DYNAMICS : THE RESIDUAL DISTRIBUTION POINT OF VIEW . A PPLICATION TO - - PowerPoint PPT Presentation

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

NON OSCILLATORY FEM-LIKE SCHEMES FOR COMPRESSIBLE FLUID

DYNAMICS: THE RESIDUAL DISTRIBUTION POINT OF VIEW.

APPLICATION TO LAMINAR AND TURBULENT FLOWS

• R. Abgrall

I-MATH Universität Zürich

May 24, 2014

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

COLLABORATIVE WORK, GOALS

Scheme: Viscous/inviscid compressible fluids simulations of possibly discontinuous flows, without tuning parameters: Dante de Santis, INRIA Meshes: Take into account curvature of solid boundaries: HO meshes : Cécile Dobrzynski, U. Bordeaux Efficiency

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

OUTLINE

1 MOTIVATING EXAMPLE 2 RESIDUAL DISTRIBUTION SCHEMES (RDS) 3 THE VISCOUS CASE 4 APPLICATION TO NS EQUATIONS 5 CONCLUSION-PERSPECTIVES

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

OVERVIEW

1 MOTIVATING EXAMPLE 2 RESIDUAL DISTRIBUTION SCHEMES (RDS) 3 THE VISCOUS CASE 4 APPLICATION TO NS EQUATIONS 5 CONCLUSION-PERSPECTIVES

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

FLOWS WITH DISCONTINUITIES

SCRAMJET LIKE, HYBRID MESH

x

10 11

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

FLOWS WITH DISCONTINUITIES

MACH NUMBER, 3RD ORDER

x y

1 2 3 4 5 6 7 8 9 10 11 12 13 1 2 3 4

limited LF plus stabilization - Mach number. Top : P2/Q2. Bottom : P1/Q1

x

8 9 10 11 12 13

limited LF plus stabilization - Mach number. Top : P2/Q2. Bottom : P1/Q1

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

M6 WING, DASSAULT MESH, SPALART-ALLMARAS TURBULENCE MODEL

M = 0.8395, α = 3.06◦. Re = 11.72 106

The M6 mesh has 77061 points and 443458 Tets

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

M6 WING, DASSAULT MESH, PRESSURE, SA TURBULENCE MODEL

M = 0.8395, α = 3.06◦. Re = 11.72 106

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

M6 WING, DASSAULT MESH, PRESSURE, SA TURBULENCE MODEL

M = 0.8395, α = 3.06◦. Re = 11.72 106

x/c

• cp

0.2 0.4 0.6 0.8 1

• 1
• 0.5

0.5 1 Numerical Experiment z/b = 20

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

M6 WING, DASSAULT MESH, PRESSURE, SA TURBULENCE MODEL

M = 0.8395, α = 3.06◦. Re = 11.72 106

x/c

• cp

0.2 0.4 0.6 0.8 1

• 1
• 0.5

0.5 1 Numerical Experiment z/b = 44

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

M6 WING, DASSAULT MESH, PRESSURE, SA TURBULENCE MODEL

M = 0.8395, α = 3.06◦. Re = 11.72 106

x/c

• cp

0.2 0.4 0.6 0.8 1

• 1
• 0.5

0.5 1 Numerical Experiment z/b = 65

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

M6 WING, DASSAULT MESH, PRESSURE, SA TURBULENCE MODEL

M = 0.8395, α = 3.06◦. Re = 11.72 106

x/c

• cp

0.2 0.4 0.6 0.8 1

• 1
• 0.5

0.5 1 Numerical Experiment z/b = 80

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

TYPICAL PROBLEM TO SOLVE

IN Ω ⊂ R2, R3,

∂W ∂t + div Fe(W) = 1 Re divFv(W, ∇W) with initial and boundary conditions, Re very large.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

TYPICAL PROBLEM TO SOLVE

IN Ω ⊂ R2, R3,

∂W ∂t + div Fe(W) = 1 Re divFv(W, ∇W) with initial and boundary conditions, Re very large.

STEADY VERSION

div Fe(W) = 1 Re divFv(W, ∇W) with BCs.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

RESIDUAL DISTRIBUTION SCHEMES

HISTORY

Ni scheme, 1981. Engineer at Bombardier Roe, 1981(−x)-today Deconinck, Ricchiuto, Nishikawa, Caraeni, . . . Strong connections with stream line diffusion method: Johnson, Hughes, etc (Not shown in the talk, there are several variational formulations of RD schemes) . . .

AIM

Combine ideas from finite volume schemes (non oscillatory, L∞ stability, upwinding with finite element methods Seen from the discrete point of view

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

OVERVIEW

1 MOTIVATING EXAMPLE 2 RESIDUAL DISTRIBUTION SCHEMES (RDS) 3 THE VISCOUS CASE 4 APPLICATION TO NS EQUATIONS 5 CONCLUSION-PERSPECTIVES

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

MODEL EQUATION: SCALAR STEADY CONVECTION-DIFFUSION div f(u) − div

• K∇u
• = 0
• n Ω ⊂ Rd boundary conditions on ∂Ω

f(u) =

• f1(u), · · · , fd(u)), fi smooth enough.

Boundary conditions: Dirichlet or inflow/outflow depending on K

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

MODEL EQUATION: SCALAR STEADY CONVECTION-DIFFUSION div f(u) − div

• K∇u
• = 0
• n Ω ⊂ Rd boundary conditions on ∂Ω

f(u) =

• f1(u), · · · , fd(u)), fi smooth enough.

Boundary conditions: Dirichlet or inflow/outflow depending on K Analysis done for non viscous problems first

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

MODEL PROBLEM, FRAMEWORK FOR SCALAR CONSERVATION LAWS. div f(u) = 0 in Ω u = g

• n

Γ−

t

• p

l

SOME NOTATIONS...

Consider Th triangulation of Ω (can do with quads...) Unknowns (Degrees of Freedom, DoF) : ui ≈ u(Mi) Mi ∈ Th a given set of nodes (vertices +other dofs) Denote by uh continuous piecewise approximation (e.g. Pk Lagrange triangles/quads, Bézier, NURBS, etc) : uh =

i

ψi ui

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

SOME SIMPLE REMARKS: 2 WAYS OF WRITING SCHEMES ∂u ∂t + ∂f(u) ∂x = 0, 2ND ORDER

FINITE VOLUMES 1D: un+1

i

= un

i − ∆t ∆x

ˆ fi+1/2 − ˆ fi−1/2

• flux : ˆ

fi+1/2 Conservation: ±

RDS: un+1

i

= un

i − ∆t ∆x

• φ−

i+1/2 + φ+ i−1/2

• .

Residuals φ−

i+1/2 = ˆ

fi+1/2 − f(ui), φ+

i−1/2 = f(ui) − ˆ

fi−1/2 Conservation : φ−

i+1/2 + φ+ i+1/2 = f(ui+1) − f(ui) =

xi+1

xi

∂f(u) ∂x dx

NON OSCILLATORY PROPERTIES

either : inputs in ˆ f,

• r tuning of numerical dissipation : symmetric TVD schemes
• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

PRINCIPLE FOR HIGHER ORDER

BACK TO 1D FOR 1 MORE SECOND.

1

∀ [xi, xi+1], φi+1/2(uh) = xi+1

xi

∂f ∂x (uh)dx

2

Distribution : φT(uh) = φ+

i+1/2(uh) + φ− i+1/2(uh)

Distribution coeff.s : φ±

i+1/2(uh) = ±ˆ

fi+1/2 ∓ f(ui)

3

Compute nodal values : solve algebraic system φ−

i+1/2 + φ+ i−1/2 = 0

∀i

i i + 1 φi+1/2 i i + 1 φ−

i+1/2

φ+

i+1/2

i i − 1 φ−

i+1/2

φ+

i−/2

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

SAME IN MULTID: REINTERPRETATION OF VF AS RD div f(u) = 0

i j k n+

ij

n−

ij

• j∈V(i)
• F(ui, uj, n+

ij ) + F(ui, uj, n− ij )

• = 0
• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

SAME IN MULTID: REINTERPRETATION OF VF AS RD div f(u) = 0

i j k n+

ij

n−

ik

• K∋i
• F(ui, uj, n+

ij ) + F(ui, uk, n− ik )

• = 0
• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

SAME IN MULTID: REINTERPRETATION OF VF AS RD div f(u) = 0

i j k n+

ij

n−

ik

• K∋i
• F(ui, uj, n+

ij ) + F(ui, uk, n− ik )

− f(ui) ·

• n+

ij + n− ik

• = 0
• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

SAME IN MULTID: REINTERPRETATION OF VF AS RD div f(u) = 0

i j k nij njk nki G

• K∋i

ΦK

i = 0

Φi := F(ui, uj, n+

ij ) + F(ui, uk, n− ik )

− f(ui) ·

• n+

ij + n− ik

• = F(ui, uj, n+

ij ) + F(ui, uk, n− ik )

− f(ui) · ni 2

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

SAME IN MULTID: REINTERPRETATION OF VF AS RD div f(u) = 0

1 2 3 nij njk nki G

Φi := F(ui, uj, n+

ij ) + F(ui, uk, n− ik )

− f(ui) · ni 2

• i∈K

Φi = −

• i

f(ui) · ni 2 =

• ∂K

fh · nd∂K

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

STREAMLINE DIFFUSION METHOD

A STANDARD REMARK

a(uh, vh) = −

• Ω

∇vh · f(uh) +

• K

hK

• K
• ∇uf(uh)∇vh
• τK
• ∇uf(uh)∇uh
• dx

+

• e∈E
• e

vhˆ fn(g, uh)dS.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

STREAMLINE DIFFUSION METHOD

A STANDARD REMARK

a(uh, vh) =

• K

K

∇vh · f(uh)dx + hK

• K
• ∇uf(uh)∇vh
• τK
• ∇uf(uh)∇uh
• dx
• +
• e∈E
• e

vh(ˆ fn(g, uh) − f(uh) · n)dS.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

PRINCIPLE FOR HIGHER ORDER

1

∀K ∈ Th compute : φK =

• ∂K

fh(uh) · n

2

Distribution : φK(uh) =

i∈K

φK

i

Distribution coeff.s : φK

i (uh) = sub-residuals

3

Compute nodal values : solve algebraic system

• K|i∈K

φK

i (uh) = 0,

∀ i ∈ Th

ΦT ΦT

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

PRINCIPLE FOR HIGHER ORDER

1

∀K ∈ Th compute : φK =

• ∂K

fh(uh) · n

2

Distribution : φK(uh) =

i∈K

φK

i

Distribution coeff.s : φK

i (uh) = sub-residuals

3

Compute nodal values : solve algebraic system

• K|i∈K

φK

i (uh) = 0,

∀ i ∈ Th un+1

i

= un

i − ωi K|i∈K

φK

i

• (uh)n
• ,

∀ i ∈ Th

ΦT ΦT

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

DESIGN PROPERTIES

STRUCTURAL CONDITIONS, BASIC PROPERTIES

Under which conditions on the φK

i s we get

Correct weak solutions (if convergent with h) Formal k th order of accuracy Monotonicity (discrete max principle) Convergence (with h, and with n !)

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

DESIGN PROPERTIES

STRUCTURAL CONDITIONS, BASIC PROPERTIES

Under which conditions on the φK

i s we get

Correct weak solutions (if convergent with h) Formal k th order of accuracy Monotonicity (discrete max principle) Convergence (with h, and with n !) Notation: DOF: σi or Mi or simply i

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

CONDITION 1 : CONSERVATION

CONSERVATION PRINCIPLE

If there is a fh, continuous approximation of f such that φK =

j∈K

φK

j =

• ∂K

fh · ˆ n example: fh = f(uh) or Lagrange interp. of f(ui) or . . .

BASIC RELATION

Scheme : for all dof i,

• K∋i

φK

i (uh) = 0

(1) introduce φGal,K

i

=

• K ψidiv f(uh)dx =
• K ∇ψi · f(uh)dx −
• ∂K ψif(uh) · ˆ

ndσ multiply (1) by test function v evaluated at i

0 =

• i

vi

K∋i

φK

i (uh)

• =
• K
• i∈K

viφK

i =

• K

i∈K

viφGal,K

i

+

• i∈K

vi

• φK

i − φGal,K i

• =
• Ω

∇vh · fh(uh)dx +

K

1 NK !

• i,j∈K

(vi − vj)

• φK

i − φGal,K i

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

CONDITION 2 : ACCURACY. uex,h interpolant of exact sol. assumed smooth Truncation error E(uex,h; v) :=

• i∈Th

vi

K| i∈K

φK

i (uex,h)

• GUIDING PRINCIPLE

E(uex,h; v) =

I ≡ EGalerkin

• Ω

∇vh · fh(uex,h) +

II

• K∈Th

1 NK!

• i,j∈K

(vi − vj)(φK

i − φGal i

)(uex,h)

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

CONDITION 2 : ACCURACY. uex,h interpolant of exact sol. assumed smooth Truncation error E(uex,h; v) :=

• i∈Th

vi

K| i∈K

φK

i (uex,h)

• GUIDING PRINCIPLE

E(uex,h; v) =

I ≡ EGalerkin

• Ω

∇vh · fh(uex,h) +

II

• K∈Th

1 NK!

• i,j∈K

(vi − vj)(φK

i − φGal i

)(uex,h)

KEY REMARK & FINAL RESULT

div f(w) = 0 = ⇒ φGal,K

i

(uex,h) =

• T ∇ψi · fh(uex,h)dx −
• ∂K ψifh(uex,h) · ˆ

ndσ = O(hk+d) Truncation error : |E(uex,h; v)| ≤ C′(Th, uex)∇v∞ hk+1 if (in d-D) |φK

i (uex,h)| ≤ C′′(Th, uex)hk+d= O(hk+d)

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

KEY REMARK

ASSUMING A REGULAR ENOUGHT EXACT SOLUTION

if fh(uex,h) − f(uex) = O(hk+1), for example Lagrange interpolation. φGal,K

i

(uex,h) =

• K

∇ψi · fh(uex,h)dx −

• ∂K

ψifh(uex,h) · ˆ ndσ =

• K

∇ψi ·

• fh(uex,h) − f(uex)
• dx −
• ∂K

ψi

• fh(uex,h) − f(uex
• · ˆ

ndσ = |K| × O(h−1) × O(hk+1) + |∂K| × O(h0) × O(hk+1) = O(hk+d)

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

CONDITION 2 : ACCURACY

LINEARITY (ACCURACY) PRESERVING SCHEMES

Since φK(wh) =

• ∂K f h(uh) · ˆ

ndl = O(hk+d) schemes for which φK

i = βK i φK

with βK

i uniformly bounded distribution coeff.s

are formally k + 1th order accurate (for k + 1th order spatial interpolation)

HOWEVER: GODUNOV’S THEOREM

The βK

i must depend on the solution : A scheme cannot be both high order accurate

and linear for a linear problem.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

CONDITION 2 : ACCURACY

LINEARITY (ACCURACY) PRESERVING SCHEMES

Since φK(wh) =

• ∂K f h(uh) · ˆ

ndl = O(hk+d) schemes for which φK

i = βK i φK

with βK

i uniformly bounded distribution coeff.s

are formally k + 1th order accurate (for k + 1th order spatial interpolation)

HOWEVER: GODUNOV’S THEOREM

The βK

i must depend on the solution : A scheme cannot be both high order accurate

and linear for a linear problem.

FUNDEMENTAL ASSUMPTION IN ALL THIS BUSINESS:

• K|i∈K

φK

i (uh) = 0,

∀ i ∈ Th has a unique solution i.e. un+1

i

= un

i − ωi

• K|i∈K

φK

i

• (uh)n
• ,

∀ i ∈ Th must converges

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

CONDITION 3: PRESERVATION OF MONOTONY + ACCURACY

SUMMARY

GOAL

Given any element K, a set of residuals {φM

i (uh)}i∈K, construct a set of residuals

{φH

i (uh)}i∈K with φH i (uex,h) = O(hk+d).

STRUIJS’ “LIMITER”

βH

i =

max(0, φM

i /φK)

• j∈K

max(0, φM

j /φK)

HIGH ORDER RESIDUALS

{φM

i (uh)}i∈K, i∈K φM i (uh) = φK

φH

i = βH i φK.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

CONDITION 3: PRESERVATION OF MONOTONY + ACCURACY

SUMMARY

GOAL

Given any element K, a set of residuals {φM

i (uh)}i∈K, construct a set of residuals

{φH

i (uh)}i∈K with φH i (uex,h) = O(hk+d).

STRUIJS’ “LIMITER”

βH

i =

max(0, φM

i /φK)

• j∈K

max(0, φM

j /φK)

HIGH ORDER RESIDUALS

{φM

i (uh)}i∈K, i∈K φM i (uh) = φK

φH

i = βH i φK+ε(uh)hK

• K
• ∇fu(uh) · ∇ϕi
• T
• ∇fu(uh) · ∇uh

dx

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

MOTIVATION FOR THIS TERM

SOLVE ∂u ∂x = 0 ON [0, 1]2

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

• 1
• 1
• 1
• 1
• 1
• 1
• 1
• 1
• 1
• 1

1 1 1 1 1 1 1 1 1 1 1 1 1

• 1
• 1
• 1
• 1
• 1
• 1
• 1
• 1
• 1
• 1
• 1
• 1

In both cases, φK = 0 : these are steady solutions when φH

i = βK i φK.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

MOTIVATION FOR THIS TERM

SOLVE ∂u ∂x = 0 ON [0, 1]2

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

• 1
• 1
• 1
• 1
• 1
• 1
• 1
• 1
• 1
• 1

1 1 1 1 1 1 1 1 1 1 1 1 1

• 1
• 1
• 1
• 1
• 1
• 1
• 1
• 1
• 1
• 1
• 1
• 1

In both cases, φK = 0 : these are steady solutions when φH

i = βK i φK.

Cure : φH,K

i

= βK

i φK −

→ βK

i φK+hK

• K
• ∇fu(uh) · ∇ϕi
• T
• ∇fu(uh) · ∇uh

dx

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

A CLEARER PICTURE ? : 1D

A LOCK AT THE ENTROPY PRODUCTION

∂f(u) ∂x = 0x ∈ [0, 1] u(0) = u0 u(1) = u1. assume f ′(u0) > 0 and f ′(u1) < 0

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

A CLEARER PICTURE ? : 1D

A LOCK AT THE ENTROPY PRODUCTION

∂f(u) ∂x = 0x ∈ [0, 1] u(0) = u0 u(1) = u1. assume f ′(u0) > 0 and f ′(u1) < 0 so that the solution is u = u0.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

0 = x0 < x1 < . . . < xn−1, xn = 1. elements: Ki+1/2 = [xi, xi+1]. Assume P1.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

0 = x0 < x1 < . . . < xn−1, xn = 1. elements: Ki+1/2 = [xi, xi+1]. Assume P1. The total residual: ΦKi +1/2 = f(ui+1) − f(ui)

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

0 = x0 < x1 < . . . < xn−1, xn = 1. elements: Ki+1/2 = [xi, xi+1]. Assume P1. The total residual: ΦKi +1/2 = f(ui+1) − f(ui) Φ

Ki+1/2 σ

= β

Ki+1/2 σ

• f(ui+1 − f(ui)
• . γ

Ki+1/2 i+1

:= β

Ki+1/2 σ

− 1

2

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

0 = x0 < x1 < . . . < xn−1, xn = 1. elements: Ki+1/2 = [xi, xi+1]. Assume P1. The total residual: ΦKi +1/2 = f(ui+1) − f(ui) Φ

Ki+1/2 σ

= β

Ki+1/2 σ

• f(ui+1 − f(ui)
• . γ

Ki+1/2 i+1

:= β

Ki+1/2 σ

− 1

2

Entropy balance for U(u) = 1

2u2:

E =

N−1

• i=0

ui

• β

Ki−1/2 i

• f(ui+1) − f(ui)
• + β

Ki+1/2 i

• f(ui+1) − f(ui)
• =

1 uh ∂f ∂x (uh)dx +

N−1

• i=0
• γ

Ki+1/2 i

ui + γ

Ki+1/2 i+1

ui+1/2

• f(ui+1) − f(ui)]big)

= 1 uh ∂f ∂x (uh)dx +

N−1

• i=0

γ

Ki+1/2 i+1

(f(ui+1) − f(ui))(ui+1 − ui).

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

0 = x0 < x1 < . . . < xn−1, xn = 1. elements: Ki+1/2 = [xi, xi+1]. Assume P1. The total residual: ΦKi +1/2 = f(ui+1) − f(ui) Φ

Ki+1/2 σ

= β

Ki+1/2 σ

• f(ui+1 − f(ui)
• . γ

Ki+1/2 i+1

:= β

Ki+1/2 σ

− 1

2

Entropy balance for U(u) = 1

2u2:

E = 1 uh ∂f ∂x (uh)dx +

N−1

• i=0

γ

Ki+1/2 i+1

(f(ui+1) − f(ui))(ui+1 − ui). Dissipative scheme: γ

Ki+1/2 i+1

(f(ui+1) − f(ui))(ui+1 − ui) ≥ 0 with a strict inequality for at least one interval, i.e. γ

Ki+1/2 i+1

f(ui+1) − f(ui) ui+1 − ui ≥ 0.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

0 = x0 < x1 < . . . < xn−1, xn = 1. elements: Ki+1/2 = [xi, xi+1]. Assume P1. The total residual: ΦKi +1/2 = f(ui+1) − f(ui) Φ

Ki+1/2 σ

= β

Ki+1/2 σ

• f(ui+1 − f(ui)
• . γ

Ki+1/2 i+1

:= β

Ki+1/2 σ

− 1

2

Entropy balance for U(u) = 1

2u2:

E = 1 uh ∂f ∂x (uh)dx +

N−1

• i=0

γ

Ki+1/2 i+1

(f(ui+1) − f(ui))(ui+1 − ui). Dissipative scheme: γ

Ki+1/2 i+1

(f(ui+1) − f(ui))(ui+1 − ui) ≥ 0 with a strict inequality for at least one interval, i.e. γ

Ki+1/2 i+1

f(ui+1) − f(ui) ui+1 − ui ≥ 0. The evaluation of β

Ki+1/2 σ

: having an L∞ stable scheme, only . . .

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

E = 1 uh ∂f ∂x (uh)dx +

N−1

• i=0

γ

Ki+1/2 i+1

(f(ui+1) − f(ui))(ui+1 − ui). Adding the streamline term, i.e. θ(ui+1 − ui) xi+1

xi

T ∂f ∂u 2 ∂ϕσ ∂x dx = (ui+1 − ui)

• ∂f

∂u

• ϕσ(xi+1) − ϕσ(xi))

will modify the entropy into E = 1 uh ∂f ∂x (uh)dx +

N−1

• i=0
• γ

Ki+1/2 i+1

f(ui+1) − f(ui) ui+1 − ui + θ

• ∂f

∂u

• )(ui+1 − ui)2

and E ≤ 1

0 uh ∂f

∂x (uh)dx provided that θ ≥ 1.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

EXAMPLES OF MONOTONE SCHEMES

MONOTONE SCHEMES : THE RUSANOV SCHEME (LOCAL LAX FRIEDRICHS)

Choice of Rusanov : not essential at all !

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

EXAMPLES OF MONOTONE SCHEMES

MONOTONE SCHEMES : THE RUSANOV SCHEME (LOCAL LAX FRIEDRICHS)

Centered linear first order distribution : φRv

i

= 1 K φK + α K

• j∈K

j=i

(ui − uj), α ≥ max

j∈K

• K

∇uf(uh) · ∇ψj

• K number of DoF per element

ψj Lagrange basis fcn. relative to node j

WHY THIS SCHEME ?

1

The Rv scheme is cheap and has general formulation

2

The Rv scheme is monotone and energy stable in the P1 case.

3

By far one of the most dissipative ones

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

NUMERICAL EXAMPLE : ROTATION

u v

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

NUMERICAL EXAMPLE : ROTATION

u v

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

GRID CONVERGENCE sum/TP1.pdfsum/TP1.pngsum/TP1.jpg sum/TP2.pdfsum h ǫL2(P1) 1/25 0.50493E-02 1/50 0.14684E-02 1/75 0.74684E-03 1/100 0.41019E-03 Ols

L2 =1.790

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

NUMERICAL EXAMPLE : BURGER’S EQ.N

∇ · u2 2 , u

• = 0

1 1 Shock Expansion Fan 1.5 −0.5 u = −0.5 u = 1.5

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

NUMERICAL EXAMPLE : BURGER’S EQ.N

LxF+PSI+Filter scheme, P 1 interpolation LxF+PSI+filter scheme, P 2 interpolation

Shock captured in 1 or 2 cells

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

FINAL REMARK The scheme is L2 stable: one can prove (linear case) error estimates similar to those of the stream-line diffusion method ||u − uex||graph norm ≤ Chk+1/2.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

FINAL REMARK The scheme is L2 stable: one can prove (linear case) error estimates similar to those of the stream-line diffusion method ||u − uex||graph norm ≤ Chk+1/2. The scheme is no longer (formaly) L∞ stable, in practice it does.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

FINAL REMARK The scheme is L2 stable: one can prove (linear case) error estimates similar to those of the stream-line diffusion method ||u − uex||graph norm ≤ Chk+1/2. The scheme is no longer (formaly) L∞ stable, in practice it does. It is however possible to construct a refined analysis, and design better limiters so that one can have both L2 + L∞, at least for triangles and second order accuracy. (work in progress)

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

ALGORITHM The scheme consists in 4 steps :

1

Evaluate the total residual, local (continuous interpolant)

2

Evaluate monotone residual (Rusanov) : local,

3

Evaluate high order residual : local

4

Gather residual : indirections, importance of good numering of the degrees of freedom The scheme is local and easy to parallelise

SOLUTION METHOD

Jacobian free + LUSGS-ILU

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

ALGORITHMIC COST loop on elements, no loop on faces (except boundary faces)

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

ALGORITHMIC COST loop on elements, no loop on faces (except boundary faces) Evaluation gradient: done before looping, small and local linear systems (precomputed and stored inverses)

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

ALGORITHMIC COST loop on elements, no loop on faces (except boundary faces) Evaluation gradient: done before looping, small and local linear systems (precomputed and stored inverses) Evaluation of total residual:

• ∂K f h · n f h interpolated by polynomial of degree p,

exact quadrature: degre p only

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

ALGORITHMIC COST loop on elements, no loop on faces (except boundary faces) Evaluation gradient: done before looping, small and local linear systems (precomputed and stored inverses) Evaluation of total residual:

• ∂K f h · n f h interpolated by polynomial of degree p,

exact quadrature: degre p only Evaluation of matrices β: done via local eigenstructure

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

ALGORITHMIC COST loop on elements, no loop on faces (except boundary faces) Evaluation gradient: done before looping, small and local linear systems (precomputed and stored inverses) Evaluation of total residual:

• ∂K f h · n f h interpolated by polynomial of degree p,

exact quadrature: degre p only Evaluation of matrices β: done via local eigenstructure Evaluation of streamline term:

• K

(∇uf(uh)·∇φσ)τ(∇uf(uh)·∇uh) ≈ |K|

• quad

ωquad(∇uf(uh)·∇φ)[xq]τ(∇uf(uh)·∇uh)[xq quadratic form

• σ

uσStream(σ) =

• quad

(∇uf(uh) · ∇uh)[xq]τ(∇uf(uh) · ∇uh)[xq] := Q

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

ALGORITHMIC COST loop on elements, no loop on faces (except boundary faces) Evaluation gradient: done before looping, small and local linear systems (precomputed and stored inverses) Evaluation of total residual:

• ∂K f h · n f h interpolated by polynomial of degree p,

exact quadrature: degre p only Evaluation of matrices β: done via local eigenstructure Evaluation of streamline term:

• K

(∇uf(uh)·∇φσ)τ(∇uf(uh)·∇uh) ≈ |K|

• quad

ωquad(∇uf(uh)·∇φ)[xq]τ(∇uf(uh)·∇uh)[xq quadratic form

• σ

uσStream(σ) =

• quad

(∇uf(uh) · ∇uh)[xq]τ(∇uf(uh) · ∇uh)[xq] := Q need to be disspiative : Q ≥ 0 and Q = 0 must imply (∇uf(uh) · ∇uh = 0 ωquad = 1/#dofs, and much less quad points (take some of the Lagrange points

• nly)
• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

ALGORITHMIC COST loop on elements, no loop on faces (except boundary faces) Evaluation gradient: done before looping, small and local linear systems (precomputed and stored inverses) Evaluation of total residual:

• ∂K f h · n f h interpolated by polynomial of degree p,

exact quadrature: degre p only Evaluation of matrices β: done via local eigenstructure Evaluation of streamline term:

• K

(∇uf(uh)·∇φσ)τ(∇uf(uh)·∇uh) ≈ |K|

• quad

ωquad(∇uf(uh)·∇φ)[xq]τ(∇uf(uh)·∇uh)[xq quadratic form

• σ

uσStream(σ) =

• quad

(∇uf(uh) · ∇uh)[xq]τ(∇uf(uh) · ∇uh)[xq] := Q need to be disspiative : Q ≥ 0 and Q = 0 must imply (∇uf(uh) · ∇uh = 0 ωquad = 1/#dofs, and much less quad points (take some of the Lagrange points

• nly)

Indirections: evaluation of

K∋σ ΦK σ

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

OVERVIEW

1 MOTIVATING EXAMPLE 2 RESIDUAL DISTRIBUTION SCHEMES (RDS) 3 THE VISCOUS CASE 4 APPLICATION TO NS EQUATIONS 5 CONCLUSION-PERSPECTIVES

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

RD WITH VISCOUS TERMS: WHAT ARE THE PROBLEMS? div fa(u) − div (K(u).∇u) = div

• fa(u) − K(u).∇u
• = 0

Accuracy: coupling of convection and diffusion: one single operator Total residual: ΦK(uh) =

• ∂K
• fa(u) − K(uh)∇uh) · nd∂K.
• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

RD WITH VISCOUS TERMS: WHAT ARE THE PROBLEMS? div fa(u) − div (K(u).∇u) = div

• fa(u) − K(u).∇u
• = 0

Accuracy: coupling of convection and diffusion: one single operator Total residual: ΦK(uh) =

• ∂K
• fa(u) − K(uh)∇uh) · nd∂K.

Major issue: ∇uh not single valued on edges.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

POTENTIAL SOLUTIONS

1

Reconstruct the gradients using out-of-element information (Caraeni, circa 2000’) ΦK(uh) =

• ∂K
• fa(u) − K(uh)

∇uh) · nd∂K. Loss of stencil compactness

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

POTENTIAL SOLUTIONS

1

Reconstruct the gradients using out-of-element information (Caraeni, circa 2000’)

2

Mixed type approach: H Nishikawa, NIA. u and q = ∇u seen as independant variables ∂u ∂τ + div fa(u) − div

• K(u)q
• = 0

∂q ∂τ + 1

T

• ∇u − q
• = 0

Hyperbolic system, explicit eigenvalues (0 is multiple eigenvalue)

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

POTENTIAL SOLUTIONS

1

Reconstruct the gradients using out-of-element information (Caraeni, circa 2000’)

2

Mixed type approach: H Nishikawa, NIA. u and q = ∇u seen as independant variables ∂u ∂τ + div fa(u) − div

• K(u)q
• = 0

∂q ∂τ + 1

T

• ∇u − q
• = 0

Hyperbolic system, explicit eigenvalues (0 is multiple eigenvalue) double the storage (for scalar), not completely clear for NS

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

POTENTIAL SOLUTIONS

1

Reconstruct the gradients using out-of-element information (Caraeni, circa 2000’)

2

Mixed type approach: H Nishikawa, NIA.

3

Gradient recovery and ∇uh globaly continuous approximation and ΦK(uh) =

• ∂K
• fa(u) − K(uh)

∇uh) · nd∂K.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

GRADIENT RECOVERY

IDEA

Obtained from super-convergent patch recovery introduced by O. C. Zienkiewicz and J. Z. Zhu, Int. J. Numer. Meth. Eng., 33, 1992 least square: the gradient are approximated with the same order as the unknowns.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

ZZ RECOVERY Assume that the numerical solution uh of the problem is known at each DOF of the grid to the k+1-th order of accuracy.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

ZZ RECOVERY Assume that the numerical solution uh of the problem is known at each DOF of the grid to the k+1-th order of accuracy. Aim : obtain the values of the solution gradient, ∇uh at all the DOFs with same

• rder of accuracy.
• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

ZZ RECOVERY Assume that the numerical solution uh of the problem is known at each DOF of the grid to the k+1-th order of accuracy. Aim : obtain the values of the solution gradient, ∇uh at all the DOFs with same

• rder of accuracy.

The components of the recovered gradient, at the generic DOF i, are written in a polynomial form as follows

• ∂uh

∂x

• i

= pKax and

• ∂uh

∂y

• i

= pKay, with pK(x) = (1, x, y, x2, . . . , xk+1, xky, . . . , y k+1), ax = (ax1, ax2, . . . , axm) and ay = (ay1, ay2, . . . , aym).

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

ZZ RECOVERY Assume that the numerical solution uh of the problem is known at each DOF of the grid to the k+1-th order of accuracy. Aim : obtain the values of the solution gradient, ∇uh at all the DOFs with same

• rder of accuracy.

The components of the recovered gradient, at the generic DOF i, are written in a polynomial form as follows

• ∂uh

∂x

• i

= pKax and

• ∂uh

∂y

• i

= pKay, with pK(x) = (1, x, y, x2, . . . , xk+1, xky, . . . , y k+1), ax = (ax1, ax2, . . . , axm) and ay = (ay1, ay2, . . . , aym). Assuming Ns sampling points, ξℓ, ℓ = 1 . . . Ni

s, are available for each DOF i,

minimize Fx =

Ni

s

• k=1

∂uh ∂x (ξk) − pK

k ax

2 and Fy =

Ni

s

• k=1

∂uh ∂y (ξk) − pK

k ay

2 , with pk = p(ξk).

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

SAMPLING RECOVERY Q1 P1 Q2 P2

• : the patch assembly points
• : where the gradient is

recovered △ : super-convergent sampling points.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

RD SCHEMES FOR VISCOUS PROBLEMS Linear scheme (Ni-LW) Φe

i = Φe

Ne

dof

+

• Ωe

A ·∇ψi Ξ

• A · ∇uh − ∇·
• K∇uh

dΩ +

• Ωe

K ∇ψi ·

• ∇uh −

∇uh

• dΩ,

Non-linear scheme ˜ Φe

i = Φe

Ne

dof

+ 1 Ne

dof

αe

j∈Σe

h

j=i

• ui − uj),

ˆ Φe

i =βe i Φe

Φe

i = ˆ

Φe

i + ε e h (uh)

• Ωe
• A·∇ψi − ∇·
• K∇ψi
• Ξ
• A · ∇uh − ∇·
• K

∇uh dΩ +

• Ωe

K ∇ψi ·

• ∇uh −

∇uh

• dΩ.
• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

RD SCHEMES FOR VISCOUS PROBLEMS Linear scheme (Ni-LW) Φe

i = Φe

Ne

dof

+

• Ωe

A ·∇ψi Ξ

• A · ∇uh − ∇·
• K∇uh

dΩ +

• Ωe

K ∇ψi ·

• ∇uh −

∇uh

• dΩ,

Non-linear scheme ˜ Φe

i = Φe

Ne

dof

+ 1 Ne

dof

αe

j∈Σe

h

j=i

• ui − uj),

ˆ Φe

i =βe i Φe

Φe

i = ˆ

Φe

i + ε e h (uh)

• Ωe
• A·∇ψi − ∇·
• K∇ψi
• Ξ
• A · ∇uh − ∇·
• K

∇uh dΩ +

• Ωe

K ∇ψi ·

• ∇uh −

∇uh

• dΩ.
• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

LINEAR ADVECTION-DIFFUSION EQUATION a·∇u = ν div

• ∇u
• ,
• n

Ω = [0, 1]2, the exact solution of the problem reads u = − cos(2πη) exp   ξ

• 1 −

√ 1 + 16π2ν2

• 2ν

  , with η = ayx − axy and ξ = axx + ayy. Here a = (0, 1)K and ν = 0.01

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

LINEAR ADVECTION-DIFFUSION EQUATION a·∇u = ν div

• ∇u
• ,
• n

Ω = [0, 1]2, the exact solution of the problem reads u = − cos(2πη) exp   ξ

• 1 −

√ 1 + 16π2ν2

• 2ν

  , with η = ayx − axy and ξ = axx + ayy. Here a = (0, 1)K and ν = 0.01 In this case Re ≈ 1, worst case scenario

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

LINEAR ADVECTION-DIFFUSION EQUATION

log(h) log(L

2 error)

• 2.2
• 2
• 1.8
• 1.6
• 1.4
• 1.2
• 5
• 4
• 3
• 2

Weighted area (2.19) L2-Projection (2.07) Least square (2.79) SPR-ZZ (3.06) log(h) log(L

2 error)

• 2.2
• 2
• 1.8
• 1.6
• 1.4
• 1.2
• 3.5
• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

Weighted area (1.01) L2-Projection (1.46) Least square (2.14) SPR-ZZ (2.24) log(h) log(L

2 error)

• 2.2
• 2
• 1.8
• 1.6
• 1.4
• 1.2
• 4.5
• 4
• 3.5
• 3
• 2.5
• 2
• 1.5
• 1

Weighted area (1.95) L2-Projection (1.45) Least square (2.05) SPR-ZZ (2.80)

L2 error in the solution of the linear advection-diffusion problem on triangular girds with quadratic elements. Error of the solution (first column), error of the x-component of the gradient (second column) error of the y-component of the gradient (third column).

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

VISCOUS BURGER EQUATION ∂ ∂x u2 2

• + ∂u

∂y = ν ∂2u ∂x2 ,

• n Ω = [0, 1]2,

with the following exact solution u = 2νπ exp(−νyπ2) sin(πx) a + exp(−νyπ2) cos(πx), with a > 1.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

VISCOUS BURGER EQUATION

log(h) log(L

2 error)

• 2
• 1.5
• 1
• 4.5
• 4
• 3.5
• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

Weighted area (1.94) SPR-ZZ (1.99) Weighted area (1.68) SPR-ZZ (2.84) log(h) log(L

2 error)

• 2
• 1.5
• 1
• 5
• 4
• 3
• 2
• 1

Weighted area (2.13) SPR-ZZ (2.16) Weighted area (1.89) SPR-ZZ (3.20) log(h) log(L

2 error)

• 2
• 1.5
• 1
• 3.5
• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

Weighted area (1.24) SPR-ZZ (1.14) Weighted area (1.45) SPR-ZZ (2.26)

L2 error in the solution of the viscous Burger problem on triangular girds with linear (dashed lines) and quadratic (solid lines) elements. Error of the solution (first column), error of the x-component of the gradient (second column) error of the y-component of the gradient (third column).

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

ANISOTROPIC PURE DIFFUSION −div

• K∇u
• = 0,
• n Ω = [0, 1]2,

with K = 1 δ

• ,

the problem has the following exact solution u = sin(2πx) e−2πy√

1/δ ,

and in the numerical simulations δ = 103.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

ANISOTROPIC DIFFUSION

log(h) log(L

2 error)

• 2
• 1.5
• 1
• 5
• 4.5
• 4
• 3.5
• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

Weighted area (1.89) SPR-ZZ (1.93) Weighted area (2.36) SPR-ZZ (2.73) log(h) log(L

2 error)

• 2
• 1.5
• 1
• 4
• 3
• 2
• 1

Weighted area (1.47) SPR-ZZ (1.47) Weighted area (1.08) SPR-ZZ (2.30) log(h) log(L

2 error)

• 2
• 1.5
• 1
• 3.5
• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

0.5 1 Weighted area (1.47) SPR-ZZ (1.65) Weighted area (1.17) SPR-ZZ (2.54)

L2 error in the solution of the anisotropic diffusion problem on triangular girds with linear (dashed lines) and quadratic (solid lines) elements. Error of the solution (first column), error of the x-component of the gradient (second column) error of the y-component of the gradient (third column).

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

ANISOTROPIC DIFFUSION

log(h) log(L

2 error)

• 2
• 1.5
• 1
• 5.5
• 5
• 4.5
• 4
• 3.5
• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

Weighted area (2.13) SPR-ZZ (2.13) Weighted area (3.28) SPR-ZZ (3.11) log(h) log(L

2 error)

• 2
• 1.5
• 1
• 5
• 4
• 3
• 2
• 1

Weighted area (2.07) SPR-ZZ (1.93) Weighted area (2.16) SPR-ZZ (3.27) log(h) log(L

2 error)

• 2.5
• 2
• 1.5
• 1
• 5
• 4.5
• 4
• 3.5
• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

0.5

Weighted area (2.12) SPR-ZZ (2.00) Weighted area (3.45) SPR-ZZ (3.16)

L2 error in the solution of the anisotropic diffusion problem on uniform, structured grid

• f quadrangles, with linear (dashed lines) and quadratic (solid lines) elements. Error of

the solution (first column), error of the x-component of the gradient (second column) error of the y-component of the gradient (third column). .

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

OVERVIEW

1 MOTIVATING EXAMPLE 2 RESIDUAL DISTRIBUTION SCHEMES (RDS) 3 THE VISCOUS CASE 4 APPLICATION TO NS EQUATIONS 5 CONCLUSION-PERSPECTIVES

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

MANUFACTURED SOLNS: ACCURACY TEST

SOL MADE OF TRIGS FUNCTIONS

Re Observed Order u

10 10

1

10

2

10

3

10

4

1.5 2 2.5 3 3.5 2nd Order GG 2nd Order SPR-ZZ 3rd Order GG 3rd Order SPR-ZZ

Re Observed Order Du

10 10

1

10

2

10

3

10

4

1 1.5 2 2.5 3 2nd Order GG 2nd Order SPR-ZZ 3rd Order GG 3rd Order SPR-ZZ

Re Observed Order u

10 10

1

10

2

10

3

10

4

1 1.5 2 2.5 3 3.5 2nd Order GG 2nd Order SPR-ZZ 3rd Order GG 3rd Order SPR-ZZ

Re Observed Order Du

10 10

1

10

2

10

3

10

4

1 1.5 2 2.5 3 2nd Order GG 2nd Order SPR-ZZ 3rd Order GG 3rd Order SPR-ZZ

Observed order Top: linear scheme; Bottom: non linear scheme Left: error on solution; Right: error on gradients

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

NACA, M=0.5, RE=5000

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

NACA, M=0.5, RE=5000

M 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 M 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Mach Number contours (top) and streamlines near the trailing edge (bottom) for the second (left) and third (right) order linear scheme.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

NACA, M=0.5, RE=5000

x/c Cp 0.2 0.4 0.6 0.8 1

• 0.5

0.5 1 P1 P2 x/c Cp 0.05 0.1 0.15 0.2

• 0.4
• 0.3
• 0.2
• 0.1

P1 P2

Pressure coefficient along the whole NACA-0012 airfoil for the second and third order simulations,with the same number of DOFs.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

NACA, M=0.5, RE=5000

x/c Cf 0.2 0.4 0.6 0.8 1

• 0.1
• 0.05

0.05 0.1 P1 P2 x/c Cf 0.7 0.75 0.8 0.85 0.9 0.95

• 0.002
• 0.001

0.001 0.002 P1 P2

Skin friction coefficient along the whole NACA-0012 airfoil for the second and third

• rder simulations, with the same number of DOFs.
• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

LAMINAR FLOW AROUND A DELTA WING

M=0.5, RE=4000, AOA=12.5◦

• A

A

t c

A − A

Λ y x σ z x

FIGURE : Left: Bottom and side views of the model of the delta wing: Λ = 75◦, σ = 60◦ and t/c = 0.024. Right: a coarse mesh of tetrahedra use for the simulations.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

LAMINAR FLOW AROUND A DELTA WING

M=0.5, RE=4000, AOA=12.5◦

Iterations L2 Residual 500 1000 10

• 10

10

• 8

10

• 6

10

• 4

10

• 2

10 ρ ρu ρv ρw ρe

t

FIGURE : Left: Streamlines and slices of Mach number contours along and behind the delta wing, for a third order simulation on a fine grid. Right: Convergence history for the order sequencing (second and third order).

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

LAMINAR FLOW AROUND A DELTA WING

M=0.5, RE=4000, AOA=12.5◦

• N. Dof
• 1/3

CD 0.01 0.02 0.03 0.04 0.17 0.18 0.19 0.2

P1 P2 Reference value

• N. Dof
• 1/3

CL 0.01 0.02 0.03 0.04 0.35 0.4 0.45

P1 P2 Reference value

FIGURE : Drag (left) and lift (right) coefficients as function of DOFs for the delta wing simulation, with linear and quadratic elements.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

LAMINAR FLOW AROUND A DELTA WING

M=0.5, RE=4000, AOA=12.5◦

• N. Dof
• 1/3

CD error 10

• 3

10

• 2

10

• 1

10

• 3

10

• 2

P1 P2

• N. Dof
• 1/3

CL error 10

• 3

10

• 2

10

• 1

10

• 2

10

• 1

P1 P2

FIGURE : Errors, respect to the reference values, of the drag (left) and lift (right) coefficients as function of DOFs for the delta wing simulation, with linear and quadratic elements.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

SHOCK-WAVE/LAMINAR BOUNDARY LAYER INTERACTION

M=2.15, θ = 30, 8◦, Re = 105

• M∞

θs

Expansion fan Compression waves Incident shock

Compression waves

Boundary layer Leading edge shock Reflected shock

• Boundary layer

Adiabatic wall Outflow Inflows Inflow∞ θs y x Incident shock Symmetry

FIGURE : Schematic representation of the waves pattern (left) and computational domain with boundary conditions (right) for the shock-wave/boundary layer interaction problem

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

SHOCK-WAVE/LAMINAR BOUNDARY LAYER INTERACTION

M=2.15, θ = 30, 8◦, Re = 105

x y

• 0.2

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.3 0.6 0.9 p: 0.68 0.74 0.8 0.86 0.92 0.98 1.04 1.1

(a)

x y

0.6 0.8 1 1.2 0.02 0.04 0.06 0.08 0.1

(b)

FIGURE : Left: contours of the pressure obtained with the third order scheme for the shock/boundary layer interaction. Right: zoom of the solution near the impinging point of the shock with the boundary layer, streamlines are also reported to show the separation bubble.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives x P

0.5 1 1.5 2 0.8 0.9 1 1.1 2nd Order 3rd Order 2nd Order Fine

(a)

x Cf

0.5 1 1.5 2 0.003 0.006 0.009 2nd Order 3rd Order 2nd Order Fine

(b)

FIGURE : Pressure (a) and skin friction (b) profiles along the flat plate for the shock/boundary layer interaction problem.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

RAE2822 AIRFOIL, TURBULENT

M=0.734, RE=6.5 106, AOA=2.79◦

Mach Pressure

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

RAE2822 AIRFOIL, TURBULENT

M=0.734, RE=6.5 106, AOA=2.79◦

Lift Drag

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

DELTA WING, TURBULENT

M=0.734, RE=6.5 106, AOA=2.79◦

Mesh Pressure

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

L1T2 AIRFOIL, TURBULENT

M=0.197, RE=3.52 106, AOA=4.01◦

Mesh Mach

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

L1T2 AIRFOIL, TURBULENT

M=0.197, RE=3.52 106, AOA=4.01◦

Mach convergence history

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

OVERVIEW

1 MOTIVATING EXAMPLE 2 RESIDUAL DISTRIBUTION SCHEMES (RDS) 3 THE VISCOUS CASE 4 APPLICATION TO NS EQUATIONS 5 CONCLUSION-PERSPECTIVES

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

CONCLUSIONS

A class of FEM-like schemes, on unstructured meshes, for steady convection-diffusion problems High order, parameter free Excellent non oscillatory properties, L∞ and L2 stability. Convection diffusion uniformly accurate for the whole range of Pe/Re number. Extension to systems with similar properties

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

CONCLUSIONS

A class of FEM-like schemes, on unstructured meshes, for steady convection-diffusion problems High order, parameter free Excellent non oscillatory properties, L∞ and L2 stability. Convection diffusion uniformly accurate for the whole range of Pe/Re number. Extension to systems with similar properties

PERSPECTIVES

Extension for unsteady problems (done for 2nd order, triangles/tets, in progress) h − p adaptation, further analysis: curved meshes (in progress) further mathematical analysis

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

ADVECTION, SPEED=1, T=150

0,2 0,4 0,6 0,8 1

• 0,5

0,5 1 1,5 Numeric Exact

Advection

50 cells, 100 dofs, T=150 0,5 0,55 0,6 0,65 0,7 0,75 0,5 1 1,5 Numeric Exact

Advection

50 cells, 100 dofs, T=150

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

ADVECTION, SPEED=1, T=150

0,2 0,4 0,6 0,8 1

• 0,5

0,5 1 1,5 Numerics Exact

Advection

100 cells, 200 dofs, T=150 0,6 0,5 1 1,5 Numerics Exact

Advection

100 cells, 200 dofs, T=150

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

CONDITION 3: PRESERVATION OF MONOTONY + ACCURACY

GOAL

Given any element K, a set of residuals {φM

i (uh)}i∈K, construct a set of residuals

{φH

i (uh)}i∈K with φH i (uex,h) = O(hk+d).

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

IDEA: GEOMETRICAL INTERPRETATION

LOW ORDER SCHEME

Set xi = (ΦK

j )L

ΦK . Conservation : x1 + x2 + x3 = 1

HIGH ORDER SCHEME

(ΦK

j )H = βK j ΦK so that

βK

j = (ΦK j )H

ΦK Conservation : β1 + β2 + β3 = 1.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

IDEA: GEOMETRICAL INTERPRETATION

PRESERVATION OF MONOTONY

First order scheme (ΦK

j )L

(ΦK

j )L =

• j∈K

cK

ij (ui − uj),

cK

ij ≥ 0

then (ΦK

j )H = (ΦK j )

(ΦK

j )L (ΦK j )L =

• j∈K

(ΦK

j )

(ΦK

j )L cK ij

• (ui − uj)

=

• j∈K
• cK

ij

⋆(ui − uj)

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

IDEA: GEOMETRICAL INTERPRETATION

PRESERVATION OF MONOTONY

First order scheme (ΦK

j )L

(ΦK

j )L =

• j∈K

cK

ij (ui − uj),

cK

ij ≥ 0

(ΦK

j )H =

• j∈K
• cK

ij

⋆(ui − uj),

• cK

ij

⋆ ≥ 0???

• cK

ij

⋆ ≥ 0 ⇐ ⇒ (ΦK

j )

ΦK × (ΦK

j )L

ΦK ≥ 0 i.e. xi × βi ≥ 0 xi = (ΦK

j )L

ΦK , βK

j = (ΦK

j )H

ΦK

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

GEOMETRICAL INTERPRETATION: SECOND ORDER xi = (ΦK

j )L

ΦK , βK

j = (ΦK

j )H

ΦK

Conservation : x1 + x2 + x3 = 1 and β1 + β2 + β3 = 1, Preservation of monotony xi × βi ≥ 0 for i = 1, 2, 3

TRICK

See the xi’s and βi’s as barycentric coordinates. Take 3 points in R2 : A1 = (1, 0), A2 = (cos(120◦), sin(120◦)) and A3 = (cos(240◦, sin(240◦)). Low order identified to L = x1A1 + x2A2 + x3A3. High order identified to H = β1A1 + β2A2 + β3A3. Find a mapping (x1, x2, x3) → (β1, β2, β3) for which xiβi ≥ 0

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

ILLUSTRATION: xi × βi ≥ 0

L = (x1, x2, x3) A2 = (0, 1, 0) A1 = (1, 0, 0) A3 = (0, 0, 1) H = (β1, β2, β3)

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

ILLUSTRATION: xi × βi ≥ 0

H = (β1, β2, β3) A2 = (0, 1, 0) A1 = (1, 0, 0) A3 = (0, 0, 1) L = (x1, x2, x3) L = H

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

EXAMPLE : PSI, STRUIJS’ “LIMITER”

MAPPING: {(ΦK

j )L} −

→ {(ΦK

j )H}

βK

j =

• (ΦK

j )L/ΦK

+

• k∈K
• (ΦK

j )L/ΦK

+

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual

Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives

EXAMPLE : PSI, STRUIJS’ “LIMITER”

MAPPING: {(ΦK

j )L} −

→ {(ΦK

j )H}

βK

j =

• (ΦK

j )L/ΦK

+

• k∈K
• (ΦK

j )L/ΦK

+ never undefined because:

• k∈K
• (ΦK

j )L/ΦK

+ ≥ 1 because

k∈K

(ΦK

j )L/ΦK = 1.

• R. Abgrall

Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual