The MOOD method for steady-state Euler equations

Transcription

1 The MOOD method for steady-state Euler equations G.J. Machado 1, S. Clain 1,2, R. Loubère 2 1 Universidade do Minho, Centre of Mathematics, Guimarães, Portugal 2 Université Paul Sabatier, Institut de Mathématique de Toulouse, Toulouse, France SHARK-FV 2015 Ofir, May, 2015, Portugal This research was financed by FEDER Funds through Programa Operacional Fatores de Competitividade COMPETE and by Portuguese Funds through FCT Fundação para a Ciência e a Tecnologia, within the Projects PEst-C/MAT/UI0013/2011 and PTDC/MAT/121185/2010

2 motivation to solve steady-state hyperbolic equations using finite volume schemes prototypes: Burgers equation (scalar case) and Euler s equation (vectorial case) regular solutions: high accuracy solutions with a shock: stability (no oscillations) approach: MOOD (Multidimensional Optimal Order Detection)

3 inviscid Burgers equation we seek the velocity function φ φpxq, solution of the 1D steady-state inviscid Burgers equation df pφq dx f, in Ω p0, 1q with Dirichlet boundary conditions φ φ lf, on x 0 φ φ rg, on x 1 with F pφq φ2 2 f fpxq

4 notation K 1 x 1 x 1 0 x x i 1 2 K i x i x i+ 1 2 x I 1 2 K I x I x I h i K i cell i I number of cells x i 1, x 2 i` 1 boundary points of cell i 2 h i length of cell i x i centroid of cell i

5 FV scheme (i) integrating equation 1 F h i` 1 F i 2 i 1 2 with F i` 1 2 F pφpx i` 1 qq 2 f i 1 h i żk i fpξq dξ let F i` 1 2 f i «f i «F i` 1 2 df pφq dx f i 0 the residual at cell K i G i 1 F h i` 1 F i 2 i 1 f i 2 f over cell K i results in

6 FV scheme (ii) goal to compute an approximation of the mean value of φ in each cell of the mesh represented by φ i the approximation to the mean value of f over cell K i, f i, will be computed by gaussian quadrature the way that the numerical fluxes F i` 1 2 characterize the FV scheme are computed first-order finite volume scheme F i` 1 1 ˆφ2 i ` φ2 i`1 maxp φ i`1, φ i q pφ i`1 φ i q 2 2 high-order finite volume scheme

7 high-order FV scheme (i) to achieve high-order numerical approximations, we introduce local polynomial reconstructions of the underlying solutions stencil: the stencil S i of cell K i is composed of the d i ` 1 closest neighbour cells excluding cell K i reconstruction: the polynomial φ p i px; d i q is based on the data associated to the stencil under a least-square technique consistency: if the underlying solution is a polynomial of degree d, the reconstruction of degree d is exact

8 high-order FV scheme (ii) define the polynomial reconstructions based on the data of the associated stencil φ i px; d i q φ i ` with M i,α 1 h i ż ÿd i α 1 R i,α rpx x i q α M i,α s K i px x i q α dx ñ 1 h i ż K i φ i pxq dx φ i for a given stencil S i, we consider the quadratic functional pr i arg min Ei p pr i q ÿ «ż ff 2 1 φ h i px; d i q dx φ j j K j jps i we set p φ i the associated polynomial that corresponds to the best approximation in the least squares sense of the data of the stencil

9 high-order FV scheme (iii) to construct a generic high-order scheme one has to substitute the left and right states in by states evaluated through high-order polynomial reconstructions let us assume that a cell polynomial degree map is given M pd 1,..., d I q T with its associated stencil map S ps 1,..., S I q T pφí φ p i px i 1 ; minpd i 1, d i qq and φì p φ p 2 i px i` 1 ; minpd i, d i`1 qq 2 pφì F i` 1 pφq ` pφí`1 2 2 max ˇˇˇp ˇ φì ˇ, ˇpφí`1ˇˇˇ pφí`1 2 pφì note that the minimal polynomial degree minpd i, d i`1 q at interface x i` 1 is mandatory to ensure that the cell is updated 2 with the first-rder scheme when d i 0 or d i`1 0

10 residual form algorithm f Ñ f i Φ pφq 1 1,...,I Ñ p φ i px; d i q Ñ F i` 1 2 f i ` F i` 1 2 Ñ G i 1 F h i` 1 F i 2 i 1 2 f i the very-high order scheme explicitly depends on the map M and the stencil S G GpΦ; M, Sq the numerical solution is given by vector Φ : pφ : i q i 1,...,I which is the solution of the nonlinear problem

11 the MOOD loop (i) pφ i px; 0q: stable/low accuracy pφ i px; 6q: oscillations if there is a discontinuity/high accuracy compromise: to locally reduce the polynomial degree in domains where the solution is discontinuous while preserving the optimal order in the smooth function domain MOOD: a posterior limiting procedure the limitation is applied after the computation of the fluxes MOOD idea: to consider the maximum polynomial degree for the reconstruction, to evaluate the flux, to compute the candidate solution and then make some corrections only if it is really necessary the detection criteria: a first filter detects all extrema (called ED) and a second filter (called u2) applied on detected extrema splits them into regular or non-regular ones for non-regular extrema, one decrements the cell polynomial degree

12 the MOOD loop (ii) k=0 Φ 0 M 0 Φ k M k IN NL SOLVER OUT k = k+1 POLYNOMIAL RECONSTRUCTION Φ k+1 Φ * DETECTION ELSE if M k+1 DECREMENTING Problematic cells CellPD map M k+1 M k or = Φ k+1 = Φ k THEN CONVERGENCE, EXIT

13 Burgers: manufactured regular solution (i) manufactured regular solution upxq sinp3πxq exppxq ` 2 then f pxq pexppxq sinp3πxq ` 2qpexppxq sinp3πxq ` 3π exppxq cosp3πxqq u lf 2 u rg 2 errors evaluation E 8 piq I max i 1 u i ū i O 8 pi 1, I 2 q logpe 8pI 1 q{e 8 pi 2 qq logpi 1 {I 2 q ū i : the exact mean value of u over cell K i initial approximation: Φ 0 p2,..., 2q T

14 Burgers: manufactured regular solution (ii) I E 8 O 8 # bad cells P E E E E P E E E E P E E E E

15 Burgers: solution with a shock data fpxq π cospπxqupxq u lf 1 u rg 0.1 analytical solution # 1 sinpπxq if 0 ď x ď x s upxq 0.1 sinpπxq if x s ď x ď 1 where x s is the location of the shock (x s ) initial approximation: : U 0 pu 0 i q i 1,...,I # u 0 1 if 0 ď x i ď 1 i if 1 4 ď x i ď 1

16 Burgers: solution with a shock/unlimited (i) the P 0 scheme solution is rather diffused the P 1 and P 5 scheme solutions seem sharper and more accurate but with numerical oscillations as expected

17 Burgers: solution with a shock/unlimited (ii)

18 Burgers: solution with a shock/unlimited (iii)

19 Burgers: solution with a shock/unlimited (iv)

20 Burgers: solution with a shock/mood we now trigger the MOOD loop to eliminate the non-physical oscillations with d max 5 the MOOD method provides a stable and sharp solution without oscillations when the mesh is refined, the solution gets better without any oscillations (I 80 and I 140) only cells around the shock were decremented

21 Burgers: solution with a shock/mood (I 80)

22 Burgers: solution with a shock/mood (I 140)

23 Burgers solution with a shock/mood I r0; 0.1s r0.3; 1s E 1 O 1 E 8 O 8 E 1 O 1 E 8 O 8 # bad cells P E E E E 02 0 P E E E E 04 2 P E E E E E E E E 05 0 MOOD E E E E d max E E E E E E E E

24 Euler s equations dfpuq dx ds, in Ω p0, 1q dx where the conservative variable U is given by U pρ, ρu, Eq T the flux is given by FpUq pρu, ρu 2 ` p, upe ` pqq T, and the source term ds dx, S pd, F, HqT pdpxq, F pxq, Hpxqq T

25 Euler: manufactured regular solution (i) manufactured regular solution ρpxq exppxq ` exppxq sinp3πxq ` 2 upxq sinpαπxq ` exppxq ppxq exppxq then ds p...,...,...qt dx U lf p...,...,...q T U rg p...,...,...q T initial approximation: analytical soluton/linear combination of the boundary conditions

26 ex3, ss, x0: exact, Rusanov, unlimited, α 1 I niter ngs E 1 pρq O 1 pρq E 1 pρuq O 1 pρuq E 1 peq O 1 peq }G} 1 }Gpx p0q q} 1 O E E E e E` E E E e E` P E E E e E` E E E e E P 1 p2q P 3 p4q P 5 p6q E E E e E E E E e E E E E e E E E E e E E E E e E E E E e E E E E e E E E E e E E E E e E E E E e E E E E e E E E E e E

27 x0: exact, unlimited, α 1, P 0

28 x0: exact, unlimited, α 1, P 1

29 x0: exact, unlimited, α 1, P 3

30 x0: exact, unlimited, α 1, P 5

31 ex3, ss, x0: bcs, Rusanov, unlimited, α 1 I niter ngs E 1 pρq O 1 pρq E 1 pρuq O 1 pρuq E 1 peq O 1 peq }G} 1 }Gpx p0q q} 1 O E E E e E` E E E e E`01 Ò P E E E e E` E E E e E` P 1 p2q P 3 p4q P 5 p6q E E E e E` E E E e E`01 Ò E E E e E`01 Ò E E E e E`01 Ò E E E e E` E E E e E`01 Ò E E E e E`01 Ò E E E e E`01 Ò E E E e E` E E E e E`01 Ò E E E e E`01 Ò E E E e E`01 Ò

32 x0: linear bc s, unlimited, α 1, P 0

33 x0: linear bc s, unlimited, α 1, P 1

34 x0: linear bc s, unlimited, α 1, P 3

35 x0: linear bc s, unlimited, α 1, P 5

36 ex3, ss, x0: exact, Rusanov, unlimited, α 3 I niter ngs E 1 pρq O 1 pρq E 1 pρuq O 1 pρuq E 1 peq O 1 peq }G} 1 }Gpx p0q q} 1 O P E`00 Ò 8.9E E`00 Ò 1.12e E` E`00 Ò 4.2E E`00 Ò 1.64e E` E`00 1.2E`00 3.4E` e E` E E E e E P 1 p2q P 3 p4q P 5 p6q E E E e E E E E e E E E E e E E E E e E E E E e E E E E e E E E E e E E E E e E E E E e E E E E e E E 04 Ò 3.5E E e E E E E e E

37 x0: exact, unlimited, α 3, P 0

38 x0: exact, unlimited, α 3, P 1

39 x0: exact, unlimited, α 3, P 3

40 x0: exact, unlimited, α 3, P 5

41 ex3, ss, x0: bcs, Rusanov, unlimited, α 3 I niter ngs E 1 pρq O 1 pρq E 1 pρuq O 1 pρuq E 1 peq O 1 peq }G} 1 }Gpx p0q q} 1 O E`00 1.2E`00 3.4E` e E` E`00 Ò 8.9E E`00 Ò 2.30e E` P E E E` e E` E E E e E` P 1 p2q P 3 p4q P 5 p6q E E E e E` E E E e E`01 Ò E E E e E`01 Ò E E E e E`01 Ò E E E e E` E E E e E`01 Ò E E E e E`01 Ò E E E e E`01 Ò E E E e E` E E E e E`01 Ò E E E e E`01 Ò E E E e E`01 Ò

42 x0: linear bc s, unlimited, α 3, P 0

43 x0: linear bc s, unlimited, α 3, P 1

44 x0: linear bc s, unlimited, α 3, P 3

45 x0: linear bc s, unlimited, α 3, P 5

46 Euler s equations From ρu D and ρu 2 ` p F, we deduce that p F D2 ρ, (1) and the pressure non negativity yields that F ě 0 and ρ ě D2 F. For F ą 0, we define the reference density (2) ˆρ 0 pd, F q D2 F, ρ 0pxq ˆρ 0 pdpxq, F pxqq (3) then we have by construction ρpxq ě ρ 0 pxq, x P Ω

47 Euler s equations On the other side, from ρu 2 ` p F, upe ` pq H, and e, we deduce that p ρpγ 1q p γ 1 γ ˆH u 1 2 ρu2 (4) Assuming that D ě 0 and arguing one more time that the pressure is non-negative, we get DH ě D4 2ρ 2 thus HD ě 0. From (4), we deduce that there exists a function ˆF with respect to ρ given by ˆF pρq γ ` 1 D 2 2γ ρ ` γ 1 γ H D ρ (5) such that F pxq ˆF pρpxqq if ρ is solution of Euler s equations

48 Euler s equations Moreover, the sonic point is given by d γ ` 1 D ˆρ son pd, Hq 3 2pγ 1q H, ρ sonpxq ˆρ son pdpxq, Hpxqq. Equation ˆF pρq γ`1 D 2 2γ polynomial in ρ of the form ρ ` γ 1 γ 2pγ 1qHρ 2 2γDF ρ ` pγ ` 1qD 3 0. H D ρ is rewritten as a quadratic Existence of solution requires that the following compatibility condition are satisfied: F ą 0, HD ě 0, γ 2 γ 2 1 ě 2DH F 2

49 Euler s equations The supersonic solution ˆρ sup pd, F, Hq writes ˆρ sup pd, F, Hq 2γF a4γ 2 F 2 4pγ ` 1qpγ ` 1qDH 2 4pγ 1q H. D After algebraic manipulations, we also have the expression rρ sup pρ 0, ρ son q ρ 0 1 a1 2 α α with α γ ` 1 ρ 0. In the same way, the subsonic solution is γ ρ son given by rρ sub pρ 0, ρ son q ρ 0 1 ` a1 2 α α

50 ex3, ss, x0: bcs, unlimited, α 1, I 320

51 ex3, ss, x0: bcs, unlimited, α 3, I 320

52 Euler: solution with a shock data f 1 pxq 0 f 2 pxq 1 f 3 pxq 0 U lf p2.4142, , q T : supersonic U rg p3.9598, , q T : subsonic unlimited with P 0, P 1, and P 5 and I 200 MOOD with M p0q p5,..., 5q T

53 Euler: solution with a shock/unlimited oscillations are clearly observed and localized around the discontinuity with MOOD: the oscillations vanish and the cell polynomial degree map is essentially equal to 5 except in the vicinity of the shock the MOOD strategy succeeds in eliminating the oscillations while preserving the high degree of reconstruction in the smooth part of the

54 Euler: solution with a shock/unlimited P0

55 Euler: solution with a shock/unlimited P1

56 Euler: solution with a shock/unlimited P5

57 Euler: solution with a shock/mood