVERTISMENT : HJ parallel Library

ut + H(t, x, ∇u, [D2u]) = 0,

x ∈ Rd u(0, x) = u0(x) C++, parallel (MPI/OpenMP) works in any dimension d (limited to machine’s capacity)

Finite Difference solver (based on ENO): MPI / OpenMP
Semi-Lagrangian schemes (1 & 2 order HJ PDE) : OpenMP
Explicit schemes, uniform grid, control-oriented equations.
Development: O. Bokanowski, H. Zidani, A. Desilles (and J. Zhao)

⇒ www.ensta.fr/∼zidani/BiNoPe-HJ/

Olivier Bokanowski SLDG for 2nd order PDEs

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I. Introduction

Olivier Bokanowski SLDG for 2nd order PDEs

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Could we consider an explicit scheme for ut − 1 2Tr(σσTD2u) + B · ∇v + rv = 0, x ∈ Rd ? PROS Only explicit is implementable for high-d ! Motivated by related HJ equations involving max operators,

r obstacle terms, application to optimal control

CONS FD, DG methods needs restrictive CFL (∆t ≤ C∆x2, or ∆t ≤ C∆x, small C) ⇒ use SL scheme Cannot be high order, and monotone ("Godunov’s theorem", Harten, Osterlee and Van Pijl 2012 "negative result") ... We shall try to bypass some of these drawbacks...

Olivier Bokanowski SLDG for 2nd order PDEs

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Could we consider an explicit scheme for ut − 1 2Tr(σσTD2u) + B · ∇v + rv = 0, x ∈ Rd ? PROS Only explicit is implementable for high-d ! Motivated by related HJ equations involving max operators,

r obstacle terms, application to optimal control

CONS FD, DG methods needs restrictive CFL (∆t ≤ C∆x2, or ∆t ≤ C∆x, small C) ⇒ use SL scheme Cannot be high order, and monotone ("Godunov’s theorem", Harten, Osterlee and Van Pijl 2012 "negative result") ... We shall try to bypass some of these drawbacks...

Olivier Bokanowski SLDG for 2nd order PDEs

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Could we consider an explicit scheme for ut − 1 2Tr(σσTD2u) + B · ∇v + rv = 0, x ∈ Rd ? PROS Only explicit is implementable for high-d ! Motivated by related HJ equations involving max operators,

r obstacle terms, application to optimal control

CONS FD, DG methods needs restrictive CFL (∆t ≤ C∆x2, or ∆t ≤ C∆x, small C) ⇒ use SL scheme Cannot be high order, and monotone ("Godunov’s theorem", Harten, Osterlee and Van Pijl 2012 "negative result") ... We shall try to bypass some of these drawbacks...

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 7

Could we consider an explicit scheme for ut − 1 2Tr(σσTD2u) + B · ∇v + rv = 0, x ∈ Rd ? PROS Only explicit is implementable for high-d ! Motivated by related HJ equations involving max operators,

r obstacle terms, application to optimal control

CONS FD, DG methods needs restrictive CFL (∆t ≤ C∆x2, or ∆t ≤ C∆x, small C) ⇒ use SL scheme Cannot be high order, and monotone ("Godunov’s theorem", Harten, Osterlee and Van Pijl 2012 "negative result") ... We shall try to bypass some of these drawbacks...

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 8

Could we consider an explicit scheme for ut − 1 2Tr(σσTD2u) + B · ∇v + rv = 0, x ∈ Rd ? PROS Only explicit is implementable for high-d ! Motivated by related HJ equations involving max operators,

r obstacle terms, application to optimal control

CONS FD, DG methods needs restrictive CFL (∆t ≤ C∆x2, or ∆t ≤ C∆x, small C) ⇒ use SL scheme Cannot be high order, and monotone ("Godunov’s theorem", Harten, Osterlee and Van Pijl 2012 "negative result") ... We shall try to bypass some of these drawbacks...

Olivier Bokanowski SLDG for 2nd order PDEs

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Could we consider an explicit scheme for ut − 1 2Tr(σσTD2u) + B · ∇v + rv = 0, x ∈ Rd ? PROS Only explicit is implementable for high-d ! Motivated by related HJ equations involving max operators,

r obstacle terms, application to optimal control

CONS FD, DG methods needs restrictive CFL (∆t ≤ C∆x2, or ∆t ≤ C∆x, small C) ⇒ use SL scheme Cannot be high order, and monotone ("Godunov’s theorem", Harten, Osterlee and Van Pijl 2012 "negative result") ... We shall try to bypass some of these drawbacks...

Olivier Bokanowski SLDG for 2nd order PDEs

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II. Schemes

Olivier Bokanowski SLDG for 2nd order PDEs

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Idea: 1d semi-lagrangian for advection convex combinations for diffusion splitting, splitting, splitting But : In general this strategy may not work practically because the semi-lagrangian scheme may not be precise enough.

Olivier Bokanowski SLDG for 2nd order PDEs

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Idea: 1d semi-lagrangian for advection convex combinations for diffusion splitting, splitting, splitting But : In general this strategy may not work practically because the semi-lagrangian scheme may not be precise enough.

Olivier Bokanowski SLDG for 2nd order PDEs

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Idea: 1d semi-lagrangian for advection convex combinations for diffusion splitting, splitting, splitting But : In general this strategy may not work practically because the semi-lagrangian scheme may not be precise enough.

Olivier Bokanowski SLDG for 2nd order PDEs

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Idea: 1d semi-lagrangian for advection convex combinations for diffusion splitting, splitting, splitting But : In general this strategy may not work practically because the semi-lagrangian scheme may not be precise enough.

Olivier Bokanowski SLDG for 2nd order PDEs

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Idea: 1d semi-lagrangian for advection convex combinations for diffusion splitting, splitting, splitting But : In general this strategy may not work practically because the semi-lagrangian scheme may not be precise enough.

Olivier Bokanowski SLDG for 2nd order PDEs

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Idea: 1d semi-lagrangian for advection convex combinations for diffusion splitting, splitting, splitting But : In general this strategy may not work practically because the semi-lagrangian scheme may not be precise enough.

Olivier Bokanowski SLDG for 2nd order PDEs

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1) SLDG schemes for first order (advection, 1d)

Morton, Priestley, Suli (1988) Crouzeillles, Mehrenberger, Vecil (2010): SLDG Qiu and Shu (2011): SLDG + splitting

Consider the 1d advection equation for t ∈ (0, T):

ut + b(x)ux = 0, x ∈ (0, 1) (with periodic b.c.)

Notice that u(t + ∆t, x) = u(t, x − b∆t) if b(x) = b = const
Introduce DG: mesh intervals Ii partition of (0, 1), and

Vk := {v ∈ L2(0, 1), v|Ii ∈ Pk for all i} "DG space" where Pk is the set of polynomials of degree ≤ k.

Olivier Bokanowski SLDG for 2nd order PDEs

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1) SLDG schemes for first order (advection, 1d)

Morton, Priestley, Suli (1988) Crouzeillles, Mehrenberger, Vecil (2010): SLDG Qiu and Shu (2011): SLDG + splitting

Consider the 1d advection equation for t ∈ (0, T):

ut + b(x)ux = 0, x ∈ (0, 1) (with periodic b.c.)

Notice that u(t + ∆t, x) = u(t, x − b∆t) if b(x) = b = const
Introduce DG: mesh intervals Ii partition of (0, 1), and

Vk := {v ∈ L2(0, 1), v|Ii ∈ Pk for all i} "DG space" where Pk is the set of polynomials of degree ≤ k.

Olivier Bokanowski SLDG for 2nd order PDEs

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1) SLDG schemes for first order (advection, 1d)

Morton, Priestley, Suli (1988) Crouzeillles, Mehrenberger, Vecil (2010): SLDG Qiu and Shu (2011): SLDG + splitting

Consider the 1d advection equation for t ∈ (0, T):

ut + b(x)ux = 0, x ∈ (0, 1) (with periodic b.c.)

Notice that u(t + ∆t, x) = u(t, x − b∆t) if b(x) = b = const
Introduce DG: mesh intervals Ii partition of (0, 1), and

Vk := {v ∈ L2(0, 1), v|Ii ∈ Pk for all i} "DG space" where Pk is the set of polynomials of degree ≤ k.

Olivier Bokanowski SLDG for 2nd order PDEs

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SLDG scheme: Find un+1 ∈ Vk such that

∀ϕ ∈ Vk,

un+1(x)ϕ(x)dx =
un(x − b∆t)ϕ(x)dx
un+1 = Π(un(· − b∆t)) where Π is the L2 projection on Vk.

Theorem (i) The scheme is exactly implementable (ii) High order: O( ∆xk+1

∆t

) (iii) L2 stable

⇒ no CFL !
"Immediate" proof
Implementation : gauss quadrature formula:

1

−1

ϕ(x)dx =

α=0,...,k

wαϕ(xα) for any ϕ ∈ P2k+1.

Olivier Bokanowski SLDG for 2nd order PDEs

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SLDG scheme: Find un+1 ∈ Vk such that

∀ϕ ∈ Vk,

un+1(x)ϕ(x)dx =
un(x − b∆t)ϕ(x)dx
un+1 = Π(un(· − b∆t)) where Π is the L2 projection on Vk.

Theorem (i) The scheme is exactly implementable (ii) High order: O( ∆xk+1

∆t

) (iii) L2 stable

⇒ no CFL !
"Immediate" proof
Implementation : gauss quadrature formula:

1

−1

ϕ(x)dx =

α=0,...,k

wαϕ(xα) for any ϕ ∈ P2k+1.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 22

SLDG scheme: Find un+1 ∈ Vk such that

∀ϕ ∈ Vk,

un+1(x)ϕ(x)dx =
un(x − b∆t)ϕ(x)dx
un+1 = Π(un(· − b∆t)) where Π is the L2 projection on Vk.

Theorem (i) The scheme is exactly implementable (ii) High order: O( ∆xk+1

∆t

) (iii) L2 stable

⇒ no CFL !
"Immediate" proof
Implementation : gauss quadrature formula:

1

−1

ϕ(x)dx =

α=0,...,k

wαϕ(xα) for any ϕ ∈ P2k+1.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 23

SLDG scheme: Find un+1 ∈ Vk such that

∀ϕ ∈ Vk,

un+1(x)ϕ(x)dx =
un(x − b∆t)ϕ(x)dx
un+1 = Π(un(· − b∆t)) where Π is the L2 projection on Vk.

Theorem (i) The scheme is exactly implementable (ii) High order: O( ∆xk+1

∆t

) (iii) L2 stable

⇒ no CFL !
"Immediate" proof
Implementation : gauss quadrature formula:

1

−1

ϕ(x)dx =

α=0,...,k

wαϕ(xα) for any ϕ ∈ P2k+1.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 24

SLDG scheme: Find un+1 ∈ Vk such that

∀ϕ ∈ Vk,

un+1(x)ϕ(x)dx =
un(x − b∆t)ϕ(x)dx
un+1 = Π(un(· − b∆t)) where Π is the L2 projection on Vk.

Theorem (i) The scheme is exactly implementable (ii) High order: O( ∆xk+1

∆t

) (iii) L2 stable

⇒ no CFL !
"Immediate" proof
Implementation : gauss quadrature formula:

1

−1

ϕ(x)dx =

α=0,...,k

wαϕ(xα) for any ϕ ∈ P2k+1.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 25

SLDG scheme: Find un+1 ∈ Vk such that

∀ϕ ∈ Vk,

un+1(x)ϕ(x)dx =
un(x − b∆t)ϕ(x)dx
un+1 = Π(un(· − b∆t)) where Π is the L2 projection on Vk.

Theorem (i) The scheme is exactly implementable (ii) High order: O( ∆xk+1

∆t

) (iii) L2 stable

⇒ no CFL !
"Immediate" proof
Implementation : gauss quadrature formula:

1

−1

ϕ(x)dx =

α=0,...,k

wαϕ(xα) for any ϕ ∈ P2k+1.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 26

SLDG scheme: Find un+1 ∈ Vk such that

∀ϕ ∈ Vk,

un+1(x)ϕ(x)dx =
un(x − b∆t)ϕ(x)dx
un+1 = Π(un(· − b∆t)) where Π is the L2 projection on Vk.

Theorem (i) The scheme is exactly implementable (ii) High order: O( ∆xk+1

∆t

) (iii) L2 stable

⇒ no CFL !
"Immediate" proof
Implementation : gauss quadrature formula:

1

−1

ϕ(x)dx =

α=0,...,k

wαϕ(xα) for any ϕ ∈ P2k+1.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 27

SLDG scheme: Find un+1 ∈ Vk such that

∀ϕ ∈ Vk,

un+1(x)ϕ(x)dx =
un(x − b∆t)ϕ(x)dx
un+1 = Π(un(· − b∆t)) where Π is the L2 projection on Vk.

Theorem (i) The scheme is exactly implementable (ii) High order: O( ∆xk+1

∆t

) (iii) L2 stable

⇒ no CFL !
"Immediate" proof
Implementation : gauss quadrature formula:

1

−1

ϕ(x)dx =

α=0,...,k

wαϕ(xα) for any ϕ ∈ P2k+1.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 28

SLDG scheme: Find un+1 ∈ Vk such that

∀ϕ ∈ Vk,

un+1(x)ϕ(x)dx =
un(x − b∆t)ϕ(x)dx
un+1 = Π(un(· − b∆t)) where Π is the L2 projection on Vk.

Theorem (i) The scheme is exactly implementable (ii) High order: O( ∆xk+1

∆t

) (iii) L2 stable

⇒ no CFL !
"Immediate" proof
Implementation : gauss quadrature formula:

1

−1

ϕ(x)dx =

α=0,...,k

wαϕ(xα) for any ϕ ∈ P2k+1.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 29

SLDG scheme: Find un+1 ∈ Vk such that

∀ϕ ∈ Vk,

un+1(x)ϕ(x)dx =
un(x − b∆t)ϕ(x)dx
un+1 = Π(un(· − b∆t)) where Π is the L2 projection on Vk.

Theorem (i) The scheme is exactly implementable (ii) High order: O( ∆xk+1

∆t

) (iii) L2 stable

⇒ no CFL !
"Immediate" proof
Implementation : gauss quadrature formula:

1

−1

ϕ(x)dx =

α=0,...,k

wαϕ(xα) for any ϕ ∈ P2k+1.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 30

SLDG scheme for non constant b(x): Find un+1 ∈ Vk,

∀ϕ ∈ Vk,

un+1(x)ϕ(x)dx =
Gauss

un(yx(−∆t))ϕ(x)dx where d

dt yx(t) = b(yx(t)), y(0) = x.

Theorem (B, Simaramata) (i) The scheme is STILL approx. implementable (Liu & Shu) (ii) order: O( ∆xk+1

∆t

) for k = 1, 2 a (iii) L2 stable for k ≥ 1, under a weak CFL.

a(here with ∆t ≤ λ∆x, λ constant)

same as Liu & Shu 2011 in the linear case.
⇒ weak CFL : ∆xk+1

∆t

≤ const. (Allows ∆t = λ∆x with large λ.)

Less immediate proof ("one-page proof")

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 31

SLDG scheme for non constant b(x): Find un+1 ∈ Vk,

∀ϕ ∈ Vk,

un+1(x)ϕ(x)dx =
Gauss

un(yx(−∆t))ϕ(x)dx where d

dt yx(t) = b(yx(t)), y(0) = x.

Theorem (B, Simaramata) (i) The scheme is STILL approx. implementable (Liu & Shu) (ii) order: O( ∆xk+1

∆t

) for k = 1, 2 a (iii) L2 stable for k ≥ 1, under a weak CFL.

a(here with ∆t ≤ λ∆x, λ constant)

same as Liu & Shu 2011 in the linear case.
⇒ weak CFL : ∆xk+1

∆t

≤ const. (Allows ∆t = λ∆x with large λ.)

Less immediate proof ("one-page proof")

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 32

SLDG scheme for non constant b(x): Find un+1 ∈ Vk,

∀ϕ ∈ Vk,

un+1(x)ϕ(x)dx =
Gauss

un(yx(−∆t))ϕ(x)dx where d

dt yx(t) = b(yx(t)), y(0) = x.

Theorem (B, Simaramata) (i) The scheme is STILL approx. implementable (Liu & Shu) (ii) order: O( ∆xk+1

∆t

) for k = 1, 2 a (iii) L2 stable for k ≥ 1, under a weak CFL.

a(here with ∆t ≤ λ∆x, λ constant)

same as Liu & Shu 2011 in the linear case.
⇒ weak CFL : ∆xk+1

∆t

≤ const. (Allows ∆t = λ∆x with large λ.)

Less immediate proof ("one-page proof")

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 33

SLDG scheme for non constant b(x): Find un+1 ∈ Vk,

∀ϕ ∈ Vk,

un+1(x)ϕ(x)dx =
Gauss

un(yx(−∆t))ϕ(x)dx where d

dt yx(t) = b(yx(t)), y(0) = x.

Theorem (B, Simaramata) (i) The scheme is STILL approx. implementable (Liu & Shu) (ii) order: O( ∆xk+1

∆t

) for k = 1, 2 a (iii) L2 stable for k ≥ 1, under a weak CFL.

a(here with ∆t ≤ λ∆x, λ constant)

same as Liu & Shu 2011 in the linear case.
⇒ weak CFL : ∆xk+1

∆t

≤ const. (Allows ∆t = λ∆x with large λ.)

Less immediate proof ("one-page proof")

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 34

SLDG scheme for non constant b(x): Find un+1 ∈ Vk,

∀ϕ ∈ Vk,

un+1(x)ϕ(x)dx =
Gauss

un(yx(−∆t))ϕ(x)dx where d

dt yx(t) = b(yx(t)), y(0) = x.

Theorem (B, Simaramata) (i) The scheme is STILL approx. implementable (Liu & Shu) (ii) order: O( ∆xk+1

∆t

) for k = 1, 2 a (iii) L2 stable for k ≥ 1, under a weak CFL.

a(here with ∆t ≤ λ∆x, λ constant)

same as Liu & Shu 2011 in the linear case.
⇒ weak CFL : ∆xk+1

∆t

≤ const. (Allows ∆t = λ∆x with large λ.)

Less immediate proof ("one-page proof")

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 35

SLDG scheme for non constant b(x): Find un+1 ∈ Vk,

∀ϕ ∈ Vk,

un+1(x)ϕ(x)dx =
Gauss

un(yx(−∆t))ϕ(x)dx where d

dt yx(t) = b(yx(t)), y(0) = x.

Theorem (B, Simaramata) (i) The scheme is STILL approx. implementable (Liu & Shu) (ii) order: O( ∆xk+1

∆t

) for k = 1, 2 a (iii) L2 stable for k ≥ 1, under a weak CFL.

a(here with ∆t ≤ λ∆x, λ constant)

same as Liu & Shu 2011 in the linear case.
⇒ weak CFL : ∆xk+1

∆t

≤ const. (Allows ∆t = λ∆x with large λ.)

Less immediate proof ("one-page proof")

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 36

SLDG scheme for non constant b(x): Find un+1 ∈ Vk,

∀ϕ ∈ Vk,

un+1(x)ϕ(x)dx =
Gauss

un(yx(−∆t))ϕ(x)dx where d

dt yx(t) = b(yx(t)), y(0) = x.

Theorem (B, Simaramata) (i) The scheme is STILL approx. implementable (Liu & Shu) (ii) order: O( ∆xk+1

∆t

) for k = 1, 2 a (iii) L2 stable for k ≥ 1, under a weak CFL.

a(here with ∆t ≤ λ∆x, λ constant)

same as Liu & Shu 2011 in the linear case.
⇒ weak CFL : ∆xk+1

∆t

≤ const. (Allows ∆t = λ∆x with large λ.)

Less immediate proof ("one-page proof")

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 37

SLDG scheme for non constant b(x): Find un+1 ∈ Vk,

∀ϕ ∈ Vk,

un+1(x)ϕ(x)dx =
Gauss

un(yx(−∆t))ϕ(x)dx where d

dt yx(t) = b(yx(t)), y(0) = x.

Theorem (B, Simaramata) (i) The scheme is STILL approx. implementable (Liu & Shu) (ii) order: O( ∆xk+1

∆t

) for k = 1, 2 a (iii) L2 stable for k ≥ 1, under a weak CFL.

a(here with ∆t ≤ λ∆x, λ constant)

same as Liu & Shu 2011 in the linear case.
⇒ weak CFL : ∆xk+1

∆t

≤ const. (Allows ∆t = λ∆x with large λ.)

Less immediate proof ("one-page proof")

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 38

SLDG scheme for non constant b(x): Find un+1 ∈ Vk,

∀ϕ ∈ Vk,

un+1(x)ϕ(x)dx =
Gauss

un(yx(−∆t))ϕ(x)dx where d

dt yx(t) = b(yx(t)), y(0) = x.

Theorem (B, Simaramata) (i) The scheme is STILL approx. implementable (Liu & Shu) (ii) order: O( ∆xk+1

∆t

) for k = 1, 2 a (iii) L2 stable for k ≥ 1, under a weak CFL.

a(here with ∆t ≤ λ∆x, λ constant)

same as Liu & Shu 2011 in the linear case.
⇒ weak CFL : ∆xk+1

∆t

≤ const. (Allows ∆t = λ∆x with large λ.)

Less immediate proof ("one-page proof")

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 39

ONE PAGE STABILITY PROOF (non constant b(x))

Scheme definition is : find un+1 ∈ Vk s.t., ∀ϕ ∈ Vk:
un+1(x)ϕ(x)dx

∼

=

un(yx(−∆t))ϕ(x)dx

+O(∆xk+1) ?

DEF

≡

i
q=0,...,pi

k

α=0

wi

q,αun(y(xi q,α))ϕ(xi q,α)

ǫi :=
Ji un(yx(−∆t))ϕ(x) − (gauss)
≤ [un(y)ϕ](2k+2)L∞(Ji)∆x2k+3
Lemma 1 : | d

dx yx(−∆t)| ≤ C, and | dq dxq yx(−∆t)| ≤ C∆t for q ≥ 2.

Lemma 2 : [un(y)](q)L∞(Ji) ≤ C∆t k

p=1 (un)(p)L∞(Ii) ∀q ≥ k + 1

[un(y)ϕ](2k+2)L∞(Ji) ≤ C k

q=0 ϕ(q)L∞(Ii)[un(y)](2k+2−q)L∞(Ii)

≤ C k

q=0 ϕ(q)L∞(Ii)∆t k p=1 (un)(p)L∞(Ii)

≤ C∆t

1 ∆xk+1/2 ϕL2(Ii) 1 ∆xk+1/2 unL2(Ii), (un, ϕ ∈ Vk)

⇒ ǫi ≤ C∆t ∆x2k+3

∆x2k+1 unL2(Ii)ϕL2(Ii) ≡ ∆t ∆x2 ≡ ∆x3 for ∆t ≤ λ∆x.

⇒

i ǫi ≤ O(∆x3), so BOUND O(∆xk+1), works for k = 1, 2 !

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 40

ONE PAGE STABILITY PROOF (non constant b(x))

Scheme definition is : find un+1 ∈ Vk s.t., ∀ϕ ∈ Vk:
un+1(x)ϕ(x)dx

∼

=

un(yx(−∆t))ϕ(x)dx

+O(∆xk+1) ?

DEF

≡

i
q=0,...,pi

k

α=0

wi

q,αun(y(xi q,α))ϕ(xi q,α)

ǫi :=
Ji un(yx(−∆t))ϕ(x) − (gauss)
≤ [un(y)ϕ](2k+2)L∞(Ji)∆x2k+3
Lemma 1 : | d

dx yx(−∆t)| ≤ C, and | dq dxq yx(−∆t)| ≤ C∆t for q ≥ 2.

Lemma 2 : [un(y)](q)L∞(Ji) ≤ C∆t k

p=1 (un)(p)L∞(Ii) ∀q ≥ k + 1

[un(y)ϕ](2k+2)L∞(Ji) ≤ C k

q=0 ϕ(q)L∞(Ii)[un(y)](2k+2−q)L∞(Ii)

≤ C k

q=0 ϕ(q)L∞(Ii)∆t k p=1 (un)(p)L∞(Ii)

≤ C∆t

1 ∆xk+1/2 ϕL2(Ii) 1 ∆xk+1/2 unL2(Ii), (un, ϕ ∈ Vk)

⇒ ǫi ≤ C∆t ∆x2k+3

∆x2k+1 unL2(Ii)ϕL2(Ii) ≡ ∆t ∆x2 ≡ ∆x3 for ∆t ≤ λ∆x.

⇒

i ǫi ≤ O(∆x3), so BOUND O(∆xk+1), works for k = 1, 2 !

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 41

ONE PAGE STABILITY PROOF (non constant b(x))

Scheme definition is : find un+1 ∈ Vk s.t., ∀ϕ ∈ Vk:
un+1(x)ϕ(x)dx

∼

=

un(yx(−∆t))ϕ(x)dx

+O(∆xk+1) ?

DEF

≡

i
q=0,...,pi

k

α=0

wi

q,αun(y(xi q,α))ϕ(xi q,α)

ǫi :=
Ji un(yx(−∆t))ϕ(x) − (gauss)
≤ [un(y)ϕ](2k+2)L∞(Ji)∆x2k+3
Lemma 1 : | d

dx yx(−∆t)| ≤ C, and | dq dxq yx(−∆t)| ≤ C∆t for q ≥ 2.

Lemma 2 : [un(y)](q)L∞(Ji) ≤ C∆t k

p=1 (un)(p)L∞(Ii) ∀q ≥ k + 1

[un(y)ϕ](2k+2)L∞(Ji) ≤ C k

q=0 ϕ(q)L∞(Ii)[un(y)](2k+2−q)L∞(Ii)

≤ C k

q=0 ϕ(q)L∞(Ii)∆t k p=1 (un)(p)L∞(Ii)

≤ C∆t

1 ∆xk+1/2 ϕL2(Ii) 1 ∆xk+1/2 unL2(Ii), (un, ϕ ∈ Vk)

⇒ ǫi ≤ C∆t ∆x2k+3

∆x2k+1 unL2(Ii)ϕL2(Ii) ≡ ∆t ∆x2 ≡ ∆x3 for ∆t ≤ λ∆x.

⇒

i ǫi ≤ O(∆x3), so BOUND O(∆xk+1), works for k = 1, 2 !

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 42

ONE PAGE STABILITY PROOF (non constant b(x))

Scheme definition is : find un+1 ∈ Vk s.t., ∀ϕ ∈ Vk:
un+1(x)ϕ(x)dx

∼

=

un(yx(−∆t))ϕ(x)dx

+O(∆xk+1) ?

DEF

≡

i
q=0,...,pi

k

α=0

wi

q,αun(y(xi q,α))ϕ(xi q,α)

ǫi :=
Ji un(yx(−∆t))ϕ(x) − (gauss)
≤ [un(y)ϕ](2k+2)L∞(Ji)∆x2k+3
Lemma 1 : | d

dx yx(−∆t)| ≤ C, and | dq dxq yx(−∆t)| ≤ C∆t for q ≥ 2.

Lemma 2 : [un(y)](q)L∞(Ji) ≤ C∆t k

p=1 (un)(p)L∞(Ii) ∀q ≥ k + 1

[un(y)ϕ](2k+2)L∞(Ji) ≤ C k

q=0 ϕ(q)L∞(Ii)[un(y)](2k+2−q)L∞(Ii)

≤ C k

q=0 ϕ(q)L∞(Ii)∆t k p=1 (un)(p)L∞(Ii)

≤ C∆t

1 ∆xk+1/2 ϕL2(Ii) 1 ∆xk+1/2 unL2(Ii), (un, ϕ ∈ Vk)

⇒ ǫi ≤ C∆t ∆x2k+3

∆x2k+1 unL2(Ii)ϕL2(Ii) ≡ ∆t ∆x2 ≡ ∆x3 for ∆t ≤ λ∆x.

⇒

i ǫi ≤ O(∆x3), so BOUND O(∆xk+1), works for k = 1, 2 !

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 43

ONE PAGE STABILITY PROOF (non constant b(x))

Scheme definition is : find un+1 ∈ Vk s.t., ∀ϕ ∈ Vk:
un+1(x)ϕ(x)dx

∼

=

un(yx(−∆t))ϕ(x)dx

+O(∆xk+1) ?

DEF

≡

i
q=0,...,pi

k

α=0

wi

q,αun(y(xi q,α))ϕ(xi q,α)

ǫi :=
Ji un(yx(−∆t))ϕ(x) − (gauss)
≤ [un(y)ϕ](2k+2)L∞(Ji)∆x2k+3
Lemma 1 : | d

dx yx(−∆t)| ≤ C, and | dq dxq yx(−∆t)| ≤ C∆t for q ≥ 2.

Lemma 2 : [un(y)](q)L∞(Ji) ≤ C∆t k

p=1 (un)(p)L∞(Ii) ∀q ≥ k + 1

[un(y)ϕ](2k+2)L∞(Ji) ≤ C k

q=0 ϕ(q)L∞(Ii)[un(y)](2k+2−q)L∞(Ii)

≤ C k

q=0 ϕ(q)L∞(Ii)∆t k p=1 (un)(p)L∞(Ii)

≤ C∆t

1 ∆xk+1/2 ϕL2(Ii) 1 ∆xk+1/2 unL2(Ii), (un, ϕ ∈ Vk)

⇒ ǫi ≤ C∆t ∆x2k+3

∆x2k+1 unL2(Ii)ϕL2(Ii) ≡ ∆t ∆x2 ≡ ∆x3 for ∆t ≤ λ∆x.

⇒

i ǫi ≤ O(∆x3), so BOUND O(∆xk+1), works for k = 1, 2 !

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 44

ONE PAGE STABILITY PROOF (non constant b(x))

Scheme definition is : find un+1 ∈ Vk s.t., ∀ϕ ∈ Vk:
un+1(x)ϕ(x)dx

∼

=

un(yx(−∆t))ϕ(x)dx

+O(∆xk+1) ?

DEF

≡

i
q=0,...,pi

k

α=0

wi

q,αun(y(xi q,α))ϕ(xi q,α)

ǫi :=
Ji un(yx(−∆t))ϕ(x) − (gauss)
≤ [un(y)ϕ](2k+2)L∞(Ji)∆x2k+3
Lemma 1 : | d

dx yx(−∆t)| ≤ C, and | dq dxq yx(−∆t)| ≤ C∆t for q ≥ 2.

Lemma 2 : [un(y)](q)L∞(Ji) ≤ C∆t k

p=1 (un)(p)L∞(Ii) ∀q ≥ k + 1

[un(y)ϕ](2k+2)L∞(Ji) ≤ C k

q=0 ϕ(q)L∞(Ii)[un(y)](2k+2−q)L∞(Ii)

≤ C k

q=0 ϕ(q)L∞(Ii)∆t k p=1 (un)(p)L∞(Ii)

≤ C∆t

1 ∆xk+1/2 ϕL2(Ii) 1 ∆xk+1/2 unL2(Ii), (un, ϕ ∈ Vk)

⇒ ǫi ≤ C∆t ∆x2k+3

∆x2k+1 unL2(Ii)ϕL2(Ii) ≡ ∆t ∆x2 ≡ ∆x3 for ∆t ≤ λ∆x.

⇒

i ǫi ≤ O(∆x3), so BOUND O(∆xk+1), works for k = 1, 2 !

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 45

ONE PAGE STABILITY PROOF (non constant b(x))

Scheme definition is : find un+1 ∈ Vk s.t., ∀ϕ ∈ Vk:
un+1(x)ϕ(x)dx

∼

=

un(yx(−∆t))ϕ(x)dx

+O(∆xk+1) ?

DEF

≡

i
q=0,...,pi

k

α=0

wi

q,αun(y(xi q,α))ϕ(xi q,α)

ǫi :=
Ji un(yx(−∆t))ϕ(x) − (gauss)
≤ [un(y)ϕ](2k+2)L∞(Ji)∆x2k+3
Lemma 1 : | d

dx yx(−∆t)| ≤ C, and | dq dxq yx(−∆t)| ≤ C∆t for q ≥ 2.

Lemma 2 : [un(y)](q)L∞(Ji) ≤ C∆t k

p=1 (un)(p)L∞(Ii) ∀q ≥ k + 1

[un(y)ϕ](2k+2)L∞(Ji) ≤ C k

q=0 ϕ(q)L∞(Ii)[un(y)](2k+2−q)L∞(Ii)

≤ C k

q=0 ϕ(q)L∞(Ii)∆t k p=1 (un)(p)L∞(Ii)

≤ C∆t

1 ∆xk+1/2 ϕL2(Ii) 1 ∆xk+1/2 unL2(Ii), (un, ϕ ∈ Vk)

⇒ ǫi ≤ C∆t ∆x2k+3

∆x2k+1 unL2(Ii)ϕL2(Ii) ≡ ∆t ∆x2 ≡ ∆x3 for ∆t ≤ λ∆x.

⇒

i ǫi ≤ O(∆x3), so BOUND O(∆xk+1), works for k = 1, 2 !

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 46

ONE PAGE STABILITY PROOF (non constant b(x))

Scheme definition is : find un+1 ∈ Vk s.t., ∀ϕ ∈ Vk:
un+1(x)ϕ(x)dx

∼

=

un(yx(−∆t))ϕ(x)dx

+O(∆xk+1) ?

DEF

≡

i
q=0,...,pi

k

α=0

wi

q,αun(y(xi q,α))ϕ(xi q,α)

ǫi :=
Ji un(yx(−∆t))ϕ(x) − (gauss)
≤ [un(y)ϕ](2k+2)L∞(Ji)∆x2k+3
Lemma 1 : | d

dx yx(−∆t)| ≤ C, and | dq dxq yx(−∆t)| ≤ C∆t for q ≥ 2.

Lemma 2 : [un(y)](q)L∞(Ji) ≤ C∆t k

p=1 (un)(p)L∞(Ii) ∀q ≥ k + 1

[un(y)ϕ](2k+2)L∞(Ji) ≤ C k

q=0 ϕ(q)L∞(Ii)[un(y)](2k+2−q)L∞(Ii)

≤ C k

q=0 ϕ(q)L∞(Ii)∆t k p=1 (un)(p)L∞(Ii)

≤ C∆t

1 ∆xk+1/2 ϕL2(Ii) 1 ∆xk+1/2 unL2(Ii), (un, ϕ ∈ Vk)

⇒ ǫi ≤ C∆t ∆x2k+3

∆x2k+1 unL2(Ii)ϕL2(Ii) ≡ ∆t ∆x2 ≡ ∆x3 for ∆t ≤ λ∆x.

⇒

i ǫi ≤ O(∆x3), so BOUND O(∆xk+1), works for k = 1, 2 !

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 47

ONE PAGE STABILITY PROOF (non constant b(x))

Scheme definition is : find un+1 ∈ Vk s.t., ∀ϕ ∈ Vk:
un+1(x)ϕ(x)dx

∼

=

un(yx(−∆t))ϕ(x)dx

+O(∆xk+1) ?

DEF

≡

i
q=0,...,pi

k

α=0

wi

q,αun(y(xi q,α))ϕ(xi q,α)

ǫi :=
Ji un(yx(−∆t))ϕ(x) − (gauss)
≤ [un(y)ϕ](2k+2)L∞(Ji)∆x2k+3
Lemma 1 : | d

dx yx(−∆t)| ≤ C, and | dq dxq yx(−∆t)| ≤ C∆t for q ≥ 2.

Lemma 2 : [un(y)](q)L∞(Ji) ≤ C∆t k

p=1 (un)(p)L∞(Ii) ∀q ≥ k + 1

[un(y)ϕ](2k+2)L∞(Ji) ≤ C k

q=0 ϕ(q)L∞(Ii)[un(y)](2k+2−q)L∞(Ii)

≤ C k

q=0 ϕ(q)L∞(Ii)∆t k p=1 (un)(p)L∞(Ii)

≤ C∆t

1 ∆xk+1/2 ϕL2(Ii) 1 ∆xk+1/2 unL2(Ii), (un, ϕ ∈ Vk)

⇒ ǫi ≤ C∆t ∆x2k+3

∆x2k+1 unL2(Ii)ϕL2(Ii) ≡ ∆t ∆x2 ≡ ∆x3 for ∆t ≤ λ∆x.

⇒

i ǫi ≤ O(∆x3), so BOUND O(∆xk+1), works for k = 1, 2 !

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 48

ONE PAGE STABILITY PROOF (non constant b(x))

Scheme definition is : find un+1 ∈ Vk s.t., ∀ϕ ∈ Vk:
un+1(x)ϕ(x)dx

∼

=

un(yx(−∆t))ϕ(x)dx

+O(∆xk+1) ?

DEF

≡

i
q=0,...,pi

k

α=0

wi

q,αun(y(xi q,α))ϕ(xi q,α)

ǫi :=
Ji un(yx(−∆t))ϕ(x) − (gauss)
≤ [un(y)ϕ](2k+2)L∞(Ji)∆x2k+3
Lemma 1 : | d

dx yx(−∆t)| ≤ C, and | dq dxq yx(−∆t)| ≤ C∆t for q ≥ 2.

Lemma 2 : [un(y)](q)L∞(Ji) ≤ C∆t k

p=1 (un)(p)L∞(Ii) ∀q ≥ k + 1

[un(y)ϕ](2k+2)L∞(Ji) ≤ C k

q=0 ϕ(q)L∞(Ii)[un(y)](2k+2−q)L∞(Ii)

≤ C k

q=0 ϕ(q)L∞(Ii)∆t k p=1 (un)(p)L∞(Ii)

≤ C∆t

1 ∆xk+1/2 ϕL2(Ii) 1 ∆xk+1/2 unL2(Ii), (un, ϕ ∈ Vk)

⇒ ǫi ≤ C∆t ∆x2k+3

∆x2k+1 unL2(Ii)ϕL2(Ii) ≡ ∆t ∆x2 ≡ ∆x3 for ∆t ≤ λ∆x.

⇒

i ǫi ≤ O(∆x3), so BOUND O(∆xk+1), works for k = 1, 2 !

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 49

ONE PAGE STABILITY PROOF (non constant b(x))

Scheme definition is : find un+1 ∈ Vk s.t., ∀ϕ ∈ Vk:
un+1(x)ϕ(x)dx

∼

=

un(yx(−∆t))ϕ(x)dx

+O(∆xk+1) ?

DEF

≡

i
q=0,...,pi

k

α=0

wi

q,αun(y(xi q,α))ϕ(xi q,α)

ǫi :=
Ji un(yx(−∆t))ϕ(x) − (gauss)
≤ [un(y)ϕ](2k+2)L∞(Ji)∆x2k+3
Lemma 1 : | d

dx yx(−∆t)| ≤ C, and | dq dxq yx(−∆t)| ≤ C∆t for q ≥ 2.

Lemma 2 : [un(y)](q)L∞(Ji) ≤ C∆t k

p=1 (un)(p)L∞(Ii) ∀q ≥ k + 1

[un(y)ϕ](2k+2)L∞(Ji) ≤ C k

q=0 ϕ(q)L∞(Ii)[un(y)](2k+2−q)L∞(Ii)

≤ C k

q=0 ϕ(q)L∞(Ii)∆t k p=1 (un)(p)L∞(Ii)

≤ C∆t

1 ∆xk+1/2 ϕL2(Ii) 1 ∆xk+1/2 unL2(Ii), (un, ϕ ∈ Vk)

⇒ ǫi ≤ C∆t ∆x2k+3

∆x2k+1 unL2(Ii)ϕL2(Ii) ≡ ∆t ∆x2 ≡ ∆x3 for ∆t ≤ λ∆x.

⇒

i ǫi ≤ O(∆x3), so BOUND O(∆xk+1), works for k = 1, 2 !

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 50

2) SLDG for diffusion equation with constant σ ∈ R:

vt − σ2 2 vxx = 0, x ∈ Ω, t ∈ (0, T), (1)

A first scheme, in semi-discrete form (denoting h ≡ ∆t)

un+1(x) = 1 2

un(x − σ

√ h) + un(x + σ √ h)

≡ S0

hun (x).

(2)

Remark: this becomes a tree method if σ

√ h = ∆x

Remark: Taking vn(x) := v(tn, x) where v is solution of (1)

the following consistency error estimate holds: vn+1 − S0

hvn

h L2 = O(hvn

4xL∞) = O(h)

SLDG-RK1 scheme := weak formulation of (1):

Find un+1 in Vk such that, for all ϕ ∈ Vk:

un+1(x)ϕ(x) =

1 2

un(x − σ

√ h) + un(x + σ √ h)

ϕ(x)
Abstract Notation: un+1 = Sh(un)

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 51

2) SLDG for diffusion equation with constant σ ∈ R:

vt − σ2 2 vxx = 0, x ∈ Ω, t ∈ (0, T), (1)

A first scheme, in semi-discrete form (denoting h ≡ ∆t)

un+1(x) = 1 2

un(x − σ

√ h) + un(x + σ √ h)

≡ S0

hun (x).

(2)

Remark: this becomes a tree method if σ

√ h = ∆x

Remark: Taking vn(x) := v(tn, x) where v is solution of (1)

the following consistency error estimate holds: vn+1 − S0

hvn

h L2 = O(hvn

4xL∞) = O(h)

SLDG-RK1 scheme := weak formulation of (1):

Find un+1 in Vk such that, for all ϕ ∈ Vk:

un+1(x)ϕ(x) =

1 2

un(x − σ

√ h) + un(x + σ √ h)

ϕ(x)
Abstract Notation: un+1 = Sh(un)

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 52

2) SLDG for diffusion equation with constant σ ∈ R:

vt − σ2 2 vxx = 0, x ∈ Ω, t ∈ (0, T), (1)

A first scheme, in semi-discrete form (denoting h ≡ ∆t)

un+1(x) = 1 2

un(x − σ

√ h) + un(x + σ √ h)

≡ S0

hun (x).

(2)

Remark: this becomes a tree method if σ

√ h = ∆x

Remark: Taking vn(x) := v(tn, x) where v is solution of (1)

the following consistency error estimate holds: vn+1 − S0

hvn

h L2 = O(hvn

4xL∞) = O(h)

SLDG-RK1 scheme := weak formulation of (1):

Find un+1 in Vk such that, for all ϕ ∈ Vk:

un+1(x)ϕ(x) =

1 2

un(x − σ

√ h) + un(x + σ √ h)

ϕ(x)
Abstract Notation: un+1 = Sh(un)

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 53

2) SLDG for diffusion equation with constant σ ∈ R:

vt − σ2 2 vxx = 0, x ∈ Ω, t ∈ (0, T), (1)

A first scheme, in semi-discrete form (denoting h ≡ ∆t)

un+1(x) = 1 2

un(x − σ

√ h) + un(x + σ √ h)

≡ S0

hun (x).

(2)

Remark: this becomes a tree method if σ

√ h = ∆x

Remark: Taking vn(x) := v(tn, x) where v is solution of (1)

the following consistency error estimate holds: vn+1 − S0

hvn

h L2 = O(hvn

4xL∞) = O(h)

SLDG-RK1 scheme := weak formulation of (1):

Find un+1 in Vk such that, for all ϕ ∈ Vk:

un+1(x)ϕ(x) =

1 2

un(x − σ

√ h) + un(x + σ √ h)

ϕ(x)
Abstract Notation: un+1 = Sh(un)

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 54

2) SLDG for diffusion equation with constant σ ∈ R:

vt − σ2 2 vxx = 0, x ∈ Ω, t ∈ (0, T), (1)

A first scheme, in semi-discrete form (denoting h ≡ ∆t)

un+1(x) = 1 2

un(x − σ

√ h) + un(x + σ √ h)

≡ S0

hun (x).

(2)

Remark: this becomes a tree method if σ

√ h = ∆x

Remark: Taking vn(x) := v(tn, x) where v is solution of (1)

the following consistency error estimate holds: vn+1 − S0

hvn

h L2 = O(hvn

4xL∞) = O(h)

SLDG-RK1 scheme := weak formulation of (1):

Find un+1 in Vk such that, for all ϕ ∈ Vk:

un+1(x)ϕ(x) =

1 2

un(x − σ

√ h) + un(x + σ √ h)

ϕ(x)
Abstract Notation: un+1 = Sh(un)

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 55

First results: (i) Implementable scheme (ii) Consistency error : O(∆t) + O( ∆xk+1

∆t

) (iii) L2 stable ⇒ We want to improve order in time

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 56

Let h := ∆t. Using Taylor expansions,

S0

hu

= u + hσ2 2 uxx + h2 σ4 24u4x + O(h3), (3) S0

hS0 hu

= u + hσ2uxx + h2 σ4 3 u4x + O(h3), (4)

On the other hand, if vn = v(tn, x) (where vt = σ2

2 vxx):

vn+1 = vn + hvt + h2 2 vtt + O(h3) (5) = vn + hσ2 2 vn

xx + h2 σ4

8 vn

4x + O(h3)

(6)

⇒ Now we look for coefficients a, b, c such that

vn+1 = avn + bS0

hvn + cS0 hS0 hvn + O(h3) system

a

+ b + c = 1

b 2

+ c =

1 2 b 24

+

c 3

=

1 8

Solution: a = b = c = 1

3.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 57

Let h := ∆t. Using Taylor expansions,

S0

hu

= u + hσ2 2 uxx + h2 σ4 24u4x + O(h3), (3) S0

hS0 hu

= u + hσ2uxx + h2 σ4 3 u4x + O(h3), (4)

On the other hand, if vn = v(tn, x) (where vt = σ2

2 vxx):

vn+1 = vn + hvt + h2 2 vtt + O(h3) (5) = vn + hσ2 2 vn

xx + h2 σ4

8 vn

4x + O(h3)

(6)

⇒ Now we look for coefficients a, b, c such that

vn+1 = avn + bS0

hvn + cS0 hS0 hvn + O(h3) system

a

+ b + c = 1

b 2

+ c =

1 2 b 24

+

c 3

=

1 8

Solution: a = b = c = 1

3.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 58

Let h := ∆t. Using Taylor expansions,

S0

hu

= u + hσ2 2 uxx + h2 σ4 24u4x + O(h3), (3) S0

hS0 hu

= u + hσ2uxx + h2 σ4 3 u4x + O(h3), (4)

On the other hand, if vn = v(tn, x) (where vt = σ2

2 vxx):

vn+1 = vn + hvt + h2 2 vtt + O(h3) (5) = vn + hσ2 2 vn

xx + h2 σ4

8 vn

4x + O(h3)

(6)

⇒ Now we look for coefficients a, b, c such that

vn+1 = avn + bS0

hvn + cS0 hS0 hvn + O(h3) system

a

+ b + c = 1

b 2

+ c =

1 2 b 24

+

c 3

=

1 8

Solution: a = b = c = 1

3.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 59

Let h := ∆t. Using Taylor expansions,

S0

hu

= u + hσ2 2 uxx + h2 σ4 24u4x + O(h3), (3) S0

hS0 hu

= u + hσ2uxx + h2 σ4 3 u4x + O(h3), (4)

On the other hand, if vn = v(tn, x) (where vt = σ2

2 vxx):

vn+1 = vn + hvt + h2 2 vtt + O(h3) (5) = vn + hσ2 2 vn

xx + h2 σ4

8 vn

4x + O(h3)

(6)

⇒ Now we look for coefficients a, b, c such that

vn+1 = avn + bS0

hvn + cS0 hS0 hvn + O(h3) system

a

+ b + c = 1

b 2

+ c =

1 2 b 24

+

c 3

=

1 8

Solution: a = b = c = 1

3.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 60

Let h := ∆t. Using Taylor expansions,

S0

hu

= u + hσ2 2 uxx + h2 σ4 24u4x + O(h3), (3) S0

hS0 hu

= u + hσ2uxx + h2 σ4 3 u4x + O(h3), (4)

On the other hand, if vn = v(tn, x) (where vt = σ2

2 vxx):

vn+1 = vn + hvt + h2 2 vtt + O(h3) (5) = vn + hσ2 2 vn

xx + h2 σ4

8 vn

4x + O(h3)

(6)

⇒ Now we look for coefficients a, b, c such that

vn+1 = avn + bS0

hvn + cS0 hS0 hvn + O(h3) system

a

+ b + c = 1

b 2

+ c =

1 2 b 24

+

c 3

=

1 8

Solution: a = b = c = 1

3.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 61

Therefore a second order scheme is now given by

SLDG-RK2 scheme un+1 = 1 3(un + S∆tun + S∆tS∆tun). (7)

In a similar way, we can identify up to the 3rd order term :

SLDG-RK3 scheme un+1 = 13 45un + 21 45S∆tvn + 9 45S∆tS∆tun + 2 45S∆tS∆tS∆tun. Theorem (B., Simarmata, 2012’) Consider SLDG-RKp, p = 1, 2, 3: (i) Consistency order: O(∆tp) + O( ∆xk+1

∆t

) (ii) L2 stable AND convex combination of

(S∆t)(q)

q=0,..,p

Prefered choice : ∆t = ∆x and k = p ∈ {1, 2, 3}.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 62

Therefore a second order scheme is now given by

SLDG-RK2 scheme un+1 = 1 3(un + S∆tun + S∆tS∆tun). (7)

In a similar way, we can identify up to the 3rd order term :

SLDG-RK3 scheme un+1 = 13 45un + 21 45S∆tvn + 9 45S∆tS∆tun + 2 45S∆tS∆tS∆tun. Theorem (B., Simarmata, 2012’) Consider SLDG-RKp, p = 1, 2, 3: (i) Consistency order: O(∆tp) + O( ∆xk+1

∆t

) (ii) L2 stable AND convex combination of

(S∆t)(q)

q=0,..,p

Prefered choice : ∆t = ∆x and k = p ∈ {1, 2, 3}.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 63

Therefore a second order scheme is now given by

SLDG-RK2 scheme un+1 = 1 3(un + S∆tun + S∆tS∆tun). (7)

In a similar way, we can identify up to the 3rd order term :

SLDG-RK3 scheme un+1 = 13 45un + 21 45S∆tvn + 9 45S∆tS∆tun + 2 45S∆tS∆tS∆tun. Theorem (B., Simarmata, 2012’) Consider SLDG-RKp, p = 1, 2, 3: (i) Consistency order: O(∆tp) + O( ∆xk+1

∆t

) (ii) L2 stable AND convex combination of

(S∆t)(q)

q=0,..,p

Prefered choice : ∆t = ∆x and k = p ∈ {1, 2, 3}.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 64

Therefore a second order scheme is now given by

SLDG-RK2 scheme un+1 = 1 3(un + S∆tun + S∆tS∆tun). (7)

In a similar way, we can identify up to the 3rd order term :

SLDG-RK3 scheme un+1 = 13 45un + 21 45S∆tvn + 9 45S∆tS∆tun + 2 45S∆tS∆tS∆tun. Theorem (B., Simarmata, 2012’) Consider SLDG-RKp, p = 1, 2, 3: (i) Consistency order: O(∆tp) + O( ∆xk+1

∆t

) (ii) L2 stable AND convex combination of

(S∆t)(q)

q=0,..,p

Prefered choice : ∆t = ∆x and k = p ∈ {1, 2, 3}.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 65

Modified SLDG-RK2∗ scheme for non constant σ(x):

Define

S−

h u(x) := Π

1 2(u(x + h) − u(x − h))

: L2-stable
"modified" SLDG-RK2 scheme

un+1 = 1 3(un + S∆tun + S∆tS∆tun) +∆t σ2(σ2)′ 12

S−

∆t1/3S− ∆t1/3S− ∆t1/3un − 1

2S−

∆t1/3S− ∆t1/3S− ∆t1/3S− ∆t1/3S− ∆t1/3un

+∆t σ2(σ2)′′ 24 S−

∆t1/2S− ∆t1/2un.

(8) Theorem L2 stable, second order convergent. Idea: Stability constant is (1 + C∆t)N ≤ eCN∆t ≤ eCT.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 66

Modified SLDG-RK2∗ scheme for non constant σ(x):

Define

S−

h u(x) := Π

1 2(u(x + h) − u(x − h))

: L2-stable
"modified" SLDG-RK2 scheme

un+1 = 1 3(un + S∆tun + S∆tS∆tun) +∆t σ2(σ2)′ 12

S−

∆t1/3S− ∆t1/3S− ∆t1/3un − 1

2S−

∆t1/3S− ∆t1/3S− ∆t1/3S− ∆t1/3S− ∆t1/3un

+∆t σ2(σ2)′′ 24 S−

∆t1/2S− ∆t1/2un.

(8) Theorem L2 stable, second order convergent. Idea: Stability constant is (1 + C∆t)N ≤ eCN∆t ≤ eCT.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 67

Modified SLDG-RK2∗ scheme for non constant σ(x):

Define

S−

h u(x) := Π

1 2(u(x + h) − u(x − h))

: L2-stable
"modified" SLDG-RK2 scheme

un+1 = 1 3(un + S∆tun + S∆tS∆tun) +∆t σ2(σ2)′ 12

S−

∆t1/3S− ∆t1/3S− ∆t1/3un − 1

2S−

∆t1/3S− ∆t1/3S− ∆t1/3S− ∆t1/3S− ∆t1/3un

+∆t σ2(σ2)′′ 24 S−

∆t1/2S− ∆t1/2un.

(8) Theorem L2 stable, second order convergent. Idea: Stability constant is (1 + C∆t)N ≤ eCN∆t ≤ eCT.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 68

Modified SLDG-RK2∗ scheme for non constant σ(x):

Define

S−

h u(x) := Π

1 2(u(x + h) − u(x − h))

: L2-stable
"modified" SLDG-RK2 scheme

un+1 = 1 3(un + S∆tun + S∆tS∆tun) +∆t σ2(σ2)′ 12

S−

∆t1/3S− ∆t1/3S− ∆t1/3un − 1

2S−

∆t1/3S− ∆t1/3S− ∆t1/3S− ∆t1/3S− ∆t1/3un

+∆t σ2(σ2)′′ 24 S−

∆t1/2S− ∆t1/2un.

(8) Theorem L2 stable, second order convergent. Idea: Stability constant is (1 + C∆t)N ≤ eCN∆t ≤ eCT.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 69

Modified SLDG-RK2∗ scheme for non constant σ(x):

Define

S−

h u(x) := Π

1 2(u(x + h) − u(x − h))

: L2-stable
"modified" SLDG-RK2 scheme

un+1 = 1 3(un + S∆tun + S∆tS∆tun) +∆t σ2(σ2)′ 12

S−

∆t1/3S− ∆t1/3S− ∆t1/3un − 1

2S−

∆t1/3S− ∆t1/3S− ∆t1/3S− ∆t1/3S− ∆t1/3un

+∆t σ2(σ2)′′ 24 S−

∆t1/2S− ∆t1/2un.

(8) Theorem L2 stable, second order convergent. Idea: Stability constant is (1 + C∆t)N ≤ eCN∆t ≤ eCT.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 70

Higher order formulas for non constant σ(x):

Kloeden & Platen 1999 (Springer, Vol 21); Debrabant
Case σ = const, second order :

un+1(x) = 1 6un(x − σ √ 3h) + 2 3un(x) + 1 6un(x + σ √ 3h).

Case σ(x) = const, second order :

un+1(x) = 1 6

∆

W=± √ 3h

un Xx(∆ W)

+ 2

3un(x) where Xx(∆ W) := x + 1 4

σ(x + σ(x)

√ h) + σ(x − σ(x) √ h) + 2σ(x)

∆

W +1 4

σ(x + σ(x)

√ h) − σ(x − σ(x) √ h) ∆ W 2 − h √ h

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 71

Higher order formulas for non constant σ(x):

Kloeden & Platen 1999 (Springer, Vol 21); Debrabant
Case σ = const, second order :

un+1(x) = 1 6un(x − σ √ 3h) + 2 3un(x) + 1 6un(x + σ √ 3h).

Case σ(x) = const, second order :

un+1(x) = 1 6

∆

W=± √ 3h

un Xx(∆ W)

+ 2

3un(x) where Xx(∆ W) := x + 1 4

σ(x + σ(x)

√ h) + σ(x − σ(x) √ h) + 2σ(x)

∆

W +1 4

σ(x + σ(x)

√ h) − σ(x − σ(x) √ h) ∆ W 2 − h √ h

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 72

Higher order formulas for non constant σ(x):

Kloeden & Platen 1999 (Springer, Vol 21); Debrabant
Case σ = const, second order :

un+1(x) = 1 6un(x − σ √ 3h) + 2 3un(x) + 1 6un(x + σ √ 3h).

Case σ(x) = const, second order :

un+1(x) = 1 6

∆

W=± √ 3h

un Xx(∆ W)

+ 2

3un(x) where Xx(∆ W) := x + 1 4

σ(x + σ(x)

√ h) + σ(x − σ(x) √ h) + 2σ(x)

∆

W +1 4

σ(x + σ(x)

√ h) − σ(x − σ(x) √ h) ∆ W 2 − h √ h

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 73

Higher order formulas for non constant σ(x):

Kloeden & Platen 1999 (Springer, Vol 21); Debrabant
Case σ = const, second order :

un+1(x) = 1 6un(x − σ √ 3h) + 2 3un(x) + 1 6un(x + σ √ 3h).

Case σ(x) = const, second order :

un+1(x) = 1 6

∆

W=± √ 3h

un Xx(∆ W)

+ 2

3un(x) where Xx(∆ W) := x + 1 4

σ(x + σ(x)

√ h) + σ(x − σ(x) √ h) + 2σ(x)

∆

W +1 4

σ(x + σ(x)

√ h) − σ(x − σ(x) √ h) ∆ W 2 − h √ h

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 74

3) d-dimensional equations:

For a given σ ∈ Rd×d, let us consider ut − 1 2Tr(σσTD2u) = 0, x ∈ Ω, t ∈ (0, T), (9a) u(0, x) = u0(x), x ∈ Ω (9b) Rem: (Debrabant and Jakobsen 2012) σ = [Σ1, . . . , Σd], Σk ∈ Rd ⇒ σσT =

d

k=1

ΣkΣT

k

Thus (9a) is equivalent to ut − 1 2

k=1,...,d

Tr(ΣkΣT

k D2u) = 0.

(10)

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 75

For the one-directional problem ut = 1 2Tr(ΣkΣk(x)TD2u) (11) we consider the cheme un+1(x) = Π1 2

un(x−Σk

√ ∆t)+un(x+Σk √ ∆t)

=: (SΣk

∆t un)(x),

and for the general problem, we can consider un+1 = 1 d

d

k=1

SΣk

d∆tun

≡: Sun OR Trotter’s splitting: un+1 = SΣd

∆t · · · SΣ2 ∆t SΣ1 ∆t un

≡: Sun

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 76

III. Numerical Examples

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 77

Example 1 : 1D diffusion

vt − 1 2σ2vxx + bvx = 0, ∀x ∈ (0, 1), ∀t ∈ (0, T) (12) v(0, x) = cos(2πx) + 1 2 cos(4πx) (13) together with periodic boundary conditions on (0, 1), σ = 0.1, T = 1 and b = 0 or b = 0.3. L2 error SLDG-RK1 SLDG-RK2 SLDG-RK3 N error

rder

error

rder

error

rder

10 3.52E-03

3.27E-05
1.37E-07
20

1.73E-03 1.02 8.05E-06 2.02 1.69E-08 3.01 40 8.61E-04 1.01 1.99E-06 2.01 2.10E-09 3.00 80 4.29E-04 1.01 4.96E-07 2.00 2.62E-10 3.00 160 2.14E-04 1.00 1.23E-07 2.00 3.27E-11 3.00

Table: [with neglictable spatial error : k = 4 and M = 1000.]

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 78

L2 error SLDG-RK1 SLDG-RK2 SLDG-RK3 M N error

rder

error

rder

error

rder

20 10 4.10E-03

4.88E-05
8.27E-07
40

20 1.75E-03 1.23 2.36E-05 1.05 5.78E-07 0.52 80 40 8.68E-04 1.01 2.23E-06 3.40 3.82E-09 7.24 160 80 4.30E-04 1.01 5.58E-07 2.00 5.10E-10 2.91 320 160 2.14E-04 1.00 1.38E-07 2.01 4.01E-11 3.67 640 320 1.07E-04 1.00 3.44E-08 2.01 4.74E-12 3.08 Table: with ∆t ≡ ∆x.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 79

k = 3, RK3, with ∆t ≫ ∆x M N L1-Error L2-Error L∞-Error 10 10 5.129E-05 1.443E-05 2.166E-05 20 20 2.465E-06 9.191E-07 1.088E-06 40 30 9.307E-08 5.002E-08 5.588E-08 80 40 6.591E-09 3.360E-09 3.823E-09 160 50 1.923E-09 1.079E-09 1.209E-09 Table: "SLDG-RK3" + P3 and large time steps ∆t ≫ ∆x.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 80

Example 2 (1D diffusion with nonconstant σ(x)). vt − 1 2σ2(x)vxx = f(t, x); σ(x) = sin(2πx) (14) v(0, x) = 0 x ∈ (0, 1), (15) with periodic boundary conditions L2 error SLDG-RK1 SLDG-RK2∗ M N error

rder

error

rder

10 10 8.60E-02

4.13E-02
20

20 3.52E-02 1.29 7.30E-03 2.50 40 40 1.59E-02 1.15 1.39E-03 2.39 80 80 7.54E-03 1.08 3.03E-04 2.20 160 160 3.67E-03 1.04 7.17E-05 2.08 320 320 1.81E-03 1.02 1.80E-05 2.00 Table: T = 0.2, ∆t ≡ ∆x.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 81

Example 3 : 2D advection with nonconstant coefficients. We consider the following rotation example: ut + 2π(−x2, x1) · ∇u = 0, x = (x1, x2) ∈ Ω, t ∈ (0, T), u(0, x) = 1 − e−20((x1−1)2+x2

2 −r 2 0 ),

with T = 0.9 and r0 = 0.25.

L2 error Trotter Strang 3rd or. split. 4th or. split. M = N error

rder

error

rder

error

rder

error

rder

10 6.89E-01

2.91E-01
1.94E+00
2.26E-02
20

3.90E-01 0.82 6.62E-02 2.13 1.81E-01 3.42 8.10E-04 4.80 40 1.92E-01 1.02 1.60E-02 2.05 1.99E-02 3.18 3.46E-05 4.55 80 9.49E-02 1.02 3.99E-03 2.01 2.45E-03 3.02 1.80E-06 4.27 160 4.71E-02 1.01 9.96E-04 2.00 3.06E-04 3.00 1.07E-07 4.07

Table: Rotation example, T = 0.9, various splittings

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 82

Example 3 : 2D advection with nonconstant coefficients. We consider the following rotation example: ut + 2π(−x2, x1) · ∇u = 0, x = (x1, x2) ∈ Ω, t ∈ (0, T), u(0, x) = 1 − e−20((x1−1)2+x2

2 −r 2 0 ),

with T = 0.9 and r0 = 0.25.

L2 error Trotter Strang 3rd or. split. 4th or. split. M = N error

rder

error

rder

error

rder

error

rder

10 6.89E-01

2.91E-01
1.94E+00
2.26E-02
20

3.90E-01 0.82 6.62E-02 2.13 1.81E-01 3.42 8.10E-04 4.80 40 1.92E-01 1.02 1.60E-02 2.05 1.99E-02 3.18 3.46E-05 4.55 80 9.49E-02 1.02 3.99E-03 2.01 2.45E-03 3.02 1.80E-06 4.27 160 4.71E-02 1.01 9.96E-04 2.00 3.06E-04 3.00 1.07E-07 4.07

Table: Rotation example, T = 0.9, various splittings

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 83

Example 4 : 2D diffusion

Set Ω := (0, 1)2 with periodic boundary conditions, and:

ut − 1 2(5uxx − 4uxy + uyy) = 0, x ∈ Ω, t ∈ (0, 1), u(0, x) = u0(x), x ∈ Ω

Non obvious initial data1
In order to define the numerical scheme, we use the fact that

A :=

5

−2 −2 1

=
k=1,2

ΣkΣT

k , with Σ1 :=

1

, Σ2 :=
2

−1

.

1u0(x) = u01(x + 2y) + u02(−y), with

u0i(x) :=

q=1,2 ci q cos(2πqx)

(ci

q = 1 i+q ) Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 84

L2 error SLDG-RK1 SLDG-RK2 SLDG-RK3 M N error

rder

error

rder

error

rder

10 10 6.66E-03

1.86E-04
2.20E-06
20

20 3.26E-03 1.02 4.52E-05 2.04 3.10E-07 2.83 40 40 1.61E-03 1.01 1.08E-05 2.06 3.20E-08 3.27 80 80 8.04E-04 1.00 2.69E-06 2.01 4.34E-09 2.88 160 160 4.01E-04 1.00 6.66E-07 2.01 4.90E-10 3.14 Table: Example 2 (2D diffusion equation), error table with ∆t ∼ ∆x.

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 85

CONCLUSION: (1)

New stability proof for SLDG with non constant b(x)
New SLDG schemes for diffusion equations, with some nice

"monotony properties".

Olivier Bokanowski SLDG for 2nd order PDEs

SLIDE 86

CONCLUSION: (2)

Generalizations: monotony / order OK up to p = 5 !

(Bokanowski/Bonnans, in progress)

Applications: american option, HJB for stochastic optimal

control, nonlinear PDE ....: for completely monotone scheme, we can replace Π by P1 interpolation : gives a building block un+1 = ˜ S(un) that is monotone and O(∆tp) + O( ∆x2

∆t ) consistent.

Potential adaptivity (polynomial degree) / parallelization
Challenging problem: develop efficient PDE solvers for HJ

for stochastic control problems in high dimension;

Olivier Bokanowski SLDG for 2nd order PDEs