DG Method for Space Physics

Transkript

1 DG Method for Space Physics Moritz B. Schily* Gregor Gassner*, Florian Hindenlang, Marvin Bohm* Joachim Saur #, Stephan Schlegel # University of Cologne - Math. Dept.*, Max Planck Institute for Plasma Physics, University of Cologne - Geo-Phys. Dept.# May 22, 2019 This Project is funded by the EU

2 Outline Motivation Model Method Tests Outlook

3 Motivation Figure: Jupiter and Io, Image by Mura et al.[2]

4 Motivation Figure: Io s footprint in the aurora, Image by Mura et al.[2]

5 Governing equations The ideal MHD equations have the form U t + F ( U) + S( U, U) = 0, (1) where U = ϱ ϱv, F = ϱv v T + p I + E ϱ v ( ) v 1 2 ϱv 2 + γp γ 1 +

6 Governing equations The ideal MHD equations have the form U t + F ( U) + S( U, U) = 0, (1) where ϱv ϱ ( ) U = ϱv B, F = ϱv v T + p I µ 0 2 B2 I B B T vb T B ( v T ) E v 1 2 ϱv 2 + γp γ 1 + S A

7 Governing equations The ideal MHD equations have the form U t + F ( U) + S( U, U) = 0, (1) where ϱv ϱ ( ) U = ϱv B, F = ϱv v T + p I µ 0 2 B2 I B B T vb T B ( v T ) E v 1 2 ϱv 2 + γp γ 1 + S A S A = 1 ( E ) B, µ 0 E = 1 2 ϱv 2 + E = v B (2) p γ µ 0 B 2 (3)

8 Governing equations The ideal MHD equations have the form U t + F ( U) + S( U, U) = 0, (1) where ϱv ϱ ( ) U = ϱv B, F = ϱv v T + p I µ 0 2 B2 I B B T vb T B ( v T ) E v 1 2 ϱv 2 + γp γ 1 + S A S A = 1 ( E ) B, µ 0 E = 1 2 ϱv 2 + E = v B (2) p γ B 2 (3) 2µ 0 S = 0, B = 0 (4)

9 Challenges B = 0 is hard to discretise as an additional constraint

10 Challenges B = 0 is hard to discretise as an additional constraint When we apply the classical model to our testcase the Alfven-wave speeds B µ0 ϱ reach values in the magnitude of 10c.

11 Altered Model To enforce the B = 0 constraint we use a generalized Lagrangian Multiplier (GLM) approach, introducing the new variable ψ.

12 Altered Model To enforce the B = 0 constraint we use a generalized Lagrangian Multiplier (GLM) approach, introducing the new variable ψ. To ensure, that the eigenvalues are smaller than c, we switch to the semi relativistic MHD model, Gombosi et al. [1] ϱ ϱv + U = B E E = 1 2 ϱv 2 + p γ µ 0 B 2 + (5)

13 Altered Model To enforce the B = 0 constraint we use a generalized Lagrangian Multiplier (GLM) approach, introducing the new variable ψ. To ensure, that the eigenvalues are smaller than c, we switch to the semi relativistic MHD model, Gombosi et al. [1] E = 1 2 ϱv 2 + U = ϱ ϱv + 1 c 2 B E ψ S A p γ µ 0 B µ 0 c 2 E ψ2 (5)

14 Altered Flux ( ϱv ) ϱv v T + p I µ 0 2 B2 I B B T + F = vb T ( Bv ) T + v 1 2 ϱv 2 + γp γ 1 + S A + (6)

15 Altered Flux ϱ ( v ) ϱv v T + p I µ 0 2 B2 I B B T + P E F = vb T B ( v T + c h ψ I ) v 1 2 ϱv 2 + γp γ 1 + S A + c h ψb c hb P E = 1 ( ) 1 µ 0 c 2 2 E 2 I E E T (6) (7)

16 Additional nonconservative- and source terms S = ( B ) 0 B v v T B vψ v ψ αψ (8)

17 Important properties of the semi relativistic model We consider the eigenvalues of the flux jacobian in x-direction with respect to the primitive variables ϱ, v, B, p, ψ and compute the speed of the Alfven waves: λ 3,4 = γ 2 A (v x + v E,x ) ± where γ 2 A v E = 1 2ϱc 2 S A, γ A = where the nonrelativistic Alfven speed is V A = ( ) VA,x 2 v x 2 + γa 4 (v x + v E,x ) 2, (9) 1 (10) 1 + V A 2 c 2 B µ0 ϱ.

18 Entropic properties of the GLM-MHD system Entropy function R( U) = ϱ r 1 γ, r = ln(pϱ γ ) (11)

19 Entropic properties of the GLM-MHD system Entropy function R( U) = ϱ r 1 γ, r = ln(pϱ γ ) (11) Entropy variables w = R U = ( γ 1 γ r βv 2, 2β v T, 2β B T, 2β, 2βψ β = ϱ 2p ) (12) (13)

20 Entropic properties of the GLM-MHD system Entropy function R( U) = ϱ r 1 γ, r = ln(pϱ γ ) (11) Entropy variables w = R U = ( γ 1 γ r βv 2, 2β v T, 2β B T, 2β, 2βψ β = ϱ 2p ) (12) (13) Entropy flux F R = vr (14)

21 Entropic properties of the GLM-MHD system Entropy flux potential Ψ = w T F F R + θ B, θ = 2β ( ) v B (15)

22 Entropic properties of the GLM-MHD system Entropy flux potential Ψ = w T F F R + θ B, θ = 2β ( ) v B (15) Entropy inequality w T ( U + F + S ) = R t + F R = 2αβψ 2 0 (16)

23 DG Approximation We first divide our computational domain Ω into elements Ω l. Then we multiply our equation (1) with a test function φ and integrate over Ω l. We transform into the reference element using a mapping x : [ 1, 1] 3 Ω l, (17) and integrate by parts to obtain the weak form 3 0 = J(u t + S) Ja [j] F φdξ + Ja [j] Fn j φds. [ 1,1] 3 ξ j j=1 [ 1,1] 3 (18) Here used the co- and contravariant vectors, the jacobi-determinant of the mapping x and and the metric identities a [i] = x ξ i, Ja [j] = a [j+1] a [j+2], J = a [j] a [j+1] a [j+2] 3 j=1 (19) ξ j Ja [j] = 0 (20)

24 DG Approximation To discretise this equation we use the Lagrange polynomials l i of the Legendre Gauss Lobatto nodes ξ 0,..., ξ N [ 1, 1]. Next we apply the corresponding quadrature rule to approximate the integral of a function f by 1 N f (ξ)dξ p(ξ i )ω i. (21) 1 We further approximate and test with U( ξ, t) N i,j,k=0 i=0 U ijk (t)l i (ξ)l j (η)l k (ζ) (22) φ = l 0 (ξ)l 0 (η)l 0 (ζ),..., l N (ξ)l N (η)l N (ζ). (23)

25 DG Approximation Using the defining property l i (ξ j ) = δ ij and the differentiation matrix D ij = l j (ξ i) yields the weak form DGSEM, where we use a two point numerical flux F on the cell interfaces. 0 = (JU t + JS) ijk ω ijk + Ja [1] F _jk ω jkδ in Ja [1] F _jk ω jkδ i0 +Ja [2] Fi_k ω ikδ jn Ja [2] Fi_k ω ikδ j0 + Ja [3] Fij_ω ij δ kn Ja [3] Fij_ω ik δ k0 N ( ) Ja [1] F pjk Dip T ω pjk + Ja [2] F ipk Djp T ω ipk + Ja [3] F ijp Dkp T ω ijp p=0 (24)

26 DG Approximation Finally we apply the SBP property Q + Q T = diag( 1, 0,..., 0, 1) of the matrix Q = diag(ω 0,..., ω N )D, to write the method in its strong form 0 = J (U t + S) ijk + N p=0 D ip ( Ja [1] F pjk ) +... ( ) ( ) + Ja [1] F_jk N Ja[1] F Njk δ in Ja [1] F_jk 0 Ja[1] F 0jk δ i (25)

27 DG Approximation To equip our scheme with additional properties, we exchange the Volume terms N p=0 D ipf p with the Flus difference 2 N p=0 D ipf # ip, where F # is a two point numerical flux. Examples: Standard DG: F # ip = 1 2 (F i + F p ) 2 Product rule: F # ip = 1 2 (G ih p + H i G p ) 2 N N D ip F # ip = D ip F p (26) p=0 p=0 N N D ip F # ip = G i D ip H p +H i D ip G p p=0 p=0 (27)

28 Averages As a final step we have to choose the surface flux F and the volume flux F #. Therefore we introduce notations for means and jumps {{a}} = 1 2 (a l + a r ), ((a b)) = 1 2 (a lb r + a r b l ) (28) a = a l a r, ((a)) ln = a ln(a) (29)

29 Entropy conservative flux Entropy conservation condition w T f = Ψ 1 {{B 1 }} θ (30)

30 Entropy conservative flux Entropy conservation condition w T f = Ψ 1 {{B 1 }} θ (30) EC flux in x direction: ((ϱ)) ln {{v 1 }} ((ϱ)) ln {{v1}} 2 {{B 1 }} 2 + p + 1 ({{ 2 B B2 2 + }}) B2 3 ((ϱ)) ln {{v 1 }} {{v 2 }} {{B 1 }} {{B 2 }} F #EC ((ϱ)) ln {{v 1 }} {{v 3 }} {{B 1 }} {{B 3 }} 1 = c h {{ψ}}, {{v 1 }} {{B 2 }} {{v 2 }} {{B 1 }} {{v 1 }} {{B 3 }} {{v 3 }} {{B 1 }} f5 EC c h {{B 1 }} (31)

31 Entropy stable flux F ES n = F #EC n 1 2 ΛH w, (32) where H discretizes the Jacobian U w and Λ controls the amount of dissipation added e.g. Λ = max( λ )I (33) local

32 Test setup We simulate in a domain close to Io, which again is simulated as a source term on the right hand side ( r c = 0, ωϱv, 0, 1 ) 2 ϱv 2, 0, (34) with ω io, if x is on io ω = ω io exp( d(x,io) d 0 ), if x in the ionosphere 0, else (35)

33 Test setup

34 Test plot

35 Test plot

36 Outlook Next steps will be:

37 Outlook Next steps will be: Divergence free implementation

38 Outlook Next steps will be: Divergence free implementation Suitable boundary conditions

39 Outlook Next steps will be: Divergence free implementation Suitable boundary conditions Semi relativistic terms

40 Thank you for your attention!

41 Tamas I Gombosi, Gábor Tóth, Darren L De Zeeuw, Kenneth C Hansen, Konstantin Kabin, and Kenneth G Powell. Semirelativistic magnetohydrodynamics and physics-based convergence acceleration. Journal of Computational Physics, 177(1): , A Mura, A Adriani, JEP Connerney, S Bolton, F Altieri, F Bagenal, Bertrand Bonfond, BM Dinelli, J-C Gérard, T Greathouse, et al. Juno observations of spot structures and a split tail in io-induced aurorae on jupiter. Science, 361(6404): , 2018.