HONOM EUROPEAN WORKSHOP on High Order Nonlinear Numerical Methods for Evolutionary PDEs, Bordeaux, March 18-22, 2013

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HONOM 2013 EUROPEAN WORKSHOP on High Order Nonlinear Numerical Methods for Evolutionary PDEs, Bordeaux, March 18-22, 2013 A priori-based mesh adaptation for third-order accurate

1 HONOM 2013 EUROPEAN WORKSHOP on High Order Nonlinear Numerical Methods for Evolutionary PDEs, Bordeaux, March 18-22, 2013 A priori-based mesh adaptation for third-order accurate Euler simulation Alexandre Carabias, Estelle Mbinky ( ) INRIA - Tropics project Sophia-Antipolis, France ( )INRIA - Gamma Project Rocquencourt, France Alexandre.Carabias@inria.fr

2 Motivation of the study Propagation of waves in large spaces. Mesh adaptation should follow the wave propagation. Adjoint-based adaptation criterion will concentrate on zone of interest for a chosen functional output. An accurate approximation scheme needs be chosen. 2 Carabias-Mbinky A priori mesh adaption for third-order...

3 Scope of the talk 1. Interpolation error analysis 2. Reconstruction error analysis 3. CENO2 Scheme 4. Approximation error analysis 5. Optimal metric 6. Resolution of optimum 7. Numerical experiments 8. Concluding remarks 3 Carabias-Mbinky A priori mesh adaption for third-order...

4 1. Interpolation error analysis (1) Π 2 : 2D Lagrange quadratic interpolation: (u Π 2 u)(δx) P u (δx). δx = (δx, δy) defines mesh size. The generic error model is a homogeneous polynomial of degree k = 3 in 2 variables : P u (x) = i=3 i=0 a(u) i are third derivatives of u. ( k i ) a(u) i x i y k i 4 Carabias-Mbinky A priori mesh adaption for third-order...

5 1. Interpolation error error analysis (2): Modeling of the quadratic order interpolation error Local optimization problem Main Idea: (P u ) 2 3 being homogeneous with a quadratic form with respect to mesh size, let us approach locally the variations of P u by a quadratic definite positive form taken at power 3 2, i.e, for (x, y) R2, P u (x, y) ( ) 3 t (x, y) Su loc 2 (x, y) Find the optimal local metric Su loc (optimal local directions and sizes) to construct a majorant of the local error model P u. The optimal local metric Su loc is the one whose unit ball is the maximum area ellipse included in the isoline 1 of P u. 5 Carabias-Mbinky A priori mesh adaption for third-order...

1. Interpolation error error analysis (3): Example P u = 6 x 3 8.594 x 2 y 6.

6 1. Interpolation error error analysis (3): Example P u = 6 x x 2 y 6.98 xy y 3 Optimal local metric in P u in ( t (x, y) S loc u (x, y)) 3 2 Superposition of the iso-lines of P u and ( t (x, y) S loc u (x, y)) Carabias-Mbinky A priori mesh adaption for third-order...

7 1. Interpolation error error analysis (4) Construction of optimal local metrics Binary decomposition by Sylvester s theorem [Brachat-Mourrain, 1999] P u (x, y) = Real case S loc u = t Q u λ λ (rank) j=1! Q u ««α1 β Q u = 1 ; α 2 β t α1 α Q u = 2 2 β 1 β 2 λ j (α j x + β j y) 3 ; β i = 1 In the complex case, λ 2 = λ 1 and α 2 = ᾱ 1. S loc u Complex case S loc u = tc Q u λ λ 2 2 3! «; t ᾱ1 c ᾱ Q u = 2 β 1 β 2 is real symmetric in both cases Q u 7 Carabias-Mbinky A priori mesh adaption for third-order...

8 1. Interpolation error error analysis (6) Optimal estimate Find the maximal ellipse inside the P u (x) = 1 level set. S u = c opt S loc u = t R u ( µ1 0 0 µ 2 ) R u c opt = 1 real case ; c opt = complex case. ( ) 3 P u (x, y) (x, y) S 2 u (x, y) 8 Carabias-Mbinky A priori mesh adaption for third-order...

9 1. Interpolation error error analysis (7) Mesh parametrization by a Riemannian metric For any x in Ω, M(x) is a symmetric matrix. M Mesh(M) Such that any edge ab of Mesh(M) has a unit M-length: 1 0 t ab M(a + tab) ab dt = 1. M(x) = d M (x)r(x)λ(x)r t (x) with det Λ = 1. The rotation R = (e ξ, e η ) parametrises the two orthogonal stretching directions of the metric. Λ is a diagonal matrix with eigenvalues λ 1 = m η /m ξ and λ 2 = m ξ /m η where m ξ and m η represent the two directional local mesh sizes in the characteristic/stretching directions of M. λ 1 /λ 2 is the stretching ratio. d M = m 1 ξ m 1 η is the node density. 9 Carabias-Mbinky A priori mesh adaption for third-order...

10 1. Interpolation error error analysis (8) Find the optimal Riemannian metric M defined on the computational domain for smallest weighted-l 1 error ) 3 min g(x) (trace(m 12 Su M 12 2 ) dx M Ω with d(m)dx = N M opt = N ( Ω ( µ1 ν 1 = µ 2 Ω ) ( 1 κ dx κ t ν1 0 R u 0 ν1 1 ) 1 2 κ = (2g µ 1 µ 2 ) 3 5. ) R u 10 Carabias-Mbinky A priori mesh adaption for third-order...

uction Given ū i on on each cell i of centroid G i, find the c i,α, α 2 s.t. R2 0 ū i

11 2. Reconstruction error error analysis Vertex, dual cell, 2-exact quadratic reconstruction Given ū i on on each cell i of centroid G i, find the c i,α, α 2 s.t. R2 0 ū i (x) = ū i + c i,α [(X G i ) α (X G i ) α dx] Cell i α 2 R2 0ū = R2 0 ū i,i dx = u i (R2 0ūi,j u j ) 2 = Min. Cell i j N(i) 11 Carabias-Mbinky A priori mesh adaption for third-order...

12 2. Reconstruction error analysis (2) Same treatment as interpolation R 2 : 2D quadratic ENO reconstruction: i=3 u(x) R2 0 u(x) ( k i )a i x i y k i i=0 ( t (x, y) S u (x, y) ) 3 2 a(u) i are third derivatives of u. 12 Carabias-Mbinky A priori mesh adaption for third-order...

13 3. CENO2 Scheme (1) Variational statement of a model problem B(u, v) = v F(u) dω Ω Γ vf Γ (u) dγ, B is linear with respect to v. Find u V such that B(u, v) = 0 v V. CENO discrete statement (after C. Groth), vertex version V 0 = {v 0, V 0 Celli = const i vertex} Find u 0 V 0 such that B(R 0 2 u 0, v 0 ) = 0 v 0 V 0 13 Carabias-Mbinky A priori mesh adaption for third-order...

14 3. CENO2 Scheme (2) 2-exact flux integration The integral on a cell interface C ij = C i C j is split into the integrals on the two segments of C ij. On each segment C (1) ij and C (2) ij a numerical integration with two Gauss points (two Riemann solvers) is applied. 14 Carabias-Mbinky A priori mesh adaption for third-order...

15 4. A priori error analysis (1) Cf. A posteriori analysis for a large set of p-order reconstruction-based Godunov methods Barth-Larson Scalar output j(u) = (g, u). Projection π 0 : V V 0, v π 0 v C i, dual cell, π 0 v Ci = vdx/meas(c i ). C i Output error δj = (g, R 0 2 π 0 u R 0 2 u 0 ). The adjoint state u 0 V 0 is the solution of: B u (R0 2 u 0 )(R 0 2 v 0, u 0) = (g, R 0 2 v 0 ), v 0 V Carabias-Mbinky A priori mesh adaption for third-order...

16 4. A priori error analysis (2) Simpler case: B(u, v) = F (v) where B is bilinear, F is linear, for example: B(u, v) = Ω vdiv(vu)dω + Γ uvv ndγ and F (v) = Γ u BvV ndγ. B(v, u0 ) = (g, v) (discr.adj. eq.) B(R2 0u 0, v) = F (v) (discr.state eq.) B(u, v) = F (v) (cont.state eq.) (g, R2 0π 0u R2 0u 0) = B(R2 0π 0u R2 0u 0, u0 ) (discr.adj. eq.) B(R2 0π 0u, u0 ) B(R0 2 u 0, u0 ) B(R2 0π 0u, u0 ) F (u 0 ) (discr.state eq.) B(R2 0π 0u, u0 ) B(u, u 0 ) (cont.state eq.) B(R2 0π 0u u, u0 ) The error is directly expressed in terms of the reconstruction error for exact solution. 16 Carabias-Mbinky A priori mesh adaption for third-order...

17 4. A priori error analysis (3) Unsteady Euler: W = (ρ, ρu, ρv, ρe) For the case of Euler eqs, we get after some calculations: B(R2 0 π 0 W W, W0 ) 2 K q (W, W ) R2 0 π 0 u q u q dω Ω q with (u q ) q=1,8 = (W, W t ), and in which the K q (W, W ) are built from space derivatives of W and Euler fluxes derivatives with respect to dependant variable W. 17 Carabias-Mbinky A priori mesh adaption for third-order...

18 5. Optimal metric (1) Using the previous reconstruction error estimate: R 0 2 π 0u q u q ( trace(m 1/2 S q M 1/2 ) ) 3 2. Optimization problem After the a priori analysis, we have to minimise the following error: = E = d 3 2 M q ( ) 3 K q (W, W ) trace(m 1/2 S q M 1/2 2 ) = ( trace(m 1/2 SM 1/2 )) 3 2 (trace[(r M Λ M R T M) 1 2 S (RM Λ M R T M) 1 2 ] ) 3 2 dxdy dxdy dxdy with constraint d M dxdy = N. 18 Carabias-Mbinky A priori mesh adaption for third-order...

19 5. Optimal metric (2) d 3 2 M Minimize E(M) = ) (trace[(r M Λ M R T M) 1 2 S (RM Λ M R T M) ] dxdy with constraint d M dxdy = N. Optimal solution: x :S = R S ΛS R T S R M opt = R S, Λ Mopt = det(s) 3 2 (d Mopt ) 5 2 = const. d Mopt = N ( det(s) 3 5 dxdy) 1 det(s) Λ S /det(s) M opt = d Mopt R Mopt Λ Mopt R T M opt 19 Carabias-Mbinky A priori mesh adaption for third-order...

20 5. Optimal metric (3): isotropic case Minimize E(M) = d 3 2 M (trace[(λ M ) 1 2 S (λm ) 1 2 ] ) 3 2 with constraint dxdy d M dxdy = N. Optimal solution: λ Mopt = trace[s] 1 /det(s) det(s) 3 2 (d Mopt ) 5 2 = const. d Mopt = N ( det(s) 3 5 dxdy) 1 det(s) 3 5 M opt = d Mopt λ Mopt 20 Carabias-Mbinky A priori mesh adaption for third-order...

21 6. Resolution of optimum, unsteady case The continuous model giving the adapted mesh involves a state system, an adjoint system and the optimality relation giving M opt. We discretise it. The time discretisation of the metric is made of coarser time intervals. We solve it. by the Global Unsteady Fixed Point algorithm of Belme et al. JCP Carabias-Mbinky A priori mesh adaption for third-order...

22 6. Resolution of optimum, unsteady case 22 Carabias-Mbinky A priori mesh adaption for third-order...

23 7. Numerical experiments We present a first series of experiments where the propagation of an acoustical perturbation is followed by the mesh adaptator in order to minimise the mesh effort. 1 Noise wall problem 2 Diffraction problem 23 Carabias-Mbinky A priori mesh adaption for third-order...

24 7. Numerical experiments: Noise wall problem (1) Propagation of an acoustic wave from a source while observing the impact on the detector. Acoustic source : r = Ae Bln(2)[x2 +y 2 ] cos(2πf ). Functional: j(w ) = T 0 M 1 2 (p p air ) 2 dmdt. 24 Carabias-Mbinky A priori mesh adaption for third-order...

25 7. Numerical experiments: Noise wall problem (2) Density field evolving in time on uniform mesh (line 1), adapted one (middle line), with corresponding adapted meshes (last line): 25 Carabias-Mbinky A priori mesh adaption for third-order...

26 7. Numerical experiments: Noise wall problem (3) Scalar output comparaison : MUSCL (-) vs CENO ( ) schemes. 5e e-08./functional_evolution_CENO.dat u 1:2./functional_evolution_MUSCL.dat u 1:2 4e e-08 Scalar output 3e e-08 2e e-08 1e-08 5e Time 26 Carabias-Mbinky A priori mesh adaption for third-order...

27 7. Numerical experiments: Diffraction problem (1) The diffraction of an acoustic wave travelling from a source location to a detector situated in a small region under the step. Acoustic source : r = Ae Bln(2)[x2 +y 2 ] cos(2πf ). Functional: j(w ) = T 0 M 1 2 (p p air ) 2 dmdt. 27 Carabias-Mbinky A priori mesh adaption for third-order...

$Diffraction$ Density field

28 7. Numerical experiments: Diffraction problem (2) Density field evolving in time on uniform mesh (line 1), adapted one (middle line), with corresponding adapted meshes (last line): 28 Carabias-Mbinky A priori mesh adaption for third-order...

29 7. Numerical experiments: Diffraction problem (3) Our goal-oriented method ensures the accuracy of the functional output j(w ) = T 1 0 M 2 (p p air ) 2 dmdt. It is thus interesting to analyse the integrand k(t) = 1 M 2 (p p air ) 2 dm of j(w ) on the micro M for different sizes of uniform vs. adapted meshes. 29 Carabias-Mbinky A priori mesh adaption for third-order...

30 7. Numerical experiments: Diffraction problem (4) Functional time integrand calculation on different sizes of non-adapted meshes (28K, 40K, 68K) vs. adapted meshes (mean sizes: 2620, 4892, 6130). 2e e-10./Uniform40K.dat u 1:2./Uniform28K.dat u 1:2./Uniform68K.dat u 1:2 3.5e-10 3e-10./Adapt2620.dat u 1:2./Adapt4892.dat u 1:2./Adapt6130.dat u 1:2 1.6e e e-10 Functional Integrand 1.2e-10 1e-10 8e-11 Functional Integrand 2e e-10 1e-10 6e-11 4e-11 5e-11 2e Time Time The convergence order is found to be 0.6 for uniform meshes and 1.98 for the adapted one. 30 Carabias-Mbinky A priori mesh adaption for third-order...

31 7. Numerical experiments: 3rd order error vs 2nd order error The previous computations where performed with 2nd-order error, made of 2nd-order derivatives, used with the 3rd-order CENO2 scheme. In a preliminary use of 3rd-order estimate, relying on 3rd-order derivatives, we compare two calculations: - one with 3rd-order scheme and 2nd-order error, - the second with 3rd-order scheme and 3rd-order error. The test case is the translation by Euler of a density Gaussian from left to right in a rectangle, with a functional observation with a Gaussian weight at rectangle center. 31 Carabias-Mbinky A priori mesh adaption for third-order...

32 7. Numerical experiments: Use of 2nd order error The integrand of the functional is a function of time which culminate when the center of density Gaussian coincides with the center of Gaussian which is the functional weight (center of rectangle). Left, 350 equivalent nodes, right 1500 equivalent nodes. 32 Carabias-Mbinky A priori mesh adaption for third-order...

33 7. Numerical experiments: Use of 3rd order error Here now are the results with the new 3rd-order estimate. Left, 350 equivalent nodes, right 1500 equivalent nodes. 33 Carabias-Mbinky A priori mesh adaption for third-order...

34 8. Concluding remarks (1) A third-order (spatially) accurate goal-oriented mesh adaptation method has been built on the basis of: The extension of Hessian analysis to higher order interpolation of Mbinky et al.. A novel a priori analysis. The Unsteady Global Fixed-Point mesh adaptation algorithm of Belme et al.. Numerical experiments are yet only 2D and at the very beginning. 34 Carabias-Mbinky A priori mesh adaption for third-order...

35 8. Concluding remarks (2) Next studies will address: Steady test cases. Flows with singularities. 3D extension. 35 Carabias-Mbinky A priori mesh adaption for third-order...

36 36 Carabias-Mbinky A priori mesh adaption for third-order...

Third Order WENO Scheme on Three Dimensional Tetrahedral Meshes

COMMUNICATIONS IN COMPUTATIONAL PHYSICS Vol. 5, No. 2-4, pp. 86-848 Commun. Comput. Phys. February 29 Third Order WENO Scheme on Three Dimensional Tetrahedral Meshes Yong-Tao Zhang 1, and Chi-Wang Shu