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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction A reconstruction-enhanced discontinuous Galerkin method for hyperbolic problems V aclav Ku cera Faculty of Mathematics and Physics Charles

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction

A reconstruction-enhanced discontinuous Galerkin method for hyperbolic problems

V´ aclav Kuˇ cera

Faculty of Mathematics and Physics Charles University Prague

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction

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Finite volume method with reconstruction Continuous Problem Space semidiscretization

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Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

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Finite volume method with reconstruction Continuous Problem Space semidiscretization

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Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

Let Ω ⊂ Rd be a bounded domain with a Lipschitz boundary ∂Ω. Continuous Problem Find u : QT = Ω×(0,T) → R such that ∂u ∂t +divf(u) = 0 in QT, u(x,0) = u0(x), x ∈ Ω, where f = (f1,··· ,fd) and fs, s = 1,...,d are Lipschitz- continuous fluxes in the direction xs, s = 1,...,d.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

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Finite volume method with reconstruction Continuous Problem Space semidiscretization

2

Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

Let Th be a partition of the closure Ω into a finite number of closed triangles K ∈ Th. By Fh we denote the set of all edges. For each Γ ∈ Fh we define a unit normal vector nΓ. For each face Γ ∈ F I

h there exist two neighbours

K (L)

Γ , K (R) Γ

∈ Th. Over Th we define the broken Sobolev space Hk(Ω,Th) = {v;v|K ∈ Hk(K) ∀K ∈ Th} and for v ∈ H1(Ω,Th) and Γ ∈ F I

h we set

v|(L)

Γ

= trace of v|K (L)

Γ

  • n Γ,

v|(R)

Γ

= trace of v|K (R)

Γ

  • n Γ,

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

Let Th be a partition of the closure Ω into a finite number of closed triangles K ∈ Th. By Fh we denote the set of all edges. For each Γ ∈ Fh we define a unit normal vector nΓ. For each face Γ ∈ F I

h there exist two neighbours

K (L)

Γ , K (R) Γ

∈ Th. Over Th we define the broken Sobolev space Hk(Ω,Th) = {v;v|K ∈ Hk(K) ∀K ∈ Th} and for v ∈ H1(Ω,Th) and Γ ∈ F I

h we set

v|(L)

Γ

= trace of v|K (L)

Γ

  • n Γ,

v|(R)

Γ

= trace of v|K (R)

Γ

  • n Γ,

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

Let Th be a partition of the closure Ω into a finite number of closed triangles K ∈ Th. By Fh we denote the set of all edges. For each Γ ∈ Fh we define a unit normal vector nΓ. For each face Γ ∈ F I

h there exist two neighbours

K (L)

Γ , K (R) Γ

∈ Th. Over Th we define the broken Sobolev space Hk(Ω,Th) = {v;v|K ∈ Hk(K) ∀K ∈ Th} and for v ∈ H1(Ω,Th) and Γ ∈ F I

h we set

v|(L)

Γ

= trace of v|K (L)

Γ

  • n Γ,

v|(R)

Γ

= trace of v|K (R)

Γ

  • n Γ,

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

Let Th be a partition of the closure Ω into a finite number of closed triangles K ∈ Th. By Fh we denote the set of all edges. For each Γ ∈ Fh we define a unit normal vector nΓ. For each face Γ ∈ F I

h there exist two neighbours

K (L)

Γ , K (R) Γ

∈ Th. Over Th we define the broken Sobolev space Hk(Ω,Th) = {v;v|K ∈ Hk(K) ∀K ∈ Th} and for v ∈ H1(Ω,Th) and Γ ∈ F I

h we set

v|(L)

Γ

= trace of v|K (L)

Γ

  • n Γ,

v|(R)

Γ

= trace of v|K (R)

Γ

  • n Γ,

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

Definition We define the space of discontinuous piecewise polynomial functions Sn

h = {v;v|K ∈ Pn(K) ∀K ∈ Th},

where Pn(K) is the set of all polynomials on K of degree ≤ n. S0

h - finite volume space,

Sn

h - discontinuous Galerkin space,

SN

h , N > n - Higher order DG reconstructions.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

Definition We define the space of discontinuous piecewise polynomial functions Sn

h = {v;v|K ∈ Pn(K) ∀K ∈ Th},

where Pn(K) is the set of all polynomials on K of degree ≤ n. S0

h - finite volume space,

Sn

h - discontinuous Galerkin space,

SN

h , N > n - Higher order DG reconstructions.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

Definition We define the space of discontinuous piecewise polynomial functions Sn

h = {v;v|K ∈ Pn(K) ∀K ∈ Th},

where Pn(K) is the set of all polynomials on K of degree ≤ n. S0

h - finite volume space,

Sn

h - discontinuous Galerkin space,

SN

h , N > n - Higher order DG reconstructions.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

Definition We define the space of discontinuous piecewise polynomial functions Sn

h = {v;v|K ∈ Pn(K) ∀K ∈ Th},

where Pn(K) is the set of all polynomials on K of degree ≤ n. S0

h - finite volume space,

Sn

h - discontinuous Galerkin space,

SN

h , N > n - Higher order DG reconstructions.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

We integrate over K ∈ Th and apply Green’s theorem d dt

  • K u(t)dx +
  • ∂K f(u)·ndS = 0.

We define ¯ uK(t) := 1 |K|

  • K u(t)dx

and obtain d dt ¯ uK(t)+ 1 |K|

  • ∂K f(u)·ndS = 0.

We assume, that there exists a piecewise polynomial function UN

h (t) ∈ SN h such that

UN

h (x,t) = u(x,t)+O(hN+1),

∀x ∈ Ω, ∀t ∈ (0,T).

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

We integrate over K ∈ Th and apply Green’s theorem d dt

  • K u(t)dx +
  • ∂K f(u)·ndS = 0.

We define ¯ uK(t) := 1 |K|

  • K u(t)dx

and obtain d dt ¯ uK(t)+ 1 |K|

  • ∂K f(u)·ndS = 0.

We assume, that there exists a piecewise polynomial function UN

h (t) ∈ SN h such that

UN

h (x,t) = u(x,t)+O(hN+1),

∀x ∈ Ω, ∀t ∈ (0,T).

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

We integrate over K ∈ Th and apply Green’s theorem d dt

  • K u(t)dx +
  • ∂K f(u)·ndS = 0.

We define ¯ uK(t) := 1 |K|

  • K u(t)dx

and obtain d dt ¯ uK(t)+ 1 |K|

  • ∂K f(u)·ndS = 0.

We assume, that there exists a piecewise polynomial function UN

h (t) ∈ SN h such that

UN

h (x,t) = u(x,t)+O(hN+1),

∀x ∈ Ω, ∀t ∈ (0,T).

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

We integrate over K ∈ Th and apply Green’s theorem d dt

  • K u(t)dx +
  • ∂K f(u)·ndS = 0.

We define ¯ uK(t) := 1 |K|

  • K u(t)dx

and obtain d dt ¯ uK(t)+ 1 |K|

  • ∂K f(u)·ndS = 0.

We assume, that there exists a piecewise polynomial function UN

h (t) ∈ SN h such that

UN

h (x,t) = u(x,t)+O(hN+1),

∀x ∈ Ω, ∀t ∈ (0,T).

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

The boundary convective terms will be treated with the aid of a numerical flux H(u,v,n):

  • Γ f(u)·ndS ≈
  • Γ H(UN,(L)

h

,UN,(R)

h

,n)dS. Lemma The averages of the exact solution u satisfy d dt ¯ uK(t)+ 1 |K|

  • ∂K H(UN,(L)

h

,UN,(R)

h

,n)dS = O(hN). Lipschitz continuity and consistency of H u −UN

h = O(hN).

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

The boundary convective terms will be treated with the aid of a numerical flux H(u,v,n):

  • Γ f(u)·ndS ≈
  • Γ H(UN,(L)

h

,UN,(R)

h

,n)dS. Lemma The averages of the exact solution u satisfy d dt ¯ uK(t)+ 1 |K|

  • ∂K H(UN,(L)

h

,UN,(R)

h

,n)dS = O(hN). Lipschitz continuity and consistency of H u −UN

h = O(hN).

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

The boundary convective terms will be treated with the aid of a numerical flux H(u,v,n):

  • Γ f(u)·ndS ≈
  • Γ H(UN,(L)

h

,UN,(R)

h

,n)dS. Lemma The averages of the exact solution u satisfy d dt ¯ uK(t)+ 1 |K|

  • ∂K H(UN,(L)

h

,UN,(R)

h

,n)dS = O(hN). Lipschitz continuity and consistency of H u −UN

h = O(hN).

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

Definition (FV reconstruction problem) Let v : Ω → R be sufficiently regular. Given ¯ vK for all K ∈ Th, find vN

h ∈ SN h such that v −vN h = O(hN+1) in Ω.

We define the corresponding reconstruction operator R : S0

h → SN h by R ¯

v := vN

h .

Definition (Reconstructed FV scheme) We seek uh(t) ∈ S0

h such that

d dt uh,K(t)+ 1 |K|

  • ∂K H
  • (Ruh)(L),(Ruh)(R),n
  • dS = 0.

Lemma The exact solution u satisfies d dt ¯ uK(t)+ 1 |K|

  • ∂K H
  • (R¯

u)(L),(R¯ u)(R),n

  • dS = O(hN).

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

Definition (FV reconstruction problem) Let v : Ω → R be sufficiently regular. Given ¯ vK for all K ∈ Th, find vN

h ∈ SN h such that v −vN h = O(hN+1) in Ω.

We define the corresponding reconstruction operator R : S0

h → SN h by R ¯

v := vN

h .

Definition (Reconstructed FV scheme) We seek uh(t) ∈ S0

h such that

d dt uh,K(t)+ 1 |K|

  • ∂K H
  • (Ruh)(L),(Ruh)(R),n
  • dS = 0.

Lemma The exact solution u satisfies d dt ¯ uK(t)+ 1 |K|

  • ∂K H
  • (R¯

u)(L),(R¯ u)(R),n

  • dS = O(hN).

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

Definition (FV reconstruction problem) Let v : Ω → R be sufficiently regular. Given ¯ vK for all K ∈ Th, find vN

h ∈ SN h such that v −vN h = O(hN+1) in Ω.

We define the corresponding reconstruction operator R : S0

h → SN h by R ¯

v := vN

h .

Definition (Reconstructed FV scheme) We seek uh(t) ∈ S0

h such that

d dt uh,K(t)+ 1 |K|

  • ∂K H
  • (Ruh)(L),(Ruh)(R),n
  • dS = 0.

Lemma The exact solution u satisfies d dt ¯ uK(t)+ 1 |K|

  • ∂K H
  • (R¯

u)(L),(R¯ u)(R),n

  • dS = O(hN).

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

Lemma The exact solution u satisfies d dt ¯ uK(t)+ 1 |K|

  • ∂K H
  • (R¯

u)(L),(R¯ u)(R),n

  • dS = O(hN).

This indicates, that we may expect u(t)−R ¯ uh(t) = O(hN), although, in principle, we have only u(t)− ¯ uh(t) = O(h). This is confirmed by numerical experiments

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

Lemma The exact solution u satisfies d dt ¯ uK(t)+ 1 |K|

  • ∂K H
  • (R¯

u)(L),(R¯ u)(R),n

  • dS = O(hN).

This indicates, that we may expect u(t)−R ¯ uh(t) = O(hN), although, in principle, we have only u(t)− ¯ uh(t) = O(h). This is confirmed by numerical experiments

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

’Standard’ FV reconstruction operator

Reconstruction stencil For each K ∈ Th we choose the reconstruction stencil SK ⊂ Th, usually some neighborhood of K. For each K ∈ Th, we seek a polynomial pSK ∈ PN(SK), s.t. 1 |K ′|

  • K ′ pSK dx = uh
  • K ′

∀K ′ ∈ SK. Finally, we define (Ruh)|K := pSK |K for all K ∈ Th.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

’Standard’ FV reconstruction operator

Reconstruction stencil For each K ∈ Th we choose the reconstruction stencil SK ⊂ Th, usually some neighborhood of K. For each K ∈ Th, we seek a polynomial pSK ∈ PN(SK), s.t. 1 |K ′|

  • K ′ pSK dx = uh
  • K ′

∀K ′ ∈ SK. Finally, we define (Ruh)|K := pSK |K for all K ∈ Th.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

Spectral FV reconstruction operator

Spectral and control volumes Let T S

h be a partition of Ω into simplices S ∈ T S h , called

spectral volumes. The FV triangulation Th is formed by subdividing each S ∈ T S

h into so-called control volumes K ⊂ S.

For each spectral volume S ∈ T S

h we seek pS ∈ PN(S), s.t.

1 |K|

  • K pS dx = uh
  • K

∀K ⊂ S, K ∈ Th. Finally, we define (Ruh)|K := pS|K for all K ⊂ S.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

Spectral FV reconstruction operator

Spectral and control volumes Let T S

h be a partition of Ω into simplices S ∈ T S h , called

spectral volumes. The FV triangulation Th is formed by subdividing each S ∈ T S

h into so-called control volumes K ⊂ S.

For each spectral volume S ∈ T S

h we seek pS ∈ PN(S), s.t.

1 |K|

  • K pS dx = uh
  • K

∀K ⊂ S, K ∈ Th. Finally, we define (Ruh)|K := pS|K for all K ⊂ S.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

’Standard’ FV R must be constructed (and stored) for each K ∈ Th independently (on unstructured meshes). Stencil size impractical for N > 2. Construction of stencils near ∂Ω. Explicit construction in 1D. Spectral FV All spectral volumes are affine equivalent ⇒ R is constructed and stored only on a reference configuration. The construction of partitions of spectral volumes into control volumes is not straightforward for higher N and 3D. No problems near boundaries. Explicit construction in 1D.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

’Standard’ FV R must be constructed (and stored) for each K ∈ Th independently (on unstructured meshes). Stencil size impractical for N > 2. Construction of stencils near ∂Ω. Explicit construction in 1D. Spectral FV All spectral volumes are affine equivalent ⇒ R is constructed and stored only on a reference configuration. The construction of partitions of spectral volumes into control volumes is not straightforward for higher N and 3D. No problems near boundaries. Explicit construction in 1D.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

’Standard’ FV R must be constructed (and stored) for each K ∈ Th independently (on unstructured meshes). Stencil size impractical for N > 2. Construction of stencils near ∂Ω. Explicit construction in 1D. Spectral FV All spectral volumes are affine equivalent ⇒ R is constructed and stored only on a reference configuration. The construction of partitions of spectral volumes into control volumes is not straightforward for higher N and 3D. No problems near boundaries. Explicit construction in 1D.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

’Standard’ FV R must be constructed (and stored) for each K ∈ Th independently (on unstructured meshes). Stencil size impractical for N > 2. Construction of stencils near ∂Ω. Explicit construction in 1D. Spectral FV All spectral volumes are affine equivalent ⇒ R is constructed and stored only on a reference configuration. The construction of partitions of spectral volumes into control volumes is not straightforward for higher N and 3D. No problems near boundaries. Explicit construction in 1D.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

’Standard’ FV R must be constructed (and stored) for each K ∈ Th independently (on unstructured meshes). Stencil size impractical for N > 2. Construction of stencils near ∂Ω. Explicit construction in 1D. Spectral FV All spectral volumes are affine equivalent ⇒ R is constructed and stored only on a reference configuration. The construction of partitions of spectral volumes into control volumes is not straightforward for higher N and 3D. No problems near boundaries. Explicit construction in 1D.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

’Standard’ FV R must be constructed (and stored) for each K ∈ Th independently (on unstructured meshes). Stencil size impractical for N > 2. Construction of stencils near ∂Ω. Explicit construction in 1D. Spectral FV All spectral volumes are affine equivalent ⇒ R is constructed and stored only on a reference configuration. The construction of partitions of spectral volumes into control volumes is not straightforward for higher N and 3D. No problems near boundaries. Explicit construction in 1D.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

’Standard’ FV R must be constructed (and stored) for each K ∈ Th independently (on unstructured meshes). Stencil size impractical for N > 2. Construction of stencils near ∂Ω. Explicit construction in 1D. Spectral FV All spectral volumes are affine equivalent ⇒ R is constructed and stored only on a reference configuration. The construction of partitions of spectral volumes into control volumes is not straightforward for higher N and 3D. No problems near boundaries. Explicit construction in 1D.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Continuous Problem Space semidiscretization

’Standard’ FV R must be constructed (and stored) for each K ∈ Th independently (on unstructured meshes). Stencil size impractical for N > 2. Construction of stencils near ∂Ω. Explicit construction in 1D. Spectral FV All spectral volumes are affine equivalent ⇒ R is constructed and stored only on a reference configuration. The construction of partitions of spectral volumes into control volumes is not straightforward for higher N and 3D. No problems near boundaries. Explicit construction in 1D.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

1

Finite volume method with reconstruction Continuous Problem Space semidiscretization

2

Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Definition Let v ∈ L2(Ω). Define by Πn

hv the L2(Ω)-projection of v on Sn h:

Πn

hv ∈ Sn h,

  • Πn

hv −v, ϕn h

  • = 0,

∀ϕn

h ∈ Sn h.

The basis of the FV schemes consisted of the identity d dt ¯ uK(t)+ 1 |K|

  • ∂K H
  • (R¯

u)(L),(R¯ u)(R),n

  • dS = O(hN)

Since ¯ u(t) = Π0

hu(t), we may view this as an identity for Π0 hu(t).

We shall generalize this relation from Π0

hu(t) to Πn hu(t).

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Definition Let v ∈ L2(Ω). Define by Πn

hv the L2(Ω)-projection of v on Sn h:

Πn

hv ∈ Sn h,

  • Πn

hv −v, ϕn h

  • = 0,

∀ϕn

h ∈ Sn h.

The basis of the FV schemes consisted of the identity d dt ¯ uK(t)+ 1 |K|

  • ∂K H
  • (R¯

u)(L),(R¯ u)(R),n

  • dS = O(hN)

Since ¯ u(t) = Π0

hu(t), we may view this as an identity for Π0 hu(t).

We shall generalize this relation from Π0

hu(t) to Πn hu(t).

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Definition Let v ∈ L2(Ω). Define by Πn

hv the L2(Ω)-projection of v on Sn h:

Πn

hv ∈ Sn h,

  • Πn

hv −v, ϕn h

  • = 0,

∀ϕn

h ∈ Sn h.

The basis of the FV schemes consisted of the identity d dt ¯ uK(t)+ 1 |K|

  • ∂K H
  • (R¯

u)(L),(R¯ u)(R),n

  • dS = O(hN)

Since ¯ u(t) = Π0

hu(t), we may view this as an identity for Π0 hu(t).

We shall generalize this relation from Π0

hu(t) to Πn hu(t).

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

We multiply our problem by an arbitrary ϕn

h ∈ Sn h, integrate over

an element K ∈ Th and apply Green’s theorem d dt

  • K u(t)ϕn

h dx +

  • ∂K f(u)·nϕn

h

  • K dS −
  • K f(u)·∇ϕn

h dx = 0.

By summing over all K ∈ Th and rearranging, we get We assume, that there exists a piecewise polynomial function UN

h (t) ∈ SN h such that

UN

h (x,t) = u(x,t)+O(hN+1),

∀x ∈ Ω, ∀t ∈ (0,T).

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

We multiply our problem by an arbitrary ϕn

h ∈ Sn h, integrate over

an element K ∈ Th and apply Green’s theorem d dt

  • K u(t)ϕn

h dx +

  • ∂K f(u)·nϕn

h

  • K dS −
  • K f(u)·∇ϕn

h dx = 0.

By summing over all K ∈ Th and rearranging, we get d dt

  • Ω u(t)ϕn

h dx + ∑ Γ∈Fh

  • Γ f(u)·n[ϕn

h]dS− ∑ K∈Th

  • K f(u)·∇ϕn

h dx = 0.

We assume, that there exists a piecewise polynomial function UN

h (t) ∈ SN h such that

UN

h (x,t) = u(x,t)+O(hN+1),

∀x ∈ Ω, ∀t ∈ (0,T).

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

We multiply our problem by an arbitrary ϕn

h ∈ Sn h, integrate over

an element K ∈ Th and apply Green’s theorem d dt

  • K u(t)ϕn

h dx +

  • ∂K f(u)·nϕn

h

  • K dS −
  • K f(u)·∇ϕn

h dx = 0.

By summing over all K ∈ Th and rearranging, we get d dt

  • Ω u(t)ϕn

h dx + ∑ Γ∈Fh

  • Γ f(u)·n[ϕn

h]dS− ∑ K∈Th

  • K f(u)·∇ϕn

h dx = 0.

We assume, that there exists a piecewise polynomial function UN

h (t) ∈ SN h such that

UN

h (x,t) = u(x,t)+O(hN+1),

∀x ∈ Ω, ∀t ∈ (0,T).

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

We multiply our problem by an arbitrary ϕn

h ∈ Sn h, integrate over

an element K ∈ Th and apply Green’s theorem d dt

  • K u(t)ϕn

h dx +

  • ∂K f(u)·nϕn

h

  • K dS −
  • K f(u)·∇ϕn

h dx = 0.

By summing over all K ∈ Th and rearranging, we get d dt

  • Ω Πn

hu(t)ϕn h dx + ∑ Γ∈Fh

  • Γ f(u)·n[ϕn

h]dS − ∑ K∈Th

  • K f(u)·∇ϕn

h dx = 0.

We assume, that there exists a piecewise polynomial function UN

h (t) ∈ SN h such that

UN

h (x,t) = u(x,t)+O(hN+1),

∀x ∈ Ω, ∀t ∈ (0,T).

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

We multiply our problem by an arbitrary ϕn

h ∈ Sn h, integrate over

an element K ∈ Th and apply Green’s theorem d dt

  • K u(t)ϕn

h dx +

  • ∂K f(u)·nϕn

h

  • K dS −
  • K f(u)·∇ϕn

h dx = 0.

By summing over all K ∈ Th and rearranging, we get d dt

  • Ω Πn

hu(t)ϕn h dx + ∑ Γ∈Fh

  • Γ f(u)·n[ϕn

h]dS − ∑ K∈Th

  • K f(u)·∇ϕn

h dx = 0.

We assume, that there exists a piecewise polynomial function UN

h (t) ∈ SN h such that

UN

h (x,t) = u(x,t)+O(hN+1),

∀x ∈ Ω, ∀t ∈ (0,T).

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Again, we introduce a numerical flux H(u,v,n):

  • Γ f(u)·n[ϕn

h]dS ≈

  • Γ H(u(L),u(R),n)[ϕn

h]dS.

Definition bh(u,ϕ) =

  • Fh

H(u(L),u(R),n)[ϕ]dS − ∑

K∈Th

  • K f(u)·∇ϕ dx.

Lemma The projections Πn

hu(t) of the exact solution satisfy

d dt

  • Πn

hu(t),ϕn h

  • +bh
  • UN

h (t),ϕn h

  • = O(hN+1)ϕn

hL2(Ω),

∀ϕn

h ∈ Sn h.

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Again, we introduce a numerical flux H(u,v,n):

  • Γ f(u)·n[ϕn

h]dS ≈

  • Γ H(u(L),u(R),n)[ϕn

h]dS.

Definition bh(u,ϕ) =

  • Fh

H(u(L),u(R),n)[ϕ]dS − ∑

K∈Th

  • K f(u)·∇ϕ dx.

Lemma The projections Πn

hu(t) of the exact solution satisfy

d dt

  • Πn

hu(t),ϕn h

  • +bh
  • UN

h (t),ϕn h

  • = O(hN+1)ϕn

hL2(Ω),

∀ϕn

h ∈ Sn h.

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Again, we introduce a numerical flux H(u,v,n):

  • Γ f(u)·n[ϕn

h]dS ≈

  • Γ H(u(L),u(R),n)[ϕn

h]dS.

Definition bh(u,ϕ) =

  • Fh

H(u(L),u(R),n)[ϕ]dS − ∑

K∈Th

  • K f(u)·∇ϕ dx.

Lemma The projections Πn

hu(t) of the exact solution satisfy

d dt

  • Πn

hu(t),ϕn h

  • +bh
  • UN

h (t),ϕn h

  • = O(hN+1)ϕn

hL2(Ω),

∀ϕn

h ∈ Sn h.

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Again, we introduce a numerical flux H(u,v,n):

  • Γ f(u)·n[ϕn

h]dS ≈

  • Γ H(u(L),u(R),n)[ϕn

h]dS.

Definition bh(u,ϕ) =

  • Fh

H(u(L),u(R),n)[ϕ]dS − ∑

K∈Th

  • K f(u)·∇ϕ dx.

Lemma The projections Πn

hu(t) of the exact solution satisfy

d dt

  • Πn

hu(t),ϕn h

  • +bh
  • UN

h (t),ϕn h

  • = O(hN+1)ϕn

hL2(Ω),

∀ϕn

h ∈ Sn h.

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Definition (DG Reconstruction problem) Let v : Ω → R be sufficiently regular. Given Πn

hv ∈ Sn h, find

vN

h ∈ SN h such that v −vN h = O(hN+1) in Ω.

We define the corresponding reconstruction operator R : Sn

h → SN h by R Πn hv := vN h .

Definition (Reconstructed DG scheme) We seek un

h ∈ Sn h such that

d dt

  • un

h(t),ϕn h

  • +bh
  • R un

h(t),ϕn h

  • = 0,

∀ϕn

h ∈ Sn h.

Lemma The exact solution u satisfies d dt

  • Πn

hu(t),ϕn h

  • +bh
  • R Πn

hu(t),ϕn h

  • = O(hN+1)ϕn

hL2(Ω).

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Definition (DG Reconstruction problem) Let v : Ω → R be sufficiently regular. Given Πn

hv ∈ Sn h, find

vN

h ∈ SN h such that v −vN h = O(hN+1) in Ω.

We define the corresponding reconstruction operator R : Sn

h → SN h by R Πn hv := vN h .

Definition (Reconstructed DG scheme) We seek un

h ∈ Sn h such that

d dt

  • un

h(t),ϕn h

  • +bh
  • R un

h(t),ϕn h

  • = 0,

∀ϕn

h ∈ Sn h.

Lemma The exact solution u satisfies d dt

  • Πn

hu(t),ϕn h

  • +bh
  • R Πn

hu(t),ϕn h

  • = O(hN+1)ϕn

hL2(Ω).

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Definition (DG Reconstruction problem) Let v : Ω → R be sufficiently regular. Given Πn

hv ∈ Sn h, find

vN

h ∈ SN h such that v −vN h = O(hN+1) in Ω.

We define the corresponding reconstruction operator R : Sn

h → SN h by R Πn hv := vN h .

Definition (Reconstructed DG scheme) We seek un

h ∈ Sn h such that

d dt

  • un

h(t),ϕn h

  • +bh
  • R un

h(t),ϕn h

  • = 0,

∀ϕn

h ∈ Sn h.

Lemma The exact solution u satisfies d dt

  • Πn

hu(t),ϕn h

  • +bh
  • R Πn

hu(t),ϕn h

  • = O(hN+1)ϕn

hL2(Ω).

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Definition (DG Reconstruction problem) Let v : Ω → R be sufficiently regular. Given Πn

hv ∈ Sn h, find

vN

h ∈ SN h such that v −vN h = O(hN+1) in Ω.

We define the corresponding reconstruction operator R : Sn

h → SN h by R Πn hv := vN h .

Definition (Reconstructed DG scheme) We seek un

h ∈ Sn h such that

d dt

  • un

h(t),ϕn h

  • +bh
  • R un

h(t),ϕn h

  • = 0,

∀ϕn

h ∈ Sn h.

Lemma The exact solution u satisfies d dt

  • Πn

hu(t),ϕn h

  • +bh
  • R Πn

hu(t),ϕn h

  • = O(hN+1)ϕn

hL2(Ω).

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Definition (DG Reconstruction problem) Let v : Ω → R be sufficiently regular. Given Πn

hv ∈ Sn h, find

vN

h ∈ SN h such that v −vN h = O(hN+1) in Ω.

We define the corresponding reconstruction operator R : Sn

h → SN h by R Πn hv := vN h .

Definition (Reconstructed DG scheme) We seek un

h ∈ Sn h such that

d dt

  • un

h(t),ϕn h

  • +bh
  • R un

h(t),ϕn h

  • = 0,

∀ϕn

h ∈ Sn h.

Lemma The exact solution u satisfies d dt

  • Πn

hu(t),ϕn h

  • +bh
  • R Πn

hu(t),ϕn h

  • = O(hN+1)ϕn

hL2(Ω).

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Lemma The exact solution u satisfies d dt

  • Πn

hu(t),ϕn h

  • +bh
  • R Πn

hu(t),ϕn h

  • = O(hN+1)ϕn

hL2(Ω).

This indicates, that we may expect u(t)−R un

h(t) = O(hN),

although, in principle, we have only u(t)−un

h(t) = O(hn).

This is confirmed by numerical experiments.

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Lemma The exact solution u satisfies d dt

  • Πn

hu(t),ϕn h

  • +bh
  • R Πn

hu(t),ϕn h

  • = O(hN+1)ϕn

hL2(Ω).

This indicates, that we may expect u(t)−R un

h(t) = O(hN),

although, in principle, we have only u(t)−un

h(t) = O(hn).

This is confirmed by numerical experiments.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Lemma The exact solution u satisfies d dt

  • Πn

hu(t),ϕn h

  • +bh
  • R Πn

hu(t),ϕn h

  • = O(hN+1)ϕn

hL2(Ω).

This indicates, that we may expect u(t)−R un

h(t) = O(hN),

although, in principle, we have only u(t)−un

h(t) = O(hn).

This is confirmed by numerical experiments.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Analogy of ’standard’ FV reconstruction operator

Reconstruction stencil For each K ∈ Th we choose the reconstruction stencil SK ⊂ Th, usually some neighborhood of K. For each K ∈ Th, we seek a polynomial pSK ∈ PN(SK), s.t.

  • Πn

hpSK

  • K ′ = un

h

  • K ′

∀K ′ ∈ SK. Finally, we define (Run

h)|K := pSK |K for all K ∈ Th.

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Analogy of ’standard’ FV reconstruction operator

Reconstruction stencil For each K ∈ Th we choose the reconstruction stencil SK ⊂ Th, usually some neighborhood of K. For each K ∈ Th, we seek a polynomial pSK ∈ PN(SK), s.t.

  • Πn

hpSK

  • K ′ = un

h

  • K ′

∀K ′ ∈ SK. Finally, we define (Run

h)|K := pSK |K for all K ∈ Th.

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Analogy of ’spectral’ FV reconstruction operator

Spectral and control volumes Let T S

h be a partition of Ω into simplices S ∈ T S h , called

spectral volumes. The DG triangulation Th is formed by subdividing each S ∈ T S

h into so-called control volumes K ⊂ S.

For each spectral volume S ∈ T S

h we seek pS ∈ PN(S), s.t.

  • Πn

hpS

  • K = un

h

  • K,

∀K ⊂ S, K ∈ Th. Finally, we define (Run

h)|K := pS|K for all K ⊂ S.

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Analogy of ’spectral’ FV reconstruction operator

Spectral and control volumes Let T S

h be a partition of Ω into simplices S ∈ T S h , called

spectral volumes. The DG triangulation Th is formed by subdividing each S ∈ T S

h into so-called control volumes K ⊂ S.

For each spectral volume S ∈ T S

h we seek pS ∈ PN(S), s.t.

  • Πn

hpS

  • K = un

h

  • K,

∀K ⊂ S, K ∈ Th. Finally, we define (Run

h)|K := pS|K for all K ⊂ S.

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

’Standard’ FV Stencil size need not be increased! To obtain higher orders we simply increase n. R must be constructed (and stored) for each K ∈ Th independently (on unstructured meshes). Construction of stencils near ∂Ω. Spectral FV The number of control volumes need not be increased! To

  • btain higher orders we simply increase n.

All spectral volumes are affine equivalent ⇒ R is constructed and stored only on a reference configuration. No problems near boundaries.

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

’Standard’ FV Stencil size need not be increased! To obtain higher orders we simply increase n. R must be constructed (and stored) for each K ∈ Th independently (on unstructured meshes). Construction of stencils near ∂Ω. Spectral FV The number of control volumes need not be increased! To

  • btain higher orders we simply increase n.

All spectral volumes are affine equivalent ⇒ R is constructed and stored only on a reference configuration. No problems near boundaries.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

’Standard’ FV Stencil size need not be increased! To obtain higher orders we simply increase n. R must be constructed (and stored) for each K ∈ Th independently (on unstructured meshes). Construction of stencils near ∂Ω. Spectral FV The number of control volumes need not be increased! To

  • btain higher orders we simply increase n.

All spectral volumes are affine equivalent ⇒ R is constructed and stored only on a reference configuration. No problems near boundaries.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

’Standard’ FV Stencil size need not be increased! To obtain higher orders we simply increase n. R must be constructed (and stored) for each K ∈ Th independently (on unstructured meshes). Construction of stencils near ∂Ω. Spectral FV The number of control volumes need not be increased! To

  • btain higher orders we simply increase n.

All spectral volumes are affine equivalent ⇒ R is constructed and stored only on a reference configuration. No problems near boundaries.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

’Standard’ FV Stencil size need not be increased! To obtain higher orders we simply increase n. R must be constructed (and stored) for each K ∈ Th independently (on unstructured meshes). Construction of stencils near ∂Ω. Spectral FV The number of control volumes need not be increased! To

  • btain higher orders we simply increase n.

All spectral volumes are affine equivalent ⇒ R is constructed and stored only on a reference configuration. No problems near boundaries.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

’Standard’ FV Stencil size need not be increased! To obtain higher orders we simply increase n. R must be constructed (and stored) for each K ∈ Th independently (on unstructured meshes). Construction of stencils near ∂Ω. Spectral FV The number of control volumes need not be increased! To

  • btain higher orders we simply increase n.

All spectral volumes are affine equivalent ⇒ R is constructed and stored only on a reference configuration. No problems near boundaries.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Reconstructed DG vs Standard DG Test functions only of order n as opposed to N. Fewer quadrature points, flux evaluations. CFL condition permits larger time steps. Mass matrices of

  • rder n ×n instead of N ×N.

The reconstruction procedure is problem-independent. The von Neumann neighborhood allows us to reconstruct: 1D: S3n+2

h

from Sn

h.

2D: S2n+1

h

from Sn

h.

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Reconstructed DG vs Standard DG Test functions only of order n as opposed to N. Fewer quadrature points, flux evaluations. CFL condition permits larger time steps. Mass matrices of

  • rder n ×n instead of N ×N.

The reconstruction procedure is problem-independent. The von Neumann neighborhood allows us to reconstruct: 1D: S3n+2

h

from Sn

h.

2D: S2n+1

h

from Sn

h.

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Reconstructed DG vs Standard DG Test functions only of order n as opposed to N. Fewer quadrature points, flux evaluations. CFL condition permits larger time steps. Mass matrices of

  • rder n ×n instead of N ×N.

The reconstruction procedure is problem-independent. The von Neumann neighborhood allows us to reconstruct: 1D: S3n+2

h

from Sn

h.

2D: S2n+1

h

from Sn

h.

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Reconstructed DG vs Standard DG Test functions only of order n as opposed to N. Fewer quadrature points, flux evaluations. CFL condition permits larger time steps. Mass matrices of

  • rder n ×n instead of N ×N.

The reconstruction procedure is problem-independent. The von Neumann neighborhood allows us to reconstruct: 1D: S3n+2

h

from Sn

h.

2D: S2n+1

h

from Sn

h.

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Reconstructed DG vs Standard DG Test functions only of order n as opposed to N. Fewer quadrature points, flux evaluations. CFL condition permits larger time steps. Mass matrices of

  • rder n ×n instead of N ×N.

The reconstruction procedure is problem-independent. The von Neumann neighborhood allows us to reconstruct: 1D: S3n+2

h

from Sn

h.

2D: S2n+1

h

from Sn

h.

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

1

Finite volume method with reconstruction Continuous Problem Space semidiscretization

2

Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Definition (Reconstructed DG scheme) We seek un,k

h

∈ Sn

h such that

un,k+1

h

−un,k

h

τk ,ϕn

h

  • +bh
  • R un,k

h ,ϕn h

  • = 0,

∀ϕn

h ∈ Sn h.

Definition (Auxiliary DG scheme) We seek uN,k

h

∈ SN

h such that

uN,k+1

h

−uN,k

h

τk ,ϕN

h

  • +bh
  • RΠn

huN,k h

,ϕN

h

  • = 0,

∀ϕN

h ∈ SN h .

Lemma un,k

h

= Πn

huN,k h

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Definition (Reconstructed DG scheme) We seek un,k

h

∈ Sn

h such that

un,k+1

h

−un,k

h

τk ,ϕn

h

  • +bh
  • R un,k

h ,ϕn h

  • = 0,

∀ϕn

h ∈ Sn h.

Definition (Auxiliary DG scheme) We seek uN,k

h

∈ SN

h such that

uN,k+1

h

−uN,k

h

τk ,ϕN

h

  • +bh
  • RΠn

huN,k h

,ϕN

h

  • = 0,

∀ϕN

h ∈ SN h .

Lemma un,k

h

= Πn

huN,k h

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Definition (Reconstructed DG scheme) We seek un,k

h

∈ Sn

h such that

un,k+1

h

−un,k

h

τk ,ϕn

h

  • +bh
  • R un,k

h ,ϕn h

  • = 0,

∀ϕn

h ∈ Sn h.

Definition (Auxiliary DG scheme) We seek uN,k

h

∈ SN

h such that

uN,k+1

h

−uN,k

h

τk ,ϕN

h

  • +bh
  • RΠn

huN,k h

,ϕN

h

  • = 0,

∀ϕN

h ∈ SN h .

Lemma un,k

h

= Πn

huN,k h

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Therefore, error estimates for the reconstructed DG scheme may be derived from estimates for the auxiliary problem Definition (Auxiliary DG scheme) We seek uN,k

h

∈ SN

h such that

uN,k+1

h

−uN,k

h

τk ,ϕN

h

  • +bh
  • RΠn

huN,k h

,ϕN

h

  • = 0,

∀ϕN

h ∈ SN h .

This is similar to the standard DG scheme Definition (Standard DG scheme) We seek ˜ uN,k

h

∈ SN

h such that

˜ uN,k+1

h

− ˜ uN,k

h

τk ,ϕN

h

  • +bh

˜ uN,k

h

,ϕN

h

  • = 0,

∀ϕN

h ∈ SN h .

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Therefore, error estimates for the reconstructed DG scheme may be derived from estimates for the auxiliary problem Definition (Auxiliary DG scheme) We seek uN,k

h

∈ SN

h such that

uN,k+1

h

−uN,k

h

τk ,ϕN

h

  • +bh
  • RΠn

huN,k h

,ϕN

h

  • = 0,

∀ϕN

h ∈ SN h .

This is similar to the standard DG scheme Definition (Standard DG scheme) We seek ˜ uN,k

h

∈ SN

h such that

˜ uN,k+1

h

− ˜ uN,k

h

τk ,ϕN

h

  • +bh

˜ uN,k

h

,ϕN

h

  • = 0,

∀ϕN

h ∈ SN h .

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Therefore, error estimates for the reconstructed DG scheme may be derived from estimates for the auxiliary problem Definition (Auxiliary DG scheme) We seek uN,k

h

∈ SN

h such that

uN,k+1

h

−uN,k

h

τk ,ϕN

h

  • +bh
  • RΠn

huN,k h

,ϕN

h

  • = 0,

∀ϕN

h ∈ SN h .

This is similar to the standard DG scheme Definition (Standard DG scheme) We seek ˜ uN,k

h

∈ SN

h such that

˜ uN,k+1

h

− ˜ uN,k

h

τk ,ϕN

h

  • +bh

˜ uN,k

h

,ϕN

h

  • = 0,

∀ϕN

h ∈ SN h .

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Therefore, error estimates for the reconstructed DG scheme may be derived from estimates for the auxiliary problem Definition (Auxiliary DG scheme) We seek uN,k

h

∈ SN

h such that

uN,k+1

h

−uN,k

h

τk ,ϕN

h

  • +bh
  • RΠn

huN,k h

,ϕN

h

  • = 0,

∀ϕN

h ∈ SN h .

This is similar to the standard DG scheme Definition (Standard DG scheme) We seek ˜ uN,k

h

∈ SN

h such that

˜ uN,k+1

h

− ˜ uN,k

h

τk ,ϕN

h

  • +bh

˜ uN,k

h

,ϕN

h

  • = 0,

∀ϕN

h ∈ SN h .

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Therefore, error estimates for the reconstructed DG scheme might possibly be derived from standard DG estimates and a thorough understanding of the operator RΠn

h : L2(Ω) → SN h .

Lemma Let v ∈ HN+1(Ω),vh ∈ SN

h . Then

v −RΠn

hvL2(Ω) ≤ ChN+1|v|HN+1(Ω),

vh −RΠn

hvhL2(Ω) ≤ C

inf

w∈HN+1(Ω)

  • hN+1|w|HN+1(Ω) +vh −wL2(Ω)
  • .

Holds for the ”spectral volume” construction of R. Holds for the ”standard” construction of R for special (trivial) cases. Based on a very general Bramble-Hilbert lemma. Estimate #2 nice, but useless.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Therefore, error estimates for the reconstructed DG scheme might possibly be derived from standard DG estimates and a thorough understanding of the operator RΠn

h : L2(Ω) → SN h .

Lemma Let v ∈ HN+1(Ω),vh ∈ SN

h . Then

v −RΠn

hvL2(Ω) ≤ ChN+1|v|HN+1(Ω),

vh −RΠn

hvhL2(Ω) ≤ C

inf

w∈HN+1(Ω)

  • hN+1|w|HN+1(Ω) +vh −wL2(Ω)
  • .

Holds for the ”spectral volume” construction of R. Holds for the ”standard” construction of R for special (trivial) cases. Based on a very general Bramble-Hilbert lemma. Estimate #2 nice, but useless.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Therefore, error estimates for the reconstructed DG scheme might possibly be derived from standard DG estimates and a thorough understanding of the operator RΠn

h : L2(Ω) → SN h .

Lemma Let v ∈ HN+1(Ω),vh ∈ SN

h . Then

v −RΠn

hvL2(Ω) ≤ ChN+1|v|HN+1(Ω),

vh −RΠn

hvhL2(Ω) ≤ C

inf

w∈HN+1(Ω)

  • hN+1|w|HN+1(Ω) +vh −wL2(Ω)
  • .

Holds for the ”spectral volume” construction of R. Holds for the ”standard” construction of R for special (trivial) cases. Based on a very general Bramble-Hilbert lemma. Estimate #2 nice, but useless.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Therefore, error estimates for the reconstructed DG scheme might possibly be derived from standard DG estimates and a thorough understanding of the operator RΠn

h : L2(Ω) → SN h .

Lemma Let v ∈ HN+1(Ω),vh ∈ SN

h . Then

v −RΠn

hvL2(Ω) ≤ ChN+1|v|HN+1(Ω),

vh −RΠn

hvhL2(Ω) ≤ C

inf

w∈HN+1(Ω)

  • hN+1|w|HN+1(Ω) +vh −wL2(Ω)
  • .

Holds for the ”spectral volume” construction of R. Holds for the ”standard” construction of R for special (trivial) cases. Based on a very general Bramble-Hilbert lemma. Estimate #2 nice, but useless.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Therefore, error estimates for the reconstructed DG scheme might possibly be derived from standard DG estimates and a thorough understanding of the operator RΠn

h : L2(Ω) → SN h .

Lemma Let v ∈ HN+1(Ω),vh ∈ SN

h . Then

v −RΠn

hvL2(Ω) ≤ ChN+1|v|HN+1(Ω),

vh −RΠn

hvhL2(Ω) ≤ C

inf

w∈HN+1(Ω)

  • hN+1|w|HN+1(Ω) +vh −wL2(Ω)
  • .

Holds for the ”spectral volume” construction of R. Holds for the ”standard” construction of R for special (trivial) cases. Based on a very general Bramble-Hilbert lemma. Estimate #2 nice, but useless.

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Therefore, error estimates for the reconstructed DG scheme might possibly be derived from standard DG estimates and a thorough understanding of the operator RΠn

h : L2(Ω) → SN h .

Lemma Let v ∈ HN+1(Ω),vh ∈ SN

h . Then

v −RΠn

hvL2(Ω) ≤ ChN+1|v|HN+1(Ω),

vh −RΠn

hvhL2(Ω) ≤ C

inf

w∈HN+1(Ω)

  • hN+1|w|HN+1(Ω) +vh −wL2(Ω)
  • .

Holds for the ”spectral volume” construction of R. Holds for the ”standard” construction of R for special (trivial) cases. Based on a very general Bramble-Hilbert lemma. Estimate #2 nice, but useless.

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Numerical experiments

N ||eh||L∞(Ω) α ||eh||L2(Ω) α |eh|H1(Ω,Th) α 4 9.30E-01 – 6.23E-01 – 4.05E+00 – 8 2.22E-01 2.07 1.55E-01 2.00 1.29E+00 1.65 16 3.25E-02 2.77 2.21E-02 2.81 2.47E-01 2.38 32 4.09E-03 2.99 2.82E-03 2.97 4.63E-02 2.41 64 5.07E-04 3.01 3.53E-04 3.00 9.46E-03 2.29 128 6.31E-05 3.01 4.41E-05 3.00 2.10E-03 2.17 256 7.86E-06 3.00 5.50E-06 3.00 4.91E-04 2.10

Table: 1D advection of sine wave, P0 elements with P2 reconstruction.

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Numerical experiments

N ||eh||L∞(Ω) α ||eh||L2(Ω) α |eh|H1(Ω,Th) α 4 5.82E-03 – 3.49E-03 – 3.65E-02 – 8 7.53E-05 6.27 4.43E-05 6,30 1.06E-03 5,11 16 9.07E-07 6.38 5.95E-07 6,22 3.58E-05 4,89 32 1.82E-08 5.64 8.70E-09 6,10 1.16E-06 4,95 64 3.41E-10 5.74 1.33E-10 6,03 3.67E-08 4,98

Table: 1D advection of sine wave, P1 elements with P5 reconstruction.

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Numerical experiments

N ||eh||L∞(Ω) α ||eh||L2(Ω) α |eh|H1(Ω,Th) α 4 2.90E-03 – 1.85E-03 – 1.63E-02 – 8 7.75E-06 8.55 3.56E-06 9.02 1.03E-04 7.30 16 2.10E-08 8.53 6.64E-09 9.07 4.34E-07 7.89 32 7.21E-11 8.18 4.02E-11 7.37 1.76E-09 7.94

Table: 1D advection of sine wave, P2 elements with P8 reconstruction.

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Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Formulation Theoretical results and numerical experiments

Thank you for your attention

V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for