FOR C ONSERVATION L AWS : N UMERICAL ANALYSIS Margarete O. Domingues - PowerPoint PPT Presentation

A DAPTIVE M ULTIRESOLUTION M ETHODS FOR C ONSERVATION L AWS : N UMERICAL ANALYSIS Margarete O. Domingues 1 , Sˆ onia M. Gomes 2 , Olivier Roussel 3 , and Kai Schneider 3 , 4 1 Instituto Nacional de Pesquisas Espaciais - LAC, Brazil 2 Universidade Estadual de Campinas - IMECC, Brazil 3 M2P2, Universit´ es d’Aix-Marseille, France 4 ´ Ecole Centrale de Marseille, France Adaptive Multiresolution Methods for Conservation Laws – p.1/80

Summary Motivation First part: MR analyses for point values and cell averages Harten’ approach Discrete and functional aspects Local regularity indicator and data compression. Stability. Lifting methodology Second part: applications to adaptive methods for PDE The SPR method The FV/MR method Examples Adaptive Multiresolution Methods for Conservation Laws – p.2/80

Basic References Books A. Cohen. Wavelet Methods in Numerical Analysis. Handbook of Numerical Analysis, Elsevier, 2000, Ciarlet, P. G. and Lions, J. L. (eds), Papers A. Harten, Multiresolution representation of data: a general framework, SIAM J. Numer. Anal. (1996),33 (3): 385-394, W. Sweldens, The lifting scheme: a construction of second generation wavelets, SIAM J. Math. Anal., (1996), 29: 511-543. Adaptive Multiresolution Methods for Conservation Laws – p.3/80

Motivation Adaptive Multiresolution Methods for Conservation Laws – p.4/80

Multiresolutions analysis - WT → f MR f J ← = ( f 0 , d 0 , . . . d J − 1 ) J f J → information of f at the  nest scale level J f 0 → information of f at coarsest level d j → difference of information (details) between levels j and j + 1 f MR → multiresolution representation J Adaptive Multiresolution Methods for Conservation Laws – p.5/80

Illustration: direct WT Adaptive Multiresolution Methods for Conservation Laws – p.6/80

Multiresolution approximations: inverse WT Adaptive Multiresolution Methods for Conservation Laws – p.7/80

Multiresolution approximations: cont.. Adaptive Multiresolution Methods for Conservation Laws – p.8/80

Multiresolution approximations: cont.. Adaptive Multiresolution Methods for Conservation Laws – p.9/80

Fundamental properties Perfect reconstruction Ef  cient algorithms Localization Polynomial cancellation Stability Multiresolution representation J � V J = V 0 + W j j =0 � � � � f J,k Φ J,k = f 0 ,k Φ 0 ,k + d j,k Ψ j,k j k k k Adaptive Multiresolution Methods for Conservation Laws – p.10/80

Brief history 1911: Haar; 1930: Littlewood-Paley; 1940: Gabor; 1960: Calderón-Zigmund 1984: Morlet e Grossmann (continuous WT ) 1985: Meyer, Mallat (multiresolution analysis) 1988: Daubechies (orthogonal wavelets) 1992: Cohen, Daubechies e Feauveau (biorthogonal wavelets ) 1992–1993: Harten; Donoho (interpolating and cell-averages MR ) 1994: Sweldens (Second generation wavelets) . . . Adaptive Multiresolution Methods for Conservation Laws – p.11/80

Goals Introduction of the principal concepts in MRA Applications in the design of adaptive schemes for PDE wavelet coef  cients as regularity indicators construction of adaptive grids: re  ned close to singularities × coarse in smooth regions Adaptive Multiresolution Methods for Conservation Laws – p.12/80

Adaptive Multiresolution Methods for Conservation Laws – p.13/80

Adaptive Multiresolution Methods for Conservation Laws – p.13/80

Adaptive Multiresolution Methods for Conservation Laws – p.13/80

Adaptive Multiresolution Methods for Conservation Laws – p.13/80

Adaptive Multiresolution Methods for Conservation Laws – p.13/80

Adaptive Multiresolution Methods for Conservation Laws – p.13/80

Adaptive Multiresolution Methods for Conservation Laws – p.13/80

Adaptive Multiresolution Methods for Conservation Laws – p.13/80

Adaptive Multiresolution Methods for Conservation Laws – p.13/80

Adaptive Multiresolution Methods for Conservation Laws – p.13/80

Adaptive Multiresolution Methods for Conservation Laws – p.13/80

Adaptive Multiresolution Methods for Conservation Laws – p.13/80

Adaptive Multiresolution Methods for Conservation Laws – p.13/80

Adaptive Multiresolution Methods for Conservation Laws – p.13/80

Adaptive Multiresolution Methods for Conservation Laws – p.13/80

Adaptive Multiresolution Methods for Conservation Laws – p.13/80

Adaptive Multiresolution Methods for Conservation Laws – p.13/80

Adaptive Multiresolution Methods for Conservation Laws – p.13/80

Adaptive Multiresolution Methods for Conservation Laws – p.13/80

Adaptive Multiresolution Methods for Conservation Laws – p.13/80

initial condition grid solution at t = 15 grid Adaptive Multiresolution Methods for Conservation Laws – p.14/80

Wavelet ideas appear in several contexts ⇓ and there are different approaches for their constructions and for the interpretation of their properties. Harten’s approach and Lifting schemes Fourier techniques are not necessary Appropriate for non-cartesian geometries Main focus: characterization of local regularity stability of the algorithms Adaptive Multiresolution Methods for Conservation Laws – p.15/80

Harten’s approach Adaptive Multiresolution Methods for Conservation Laws – p.16/80

MR analysis for point values Dyadic grids on the interval X 4 X 3 X 2 X 1 X 0 X j = { x j,k = k 2 − j , 0 ≤ k ≤ 2 j } ⊂ Ω = [0 , 1] Adaptive Multiresolution Methods for Conservation Laws – p.17/80

X j ⊂ X j +1 x j,k = x j +1 , 2 k – old points X j +1 re  nement of X j x j +1 , 2 k +1 ∈ X j +1 \ X j – new points Discretization by point values at X j : D j : f → f j f j,k = f ( x j,k ) Adaptive Multiresolution Methods for Conservation Laws – p.18/80

Goal: To construct a multiresolution analysis for f J . WT → f MR f J − = ( f 0 , d 0 , . . . d J − 1 ) J Principal tool: interpolating predicton operator to approximate the point values of f at level j + 1 from the information at the coarser level j . P j → j +1 : f j → ˜ f j +1 ˜ f j +1 , 2 k = f j,k ˜ f j +1 , 2 k +1 ≈ f j +1 , 2 k +1 Wavelets are prediction errors d j,k = f j +1 , 2 k +1 − ˜ f j +1 , 2 k +1 Adaptive Multiresolution Methods for Conservation Laws – p.19/80

Simple Example: linear interpolation 1 = f j,k + f j,k ˜ f j +1 , 2 k − 2 Adaptive Multiresolution Methods for Conservation Laws – p.20/80

Cubic interpolation f j +1 , 2 k − 1 = 9 16 [ f j,k + f j,k +1 ] − 1 ˜ 16 [ f j,k − 1 + f j,k 2 ] + Adaptive Multiresolution Methods for Conservation Laws – p.21/80

Interpolation errors d j,k = f j +1 , 2 k +1 − ˜ f j +1 , 2 k +1 d j,k x j+1,2k+1 Adaptive Multiresolution Methods for Conservation Laws – p.22/80

Wavelet coef  cients Prediction errors d j,k = f j +1 , 2 k +1 − ˜ f j +1 , 2 k +1 d j = ( d j,k ) measures the difference of information between discretization levels j and j + 1 . Polynomial cancellation: if f is a polynomial of degree ≤ r , then using interpolation of degree r d j,k = 0 Adaptive Multiresolution Methods for Conservation Laws – p.23/80

Analysis algorithm f 0 f J f J ! 1 f J ! 2 ... d d d J ! 1 J ! 2 0 For j = J − 1 , . . . , 0 , given f j +1 Do decimation: f j,k = f j +1 , 2 k +1 Do prediction: ˜ f j +1 , 2 k +1 = [ P j → j +1 f j ](2 k + 1) Compute: d j,k = f j +1 , 2 k +1 − ˜ f j +1 , 2 k +1 Adaptive Multiresolution Methods for Conservation Laws – p.24/80

Synthesis algorithm f ... f f 1 f 0 J J ! 1 d d d 1 J ! 1 0 For j = 0 , . . . , J − 1 , given f j and d j Do prediction: ˜ f j +1 , 2 k +1 = [ P j → j +1 f j ](2 k + 1) Compute: f j +1 , 2 k = f j,k d j,k + ˜ f j +1 , 2 k +1 = f j +1 , 2 k +1 Adaptive Multiresolution Methods for Conservation Laws – p.25/80

Local regularity analysis Polynomial Lagrange interpolation error For f ∈ C s ( I j,k ) , 1 ≤ s ≤ r , with bounded derivative f ( s +1) , | d j,k | = | f ( x j +1 , 2 k +1 ) − p ( x j +1 , 2 k +1 ) | K 2 − ( s +1) j max | f ( s +1) ( ξ ) | ≤ ξ ∈ I j,k where r is the polynomial degree, I j,k is the interval containing the interpolation stencil, K = K ( s, r ) . Adaptive Multiresolution Methods for Conservation Laws – p.26/80

Consequence There is a relation between the decay rate of the wavelet coef  cients and the local regularity of the function. For high interpolation accuracy, a fast decay occurs on smooth regions Wavelet coef  cients d j,k can be used as local regularity indicators Adaptive Multiresolution Methods for Conservation Laws – p.27/80

Example  8 . 1 e 1 / 4 e −| x − 1 / 2 | , x ≤ 1 / 4 ,          9 e −| x − 1 / 2 | , f ( x ) = 1 / 4 ≤ x ≤ 3 / 4 ,       e −| x − 1 / 2 | (16 x 2 − 24 x + 18) ,   x ≥ 3 / 4 ,  ξ = 1 / 4 → f has a jump discontinuity ξ = 1 / 2 → f (1) has a jump discontinuity ξ = 3 / 4 → f (2) has a jump discontinuity Adaptive Multiresolution Methods for Conservation Laws – p.28/80