Gaining insight of accuracy enhancement through divided difference - - PowerPoint PPT Presentation

Gaining insight of accuracy enhancement through divided difference estimates

Jennifer K. Ryan

University of East Anglia, Norwich, UK Heinrich-Heine University, Düsseldorf, Germany

Advances in PDEs: Theory, Computation and Application to CFD 20-24 August 2018

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Joint work with Julia Docampo (MIT) and Xiong Meng (Harbin)

WHY ARE DIVIDED DIFFERENCES IMPORTANT?

➤ Usual Answer: ➤ Accuracy in derivative approximations. ➤ Other advantages: ➤ Gives insight into design of numerical schemes ➤ Accuracy enhancement: ➤ Tells us how to choose filter scalings ➤ Gives insight into reducing computational cost ➤ By using a reduced amount of information.

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ACCURACY ENHANCEMENT

OUTLINE

➤ Background ➤ Accuracy enhancement through post-processing: ➤ Smoothness-Increasing Accuracy-Conserving (SIAC)

filter

➤ DG and divided difference error estimates ➤ Linear divided difference estimates ➤ Compact multi-dimensional SIAC (Line SIAC) ➤ Nonlinear extension ➤ Summary and Future work

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Constructing accurate post-processing techniques: Smoothness-Increasing Accuracy-Conserving (SIAC) Filters

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EXTRACTING EXTRA ACCURACY

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Post-Processed Solution Post-Processor Numerical Information

Consistency requirements (Moment conditions/ polynomial reproduction)

➤ Geometry of the information ➤ How the information was generated ➤ PDE ➤ Numerical Method

USEFUL REFERENCES

Basic properties of dg and beyond

1.J.H. Bramble and A.H. Schatz, ” Higher order local accuracy by averaging in the finite element method”, Mathematics of Computation, 31 (1977), pp.94–111. 2.B. Cockburn, M. Luskin, C.-W. Shu, and E. Suli, ” Enhanced accuracy by post-processing for finite element methods for hyperbolic equations”, Mathematics of Computation, 72 (2003), pp.577–606. 3.M.S. Mock and P .D. Lax, ”The computation of discontinuous solutions of linear hyperbolic equations”, Communications on Pure and Applied Mathematics, 31 (1978), pp.423–430. (Moment preservation provided data is suitably preprocessed). 4.V. Thomee, ” High order local approximations to derivatives in the finite element method”, Mathematics of Computation, 31 (1977), pp. 652–660. 5.D. Gottlieb and E. Tadmor, “Recovering pointwise values of discontinuous data within spectral accuracy”, Proceedings of U.S.-Israel Workshop, Progress in Scientific Computing, vol. 6, Birkhauser Boston Inc., 1985, pp. 357–375. (Spectrally accurate post-processor). 6.A.Majda and S. Osher, Propagation of error into regions of smoothness for accurate difference approximate solutions to hyperbolic equations, Comm. Pure Appl. Math. 30 (1977), 671–705.

➤ Assume our post-processed solution is defined

through a convolution:

➤ SIAC kernel:

SMOOTHNESS-INCREASING ACCURACY-CONSERVING (SIAC) FILTER

Chosen to maintain

• Consistency
• 2r moments

Function chosen based on

• Given information
• Computational efficiency

Data∗(x) = Z

R

K(y − x)Data(y) dy

K(x) =

r

X

γ=−r

cγ ψ(m+1)(x − γ)

−2 −1 1 2 1

ψ(1) ψ(2) ψ(3)

➤ Choosing j(m+1) to be a B-spline of order m+1: ➤ B-spline properties: ➤ Compact support ➤ Smoothness of m - 1. ➤ Derivatives can be written as divided

differences

SIAC KERNEL

K(x) =

r

X

γ=−r

cγ ψ(m+1)(x − γ)

ψ(1)(x) =χ[− 1

2 , 1 2] =

( 1, x ∈ ⇥ − 1

2, 1 2

⇤ 0, else ψ(m+1)(x) = 1 m ✓ x + m + 1 2 ◆ ψ(m) ✓ x + 1 2 ◆ + ✓m + 1 2 − x ◆ ψ(m) ✓ x − 1 2 ◆ , m ≥ 1.

SIAC KERNEL: LINEAR COMBINATION OF B-SPLINES

superconvergence extraction through siac filtering

r=m=2 Kernel:

• Preserves 4 moments
• C1 continuity
• Support of length 7

r=m=1 Kernel:

• Preserves 2 moments
• Continuous
• Support of length 5

SIAC KERNEL: FOURIER SPACE

➤ In physical space, the filter is ➤ In Fourier space this is:

ˆ K(k) = ✓sin(kπ) kπ ◆m+1 c0 + 2

r

X

γ=0

cγ cos(γkπ) ! K(x) =

r

X

γ=−r

cγψ(m+1)(x − γ)

Thomee, Math. Comp. (1977) Ji & Ryan, ICOSAHOM 2014 Proceedings

superconvergence extraction through siac filtering

SMOOTHNESS-INCREASING ACCURACY-CONSERVING (SIAC) FILTER

DG data accuracy: p+1 SIAC DG order: 2p+1

SIAC filtering allows:

• Global enhanced accuracy
• Global smoothness

Example: p=r=m=2

EXTRACTING EXTRA INFORMATION OUT OF DATA

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Post-Processed Solution Post-Processor Numerical Information

➤ Geometry of the information ➤ How the information is generated ➤ PDE ➤ Numerical Method

IMPORTANT COMPONENTS OF DG

➤ Approximation Space: Consists of piecewise polynomials ➤ Based on a Variational formulation ➤ Weak continuity is enforced through the numerical flux

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Z

τe

(uh)tvh(x) dx −

2

X

d=1

Z

τe

f(uh)(vh)xi(x) dx +

2

X

d=1

Z

∂τe

ˆ fiˆ nivh ds = 0

DISCONTINUOUS GALERKIN ERRORS:

➤ L2-Error Estimate: ➤ Negative-Order Norm Estimate: ➤ For linear hyperbolic equations

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ku uhk0  Chp+1|u0|Hp+1

k∂α

h (u uh)k−(p+1)  Ch2p+1|u0|Hp+2(DΩ1)

What extra information to we get from post-processing?

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SIAC FILTERED DG

SIAC filtered solution: SIAC filtered error:

u∗

h(x, t) = 1

H Z

R

K ✓x − y H ◆ uh(y, t) dy

(Uniform) k(u K(2p+1,p+1)

h

⇤ uh)(T)k0 Ch2p+1 (Non-uniform) k(u K(2p+1,p+1)

H

⇤ uh)(T)k0 Ch

2 3 (2p+1)

Bramble & Schatz, Math. Comp (1977) Cockburn, Luskin, Shu, & Süli, Math. Comp (2003) (❩L-inf estimates)❪ Ji, Xu, Ryan, Math. Comp. (2012)

H depends on mesh

SIAC FILTER: ERROR ESTIMATES

For the first term:

➤ Kernel reproduces polynomials of degree up to 2r. ➤ Equivalent to consistency and moment conditions:

ku u∗k0  ku u∗k0 + kKh ⇤ (u uh)k0

ku u∗k0  C∆x2r+1

Z

R

K(x − y)ym dy = xm, m = 0, 1, . . . , 2r. Z

R

K(x − y)dy = 1, Z

R

K(y)ym dy = 0, m = 1, . . . 2r.

SIAC FILTER: ERROR ESTIMATES

For the second term: where the negative-order norm is given by

kKh ⇤ (u uh)k0 C X

|α|≤m+1

kDα(u uh)k−(m+1) kvk−(m+1),Ω = sup

φ∈C∞

0 (Ω)

(v, φ) kφkm+1,Ω

SIAC FILTER: ERROR ESTIMATES

Second term (cont.)

Using properties of B-splines and convolution Properties of the scheme

kKH ⇤ (u uh)k0  X

|α|≤m+1

kDα(KH ⇤ (u uh))k0  X

|α|≤m+1

k(∂α

HKH) ⇤ (u uh)k0

 X

|α|≤m+1

kKH ⇤ ∂α

H(u uh)k0

 X

|α|≤m+1

kKHk1k∂α

H(u uh)k−(m+1)

SIAC FILTER: ERROR ESTIMATES

For linear hyperbolic equations:

(∂α

h (u − uh), φ)(T) =(∂α h (u − uh), φ)(0) −

Z T d dt(∂α

h uh, φ) dt

=(∂α

h (u − uh), φ)(0) −

Z T B @

N

X

j=1

[∂α

h uh]j+1/2

| {z }

jump in uh

(φ − vh)(x+

j+1/2)

1 C A dt

Initial projection interelement flux values

kvk−(m+1),Ω = sup

φ∈C∞

0 (Ω)

(v, φ) kφkm+1,Ω

SIAC FILTER: KEY INGREDIENTS FOR THE LINEAR ESTIMATE

• 1. The dual equation. For the equation ut + ux = 0, we require
• 2. Relation between derivatives and divided differences
• 3. DG divided difference estimate:

d dt(u, ϕ) = 0 ⇒ ϕt + ϕx = 0.

k∂α

h (u uh)k0  Chp+1

Dαψ(m+1)(x) = ∂α

Hψ(m+1−α)(x)

AIM

• 1. Exploiting the information to make the SIAC filter more

computationally efficient

• 2. Extension to nonlinear hyperbolic conservation laws

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Exploiting information for computational efficiency

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Julia Docampo Sánchez (MIT)

LINE SIAC FILTER FOR MULTI-DIMENSIONAL FILTERING

➤ Cartesian-aligned filter vs. Rotated filter

h is the uniform DG element size, H is the kernel scaling

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LINE SIAC FILTER FOR MULTI-DIMENSIONAL FILTERING

➤ Recall: For the estimates to work, we require:

• 1. Suitable dual equation
• 2. Relation between derivatives and divided differences
• 3. Order p+1 for the divided difference errors.

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D↵ψ(`)(x) = ∂↵

h ψ(`−↵)(x),

α = αx + αy

coordinate aligned

LINE SIAC FILTER FOR MULTI-DIMENSIONAL FILTERING

➤ For rotated SIAC need to define directional divided

differences.

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SIAC FILTER: DIRECTIONAL DIVIDED DIFFERENCE ESTIMATES

➤ Direction vector: u=(ux,uy). ➤ Scaled directional divided difference with respect to u: ➤ a-th directional divided difference:

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∂u,Hf(t) = 1 H ✓ f ✓ x + H 2 ux, y + H 2 uy ◆ − f ✓ x − H 2 ux, y − H 2 uy ◆◆ =∂ux,Hf ✓ x, y + H 2 uy ◆ + ∂uy,Hf ✓ x − H 2 ux, y ◆ . ∂α

u,Hf(x, y) = ∂u,H

• ∂α−1

u,H f(x, y)

• ,

α > 1.

SIAC FILTER: DIVIDED DIFFERENCE ESTIMATES

➤ We can relate the directional divided difference to the

coordinate aligned divided difference

➤ As long as q s 0, p/2, then we have superconvergence! ➤ Leads to a reduced error constant.

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D↵ψ(`)(t) = (cos θ)↵x(sin θ)↵y ∂↵

h ψ(`−↵)(x)

coordinate aligned

θ = arctan ✓∆y ∆x ◆

LINE SIAC FILTER FOR MULTI-DIMENSIONAL FILTERING

➤ Reducing the support to a line: axis-aligned vs. rotated

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Tensor Product SIAC filter Line SIAC filter

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DG Errors Tensor Product Filter Line Filter p/4 rotation Line Filter 3p/4 rotation

LINE SIAC FILTER FOR MULTI-DIMENSIONAL FILTERING

SISC (2017)

ut + ux + uy = 0, u(x, y, 0) = sin(x + y)

LINE SIAC FILTER FOR MULTI-DIMENSIONAL FILTERING

➤ Numerical test: 2D advection equation, u(x,u,0)=sin(x+y) ➤ Extra accuracy and error reduction!

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Unfiltered Line Filtering 2D Filter θ = 3π/4 θ = π/4 θ = 0 N L2-error Order L2-error Order L2-error Order L2-error P1 20 9.7e-03

• 1.5e-03
• 2.7e-03
• 1.6e-03

40 2.4e-03 2.02 1.9e-04 2.98 2.6e-04 3.33 2.0e-04 80 5.9e-04 2.01 2.4e-05 2.99 2.8e-05 3.21 P2 20 2.4e-04

• 1.5e-06
• 1.4e-04
• 6.1e-06

40 2.9e-05 3.01 4.7e-08 4.99 2.3e-06 5.91 1.2e-07 80 3.6e-06 3.01 1.5e-09 5.00 3.7e-08 5.95

• P3

20 4.5e-06

• 7.7e-10
• 1.6e-05
• 1.4e-07

40 2.8e-07 4.01 6.9e-12 6.79 6.9e-08 7.87 5.6e-10

LSIAC FILTER: SMOOTHNESS

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LSIAC FILTER: COMPUTATIONAL COST

➤ Total operations per point. ➤ Elapsed time per point.

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Filter Type Intersection Scans Integrals Quadrature Sums Line Filter 4 10 10 2D Rotated Filter 64 93 8649 2D No Rotation 64 63 3969

• No. of Splines and degree

Line Filter 2D Rotated Filter 2D No Rotation 3, 1 0.09 0.87 0.68 5, 2 0.35 3.49 2.60 7, 3 0.41 10.42 6.75

The nonlinear extension

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Xiong Meng (Harbin)

SIAC FILTER: CHALLENGES FOR NONLINEAR ERROR ESTIMATES

Consider Difficulties in extending the estimates:

• 1. Defining the dual equation
• 2. The DG divided difference estimate.

Because

ut + f(u)x = 0, (x, t) ∈ Ω × [0, T).

∂α

h f(uh) 6= f (∂α h uh)

• Num. Math. (2017)/IMAJNA (2018)

NONLINEAR DIVIDED DIFFERENCE ESTIMATE: DUAL EQUATION

For the dual equation, we take This gives where

∂α

h ϕt + f 0(u)∂α h ϕx =0,

(x, t) ∈ Ω × [0, T), ϕ(x, T) =Φ(x), x ∈ Ω

d dt(∂α

h u, ϕ) + F(u; ϕ) = 0,

F(u; ϕ) = (−1)α(f 0(u)u − f(u), ∂α

h ϕx).

NONLINEAR DG DIVIDED DIFFERENCE ESTIMATE

Consider ut + f(u)x = 0. The divided difference equation is The DG scheme is where

∂α

h ut + ∂α h f(u)x = 0

(∂α

h ut, v)j0 = Hj0(∂α h f(u)

| {z }

6=f(∂α

h u)

, v), ∀v ∈ P p(Ij0)

Hj(p, q) = (p, qx)j − (p−q−)j+1/2 + (p−q+)j−1/2

NONLINEAR DG DIVIDED DIFFERENCE ESTIMATE

Denote e = u - uh and a = 1. Then the error equation is Let e = (Pu - uh) + (u - Pu) = x + h, then where

(∂het, v) = H(∂h(f(u) − f(uh)), v)

H(∂h(f(u) − f(uh)), v) = H(∂h(f 0(u)ξ), v) | {z }

H(f 0(u(x+h/2))¯ ξ,¯ ξ)+H(∂hf 0(u)ξ,¯ ξ)

+ H(∂h(f 0(u)η), v) | {z }

projection terms

− H(Re2, v) | {z }

high order terms

v = ∂hξ := ¯ ξ

NONLINEAR DIVIDED DIFFERENCE ESTIMATE: DUALITY ESTIMATE

NONLINEAR DIVIDED DIFFERENCE ESTIMATE: DUALITY ESTIMATE

Projection Estimate:

NONLINEAR DIVIDED DIFFERENCE ESTIMATE: DUALITY ESTIMATE

• X. Meng, C.-W. Shu, Q. Zhang and B. Wu, SINUM(2012).

NONLINEAR DIVIDED DIFFERENCE ESTIMATE: DUALITY ESTIMATE

NONLINEAR DIVIDED DIFFERENCE ESTIMATE: DUALITY ESTIMATE

Residual Estimate:

NONLINEAR DIVIDED DIFFERENCE ESTIMATE: DUALITY ESTIMATE

Consistency Estimate:

SIAC NONLINEAR ESTIMATE

NUMERICAL RESULTS: BURGERS EQUATION

NUMERICAL RESULTS: BURGERS EQUATION

NUMERICAL RESULTS: STRONG NONLINEARITY

NUMERICAL RESULTS: STRONG NONLINEARITY

SUMMARY: DIVIDED DIFFERENCE ESTIMATES

➤ Important for ensuring accuracy enhancement ➤ Can reduce the computational cost by utilising information

contained in the directional divided difference

➤ Can obtain non-optimal estimates in the 1D nonlinear case

for both scalar equations and systems

➤ Numerically, we see optimal accuracy for smooth solutions ➤ Future work ➤ Optimal nonlinear estimates, including the 2D extension ➤ Can line filtering work for derivatives?

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REFERENCES

Basic properties of dg and beyond

• 1. J.H. Bramble and A.H. Schatz, ” Higher order local accuracy by averaging in the finite element

method”, Mathematics of Computation, 31 (1977), pp.94–111.

• 2. B. Cockburn, M. Luskin, C.-W. Shu, and E. Suli, ” Enhanced accuracy by post-processing for finite

element methods for hyperbolic equations”, Mathematics of Computation, 72 (2003), pp.577–606.

• 3. J. King, H. Mirzaee, J.K. Ryan, and R.M. Kirby, ”Smoothness-Increasing Accuracy-Conserving (SIAC)

Filtering for discontinuous Galerkin Solutions: Improved Errors Versus Higher-Order Accuracy”, Journal

• f Scientific Computing, 53 (2012), 129–149.
• 4. X. Meng and J.K. Ryan, "Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation

laws: divided difference estimates and accuracy enhancement." Numerische Mathematik, 136 (2017), 27–73.

• 5. H. Mirzaee, J.K. Ryan, and R.M. Kirby, “Efficient Implementation of Smoothness-Increasing Accuracy-

Conserving (SIAC) Filters for Discontinuous Galerkin Solutions”, Journal of Scientific Computing, vol. 52 (2012), pp. 85–112.

• 6. M.S. Mock and P

.D. Lax, ”The computation of discontinuous solutions of linear hyperbolic equations”, Communications on Pure and Applied Mathematics, 31 (1978), pp.423–430.

• 7. J.K. Ryan, "Exploiting Superconvergence through Smoothness-Increasing Accuracy-Conserving (SIAC)

Filtering”, Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014, Salt Lake City, Utah. Lecture Notes in Computational Science and Engineering, Springer, 106 (2015), pp 87–102.

• 8. V. Thomee, ” High order local approximations to derivatives in the finite element method”,

Mathematics of Computation, 31 (1977), pp. 652–660.