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

Gaining insight of accuracy enhancement through divided difference estimates Joint work with Julia Docampo (MIT) and Xiong Meng (Harbin) 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 1

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. 2

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 4

Constructing accurate post-processing techniques: Smoothness-Increasing Accuracy-Conserving (SIAC) Filters 5

EXTRACTING EXTRA ACCURACY Post-Processed Numerical Post-Processor Solution Information Consistency requirements (Moment conditions/ ➤ Geometry of the information polynomial reproduction) ➤ How the information was generated ➤ PDE ➤ Numerical Method 6

USEFUL REFERENCES 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 Basic properties of dg and beyond 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.

SMOOTHNESS-INCREASING ACCURACY-CONSERVING (SIAC) FILTER ➤ Assume our post-processed solution is defined through a convolution: Z Data ∗ ( x ) = K ( y − x )Data( y ) dy R ➤ SIAC kernel: r X c γ ψ ( m +1) ( x − γ ) K ( x ) = γ = − r Function Chosen to maintain chosen based on Consistency • • Given information 2r moments • • Computational efficiency

r X c γ ψ ( m +1) ( x − γ ) K ( x ) = SIAC KERNEL γ = − r ➤ Choosing j ( m +1) to be a B-spline of order m+1 : ( − 1 2 , 1 ⇥ ⇤ 1 , x ∈ ψ (1) ( x ) = χ [ − 1 2 2 ] = 2 , 1 0 , else ψ ( m +1) ( x ) = 1 ✓ x + m + 1 ◆ ✓ x + 1 ◆ ✓ m + 1 ◆ ✓ x − 1 ◆� ψ ( m ) ψ ( m ) + − x , 2 2 2 2 m ➤ B-spline properties: m ≥ 1 . ψ (1) 1 ψ (2) ➤ Compact support ψ (3) ➤ Smoothness of m - 1 . ➤ Derivatives can be written as divided differences 0 − 2 − 1 0 1 2

SIAC KERNEL: LINEAR COMBINATION OF B-SPLINES r=m=1 Kernel: • Preserves 2 moments • Continuous • Support of length 5 superconvergence extraction through siac filtering r=m=2 Kernel: • Preserves 4 moments • C 1 continuity • Support of length 7

SIAC KERNEL: FOURIER SPACE ➤ In physical space, the filter is r X c γ ψ ( m +1) ( x − γ ) K ( x ) = γ = − r ➤ In Fourier space this is: ◆ m +1 ! r ✓ sin( k π ) ˆ X K ( k ) = c 0 + 2 c γ cos( γ k π ) k π γ =0 Thomee, Math. Comp. (1977) Ji & Ryan, ICOSAHOM 2014 Proceedings

SMOOTHNESS-INCREASING ACCURACY-CONSERVING (SIAC) FILTER SIAC filtering allows: • Global enhanced accuracy • Global smoothness superconvergence extraction through siac filtering DG data accuracy: p+1 SIAC DG order: 2p+1 Example: p=r=m=2

EXTRACTING EXTRA INFORMATION OUT OF DATA Post-Processed Numerical Post-Processor Solution Information ➤ Geometry of the information ➤ How the information is generated ➤ PDE ➤ Numerical Method 13

IMPORTANT COMPONENTS OF DG ➤ Approximation Space: Consists of piecewise polynomials ➤ Based on a Variational formulation 2 2 Z Z Z ˆ X X ( u h ) t v h ( x ) dx − f ( u h )( v h ) x i ( x ) dx + f i ˆ n i v h ds = 0 τ e τ e ∂τ e d =1 d =1 ➤ Weak continuity is enforced through the numerical flux 14

DISCONTINUOUS GALERKIN ERRORS: ➤ L 2 -Error Estimate: k u � u h k 0  Ch p +1 | u 0 | H p +1 ➤ Negative-Order Norm Estimate: h ( u � u h ) k − ( p +1)  Ch 2 p +1 | u 0 | H p +2 ( D Ω 1 ) k ∂ α ➤ For linear hyperbolic equations 15

What extra information to we get from post-processing? 16

SIAC FILTERED DG SIAC filtered solution: H depends on mesh ✓ x − y ◆ h ( x, t ) = 1 Z u ∗ u h ( y, t ) dy K H H R SIAC filtered error: k ( u � K (2 p +1 ,p +1) ⇤ u h )( T ) k 0  Ch 2 p +1 (Uniform) h k ( u � K (2 p +1 ,p +1) 2 3 (2 p +1) (Non-uniform) ⇤ u h )( T ) k 0  Ch H Bramble & Schatz, Math. Comp (1977) Cockburn, Luskin, Shu, & Süli, Math. Comp (2003) (❩ L-inf estimates )❪ Ji, Xu, Ryan, Math. Comp. (2012)

SIAC FILTER: ERROR ESTIMATES k u � u ∗ k 0  k u � u ∗ k 0 + k K h ⇤ ( u � u h ) k 0 For the first term: k u � u ∗ k 0  C ∆ x 2 r +1 ➤ Kernel reproduces polynomials of degree up to 2r. Z K ( x − y ) y m dy = x m , m = 0 , 1 , . . . , 2 r. R ➤ Equivalent to consistency and moment conditions: Z Z K ( y ) y m dy = 0 , m = 1 , . . . 2 r. K ( x − y ) dy = 1 , R R

SIAC FILTER: ERROR ESTIMATES For the second term: X k K h ⇤ ( u � u h ) k 0  C k D α ( u � u h ) k − ( m +1) | α | ≤ m +1 where the negative-order norm is given by ( v, φ ) k v k − ( m +1) , Ω = sup k φ k m +1 , Ω 0 ( Ω ) φ ∈ C ∞

SIAC FILTER: ERROR ESTIMATES Second term (cont.) Using properties of B-splines and convolution X k K H ⇤ ( u � u h ) k 0  k D α ( K H ⇤ ( u � u h )) k 0 | α | ≤ m +1 X  k ( ∂ α H K H ) ⇤ ( u � u h ) k 0 | α | ≤ m +1 X  k K H ⇤ ∂ α H ( u � u h ) k 0 | α | ≤ m +1 X  k K H k 1 k ∂ α H ( u � u h ) k − ( m +1) | α | ≤ m +1 Properties of the scheme

SIAC FILTER: ERROR ESTIMATES ( v, φ ) k v k − ( m +1) , Ω = sup k φ k m +1 , Ω 0 ( Ω ) φ ∈ C ∞ For linear hyperbolic equations: Z T d ( ∂ α h ( u − u h ) , φ )( T ) =( ∂ α h ( u − u h ) , φ )(0) − dt ( ∂ α h u h , φ ) dt 0 0 1 Z T N X ( φ − v h )( x + B C =( ∂ α h ( u − u h ) , φ )(0) − [ ∂ α h u h ] j +1 / 2 j +1 / 2 ) A dt @ 0 | {z } j =1 jump in u h Initial projection interelement flux values

SIAC FILTER: KEY INGREDIENTS FOR THE LINEAR ESTIMATE 1. The dual equation. For the equation u t + u x = 0, we require d dt ( u, ϕ ) = 0 ϕ t + ϕ x = 0 . ⇒ 2. Relation between derivatives and divided differences D α ψ ( m +1) ( x ) = ∂ α H ψ ( m +1 − α ) ( x ) 3. DG divided difference estimate: h ( u � u h ) k 0  Ch p +1 k ∂ α

AIM 1. Exploiting the information to make the SIAC filter more computationally efficient 2. Extension to nonlinear hyperbolic conservation laws 23

Exploiting information for computational efficiency Julia Docampo Sánchez (MIT) 24

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 25

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. D ↵ ψ ( ` ) ( x ) = ∂ ↵ h ψ ( ` − ↵ ) ( x ) , α = α x + α y coordinate aligned 26

LINE SIAC FILTER FOR MULTI-DIMENSIONAL FILTERING ➤ For rotated SIAC need to define directional divided differences. 27

SIAC FILTER: DIRECTIONAL DIVIDED DIFFERENCE ESTIMATES ➤ Direction vector: u =(u x ,u y ). ➤ Scaled directional divided difference with respect to u : ✓ ✓ ◆ ✓ ◆◆ ∂ u , H f ( t ) = 1 x + H 2 u x , y + H x − H 2 u x , y − H f 2 u y − f 2 u y H ✓ ◆ ✓ ◆ x, y + H x − H = ∂ u x , H f + ∂ u y , H f 2 u y 2 u x , y . ➤ a - th directional divided difference: ∂ α − 1 � � ∂ α u , H f ( x, y ) = ∂ u , H u , H f ( x, y ) α > 1 . , 28

SIAC FILTER: DIVIDED DIFFERENCE ESTIMATES ➤ We can relate the directional divided difference to the coordinate aligned divided difference D ↵ ψ ( ` ) ( t ) = (cos θ ) ↵ x (sin θ ) ↵ y ∂ ↵ h ψ ( ` − ↵ ) ( x ) coordinate aligned ➤ As long as q s 0, p /2 , then we have superconvergence! ➤ Leads to a reduced error constant. ✓ ∆ y ◆ θ = arctan ∆ x 29

LINE SIAC FILTER FOR MULTI-DIMENSIONAL FILTERING ➤ Reducing the support to a line: axis-aligned vs. rotated Tensor Product SIAC filter Line SIAC filter 30