Lecture 5) Eigenstructure and Approximate Riemann Solvers for Hyperbolic Conservation Laws

Transcription

1 ecture 5) Eigenstructure and Approimate iemann Solvers for Hperbolic Conservation aws B Prof. Dinshaw S. Balsara (dbalsara@nd.edu) es Houches Summer School in Computational Astrophsics 1

2 6.1) Introduction ecture 2 has shown us that monotonicit preserving reconstruction and iemann solvers are important for linear hperbolic equations. We studied the eigenstructure of the linear sstems. ecture 3 has shown us the kinds of simple waves -- shocks and rarefactions -- that we get in a non-linear, scalar conservation law. We also got our first glimpse of an approimate iemann solver. ecture 3 also showed that the same simple wave structures find analogues in sstems of hperbolic conservation laws. We described a eact and approimate (two-shock) iemann solvers. Here we stud the eigenstructure of the Euler and MHD sstems. The choice of these sstems is based on utilit. el. hdro & MHD are similar First goal : Stud eigenstructure for these sstems. Different waves/discontinuities behave differentl; can we understand them? 2

3 Eact iemann solvers have been designed for all these sstems. The barrier to their practical utilization in codes is computational compleit. Much of the information generated b such iemann solvers is never used in the computation of a numerical flu. As the hperbolic sstem gets larger, an increasing amount of information that is generated b the iemann solver is discarded. Second Goal: Stud linearized iemann solvers. These are approimate iemann solvers. 3

4 What are the essential elements of an eact or approimate iemann solver? 1) A Self-Similar Wave Model. (Wave Model == how we conceptualize the waves in the iemann solver. Ma not be the same as the eact S.) 2) Consistenc with the Conservation aw: Not just for infinitesimal fluctuations but also for isolated discontinuities of finite strength. 3) Entrop Enforcement: Arbitrar discontinuities to be resolved correctl. Need for conservation and entrop generation. 4) Preservation of Discontinities: Isolated contact discontinuities should not be smeared on the mesh. Shocks are self-steepening ok to smear them. Contacts are not must keep them intact as much as possible. All of the iemann Solvers discussed here are one dimensional. All multidimensional flow features are treated dimension-b-dimension. Imparts mesh imprinting. 4

5 inearized (Approimate) iemann Solvers: Tr to turn U t + F(U) = 0 into a suitable linear problem U t + Ā U = 0, Ā is a matri. It is eas to stud its eigenstructure. Ā(U, U ) depends on both the input states! A) When the difference between the left and right state is suitabl small, can obtain : ( ) U + A U = 0 with A F U U i.e. problem linearizes easil consistenc. t B) When left and right states differ b much, we still want a matri U + A U = 0 that mimics the above equation U + A U = 0. t t equation : C) The eigenstructure of A should have parallels to that of A in the linear regime. Question: What does this mean for eigenstructure? elate it to linear sstems (ect 3). D) When isolated discontinuities are present, the structure of A should be such that the can propagate at the "correct" spee d. Question: What about self-similarit? Question: What does that give us? Can ou relate it to what we learned in Chp 3? 5

6 ecall that the linearized iemann solver preserves all isolated discontinuities eactl. Sometimes, we ma not want to preserve all this structure. In retrospect, the linearized S is not positivit preserving; H S is positivit preserving. Which is wh we have several favorable variants of H. The H iemann solver, from ect 3, is an eample of such a iemann solver that washes out some structure et gives stabilit & positivit! inearized iemann solvers can also be temperamental performers, especiall in the vicinit of strong shocks. H iemann solvers are robust and stable, so are its variants. The variants of the H iemann solver HC, HI, HD can even capture isolated discontinuities eactl. Third goal: Design H, HC, HI, F iemann solvers and their variants for sstems. 6

7 6.2) The Eigenstructure of the Euler Equations 6.2.1) Derivation of the Eigensstem for the Euler Equations Write the Euler equations as: ( ) ( ) ( ) U + F U + G U + H U = 0 t z ρ ρ v ρ v ρ vz 2 v v + P ρ v v ρ ρ 2 ρ v vz ρ v v + ρ v + ρ v + P + ρ v v z = 0 t z 2 ρ v ρ v z vz ρ v vz ρ v z + P ε ( ε +P) v ( ε +P) v ( ε +P) v z ε 1 P ρ v 2 Γ 1 2 = e + with e 7

8 Write the -directional variation as: U + A U = 0 with A F U U given b t ρ 2 ( Γ 1) ρ 2 v + v 2v ( Γ 1v ) ( Γ 1v ) ( Γ 1v ) z ( Γ 1 v ) ρ 2 ρ v ρ v + vv v v 0 0 ρ v = 0 t ρ vz vv z vz 0 v 0 ρ vz ε ( Γ 1 ) ε 2 2 vh + v v H ( Γ 1) v ( 1vv ) ( 1vv ) z v Γ Γ Γ 2 with the total enthalp " H" defined b: ( 1) 1 2 Γ ρ H e + P + ρ v P = ρ H ρ ( 1 ) ε ρ 2 v = Γ Γ 2 v 2 ( ) 8

9 It is simpler to stud this sstem in terms of the vector of primitive variables: ( ρ z ) V v v v P T Question: If so, wh do we still want to stud the sstem in conservation form? This is most easil done b recasting the sstem with the Jacobian matrices : U V and (see tet). V U 9

10 More familiar primitive form: ρ v ρ ρ v 0 v ρ v v v 0 0 v = 0 t v v 0 v z z P 0 Γ P 0 0 v P { } Eigenvalues: v c, v, v, v, v + c where c s s s Γ P ρ Matri of ight Eigenvectors: Matri of Orthonormal eft Eigenvectors: 2 ( s) ( s) ρ 2c c c ρ c ρ c p = p = Question: Can ou phsicall interpret the above eigenvectors? s s s 2 2 cs cs 2 0 ρ ( 2cs) ( 2cs) 10

11 6.3) inearized iemann Solver for the Euler Equations Finding a linearization of the non-linear hperbolic sstem is tantamount to saing that we want to replace U t + F(U) = 0 b U + A U = 0 Task: For general left & right states U and U, find ( ) ( ) ( ) ( ) ( 11 = = ) t ( ) a matri A U,U (i) It is a linear mapping to the vector space "U" to the vector space "F". (ii) As U U U, we have A U,U (iii) For all U and U we have ( ) A( U). Yields co ( ) ( ) ( ) ( ) A U,U U U = F U F U Ensures that isolated discontinuities propagate at the right speed. et's see wh: nsistenc Assume U and U are left and right states of an isolated discontinuit that moves with speed "S". ( ) ( ) = ( ) ( ) ( ) ( ) Nonlinear cons. law gives : F U F U S U U inearized sstem should give : A U, U U U = S U U such that: of flu. I.e. the two sstems should predict the same propagation speed for the discontinuit. This is ensured if : F U F U S U U A U, U U U

12 ( ) linearl independent ( ) ( ) (iv) The eigenvectors of A U,U are. An jump U U can be projected into the eigenspace of A U,U. This is needed if we are to used methods drawn from linear hp. sstems in formulating a iemann solver. Question: How general can U and U be? Collectivel these four properties are known as Propert U, because the endow the iemann solver with uniform validit at discontinuities. Note that the linearized S is not positivit preserving. So in principle, the (iv) propert cannot alwas be guaranteed. Even so, the linearized S has some ver desirable theoretical properties it reduces dissipation to the minimum level that is allowed. That is wh we continue to stud it. The HI S can reproduce this propert of minimizing dissipation for intermediate waves. 12

13 There are man different was to get Propert U. Easiest approach is via parameter vectors: W T 1/2 ( w 1, w 2, w 3, w 4, w5) ρ ( 1, v, v, vz, H) ( ρ, v ) ρ, v ρ, v z ρ, H ρ T 1/2 T ( ) ρ ( z ) T 1/2 ( ) ρ ( z ) W w, w, w, w, w 1, v, v, v, H W w, w, w, w, w 1, v, v, v, H ecalling ε = 1 ρ H + Γ T ( Γ 1) 2 ρ v, we can write the vectors "U" and "F" as: 2Γ 2 w 1 ρ w w 1 2 ρ v w 1 w 3 ρ v w 1 w4 ρ v z U= = ( 1) ε 1 Γ w ( ) 1 w 5 + w 2 + w 3 + w 4 Γ 2Γ T T 13

14 T 1/2 ( ) ρ ( ) W w, w, w, w, w 1, v, v, v, H z ( ρ, v ) ρ, v ρ, v z ρ, H ρ ( 1) 1 2 Γ 1 2 ρ H e + P + ρ v P = ρ H ρ 2 v Γ 2 w 1 w2 ρ v 2 ( 1) 2 ρ v + P Γ ( Γ 1) w 1 w 5 + w 2 ( w ) 2 + w 3 + w 4 Γ 2Γ F= ρ v v = w 2 w 3 ρ v v z ρ H v w 2 w 4 w 2 w5 Notice that "U" and "F" are quadratic in the components of the parameter vectors. T T 14

15 Now realize the useful trick: = p q + q p ( p q ) p q p q = ( p p )( q q ) + ( q + q )( p p ) 1 1 where p ( p p ) and q ( q q ) From Calculus, recall that :- ( p q) d = p dq + q dp See the analog? We also define the oe - averaged variables : w ρ v + ρ v w w w v = ; v ; v ; H ; z w1 ρ w1 w1 w + ρ 1 ρ ρρ ( ) ( ) These variables enable us to represent U U and F F in ( ) terms of W W. et us see how it is done. 15

16 2 w 1 ρ w w 1 2 ρ v w 1 w 3 ρ v w 1 w4 ρ v z U= = 1 ( 1) ε Γ w ( w 5 + w 2 + w 3 + w 4 2 ) Γ Γ 2w w2 w w 0 w 0 0 U U = B W W = W W w4 0 0 w1 0 1 Γ 1 Γ 1 Γ 1 1 w5 w2 w3 w4 w1 Γ Γ Γ Γ Γ 3 1 ( ) ( ) 1 ( ) ( ) ( ) It is also useful to write :- W W = B U U 16

17 2w w2 w w 0 w 0 0 U U = B W W = W W w4 0 0 w1 0 1 Γ 1 Γ 1 Γ 1 1 w5 w2 w3 w4 w1 Γ Γ Γ Γ Γ v v = w 1 ( W W) vz Γ 1 Γ 1 Γ 1 1 H v v vz Γ Γ Γ Γ Γ 3 1 ( ) ( ) This defines the matri B. Notice that B is lower triangular, therefore, it is eas to invert. 17

18 w2 w Γ 1 Γ+ 1 Γ 1 Γ 1 Γ 1 w5 w2 w3 w4 w 1 Γ Γ Γ Γ Γ F F = C W W = W W 0 w w w4 0 w2 0 0 w5 0 0 w 2 v Γ 1 Γ+ 1 Γ 1 Γ 1 Γ 1 H v v v z Γ Γ Γ Γ Γ = w 1 0 v v 0 0 W W 0 vz 0 v 0 0 H 0 0 v ( ) ( ) 3 2 This defines the matri C. 1 ( ) ( ) ( ) From the previous page, recall that :- W W = B U U 1 ( ) ( ) ( ) We then have from this page that :- F F = C W W = C B U U ( ) This enables us to identif : A U,U ( ) C ( B) 1 18

19 ( Γ 1) 2 v + v 2v Γ 1v Γ 1v Γ 1vz Γ A( U, U) = C ( B) = vv v v 0 0 vv z vz 0 v 0 ( Γ 1) 2 2 vh + v v H ( Γ 1v ) ( Γ 1vv ) ( Γ 1vv ) z Γ v 2 ( ) ( ) ( ) ( ) The amazing fact is that this matri is formall just the same as the original characteristic matri!! We have no reason to epect this, since the variables in the above matri have no phsical standing; the onl have a formal standing. ( ) Γ We make the connection more concrete b defining : P ρ H ( v + v + vz) Γ 2 ecall that phsicall speaking we indeed do have : P = ( ) Γ ρ H v Γ 2 We can thus epress the eigenvalues of the above matri as: 2 Γ P { v c, v, v, v, v + c } where c s s s ρ 19

20 This eact, formal analog can be used to derive the eigenvectors too. We just make the formal transcriptions: ρ ρ ; v v ; v v ; v v ; H H ; c c z z s s Notice two important points of difference: A) If U and U correspond to the eact jump in a right-going shock then v c + is the right-going shock speed. One cannot sa the same about v + c. B) For an phsical state, P is guaranteed to be positive. We can't guarantee positivit of P for an two phsical states U and U. s s Notice too that nothing has been said so far on the entrop fi. From the in ecture 3, we know that we must have an entrop fi. discussion The linearized iemann solver has also been formulated for real gases. 20

21 6.4) Entrop Fies for inearized iemann Solvers B linearizing a non-linear sstem of conservation laws, we wish to replace the non-linear sstem with an equivalent linear sstem. We have studied linear hperbolic sstems. We, therefore, wish to draw upon insights from that stud in Section 3.4. ( ) Having obtained an M M matri A U, U, we perform its characteristic analsis: m m m { λ m= M} { l m= M} { r m= M} I.e. we obtain : 1,...,, : 1,...,, : 1,...,. ( ) Given an arbitrar jump across a zone boundar U U, we can project it onto the space of right eigenvector s: M m m m m = α r α l m= 1 U U where U ( U ) This just represents each simple wave discontinuit as a discrete jump in the solution without eamining whether it is a phsical compressive shock or an unphsical rarefaction shock. It is up to us to go back and 21 provide an entrop fi to the rarefaction shocks.

22 As with linear hperbolic sstems, we get M + 1 constant states in space-time: ( t) U, = U for m M m p p p p + α r = α r p= 1 p= m+ 1 ( ) = U U U = U for λ And their associated flues: t 1 < λ m m+ 1 for, 1,..., 1 λ M < < λ m= M t < t ( t) F, = F m M ( m) p p p p p p m m 1 = F F + λ α r = F λ α r for λ < < λ +, m= 1,..., M 1 p= 1 p= m+ 1 t = F for t 1 < λ M for λ < t 22

23 Find 0 m0 m0 λ λ + m 0 m < 0 such that: 1 helps us find resolved state and numerical flu. ( m 1) ( m ) ( m + 1) To appl an entrop fi, find the three states U, U and U. Whether or not an entrop fi is needed is decided b eamining wave speeds in those three s tates, as shown net. 23

24 We onl show the characteristics associated with the m and m + 1 wave families. 0 0 Questions: Which of these waves is a shock, which of them are rarefactions? Which of of the wave families in each panel need an entrop fi? Assume non-moving boundaries. 24

25 Further caveats: (i) Eamine the resolved states for positivit, linearized iemann solvers have problems when it comes to maintaining positivit. m0 0 (ii) One usuall uses a shortcut b defining U λ m m0 m0 λ ( ) and λ λ ( U ) m0 m0 and using the condition λ < 0 < λ to identif the formation of a rarefaction fan in the m th 0 wave famil. We provide two entrop fies: Entrop Fi # 1: m0 m0 If λ < 0 < λ our resolved flu sh ( m 1) ( m ) 0 0 F and F. ( ) ( 1) Here we have F F so that the full contribution from th m0 m0 m0 the wave famil is. m 0 m m m m m = + λ λ α r α ould be a linear combination of the flues r 25

26 With 0< β <1, let the contribution from the left of the zone boundar be : βλ α m m m ( ) Similarl, let the contribution from the right of the zone boundar be : 1 β λ α r r m m m The two contributions should add up to the full contribution that we identified before: ( ) β λ α r + 1 β λ α r = λ α r m m m m m m m m m so that: β = λ λ m m λ λ m 0 0 m 0 0 With the above definition, and several other ancillar definitions ( see below), we are read to obtain closed form epressions for the conserved variables and flues. When 0, we make the following definitions:- ( ) ( ) ( ) ( ) λ ma λ,0 ; λ min λ,0 ; α α H λ ; α α H λ +, m m, m m +, m m m, m m m When λ λ m m λ λ m m λ λ λ λ λ < 0, we make the following definitions:- m m m m m m m m +, m m, m m +, m m, m m ; λ ; ; m m α α α α m m m m m m λ λ λ λ λ λ λ λ Where we also define:- λ λ λ λ λ λ λ λ λ m +, m, m 26

27 M ( S ), m m lin + α r m= 1 U = U = U M m= 1 α +, m 1 1 = U U 2 2 r m M +,, ( + ) ( α α ) m= 1 r m m m M ( S ), m m m F lin = F + λ α r = F + A U U m= 1 ( ) M +, m m m + = F λ α r = F A U U m= = ( F + F ) 2 2 M m= 1 λ α r m m m ( ) 1 1 = F F A U U 2 2 ( + ) ( ) This is ver reminiscent of the linear case, ecept that the above definitions build in the entrop fi. The book provides further detail. 27

28 All is not perfect with linearized iemann solvers: High speed flows can be especiall problematical. Question: Wh? We have a need for positivel conservative schemes, i.e. schemes that can guarantee that the pressure and densit remain non-negative. The linearized iemann solver is not positivel conservative. H is. The states in the net figure can cause a linearized iemann solver to choke! 28

29 The are also susceptible to a carbuncle instabilit when strong shocks move slowl relative to the mesh. 29

30 6.5) H, HC and HI iemann Solvers Consider the states in the hdrodnamical iemann problem, shown below. Below we also show the simplifications in the wave model that are made for the H and HC iemann solvers. The C stands for contact. 30

31 6.5.1) H iemann Solver Question: For the scalar conservation law, which structure did the wave model of the H iemann solver average over? Question: If we make a similar wave model for the Euler equations, which structures are we averaging over? U if S > 0 UH = U if S 0 S with U = U if S < 0 ( S ) S U S U ( F F) S S F if S > 0 ( S ) FH = F if S 0 S F if S < 0 S S S S with F = F F + U U S S S S S S 1 M ( λ ( ) ) λ ( ) ( ) ( ) and S min U, v c ; S ma U, v + c s s 31

32 For a scalar conservation law, we have no further requirements. For Euler equations, we would make some further demands: 1) Notice that if we have an isolated right-going shock, the H iemann solver picks out the correct flu. Question: For a transonic shock (with S < 0 < S ) this is non-trivial. Can ou prove it? 2) For open rarefaction fans that straddle the zone boundar, the entrop fi is also naturall built in. 3) The H iemann solver also maintains positivit of the resolved state U *. A result from Einfeldt et al (1991) then claims that a scheme that uses such a iemann solver will also keep the pressure positive. This guarantee is onl true in 1D; it does not etend to 2D and 3D. 32

33 6.5.2) HC iemann Solver (Philosoph:- Start with H & build on top of it.) The HC iemann solver is designed to overcome the one failing of the H iemann solver the inabilit to capture contact discontinuities. The resolution is to put two constant states between the left and right states that are separated b a contact discontinuit moving with a speed S M. This is an improvement of our wave model. U U if S > 0 U if S 0 S ( S ) M HC = U if SM 0 S U if S < 0 F F if S > 0 F if S 0 S ( S ) M HC = F if SM 0 S F if S < 0 For the contact discontinuit to have the properties of an actual contact discontinuit, we demand: S = v = v = v 33 M

34 Question: But how do we obtain S? Answer: Use the components of U from the H iemann solver. S M ( ) ρ ( ) ρ ( S v ) ρ ( S v ) U ρ v S v v S v + P P * 2 M = v = v = v = = * U1 * * A pleasant corollar of making this choice is that the pressure P is a constant * * * across the contact. I.e. P is the same in U and U. * * Question: But how do we obtain the post-shock states, U and U? Answer: Jump conditions at discontinuities: ( ) eft: F F = S U U SU F = SU F ight: F F = S ( U U ) SU F = SU F 34

35 eft: S ρ S ρ ρ ρ ρ v ρ M ρ 2 S S P v v v P M ρ M + ρ ρ + v ρ S v M = S v v v ρ ρ v ρ S v z M z v ρ v z v ρ z ε P SM P v ( ε ) + ε ( ε + ) ight: S ρ S ρ ρ ρ ρ v ρ M ρ 2 S S P v v v P M ρ M + ρ ρ + v ρ S v M = S v v v ρ ρ v ρ S v z M z v ρ v z v ρ z ε v ( ε P) S + M ε ( ε + P ) S v S v = = First ow gives: ρ ρ ; ρ ρ ; Notice ρ ρ. Wh is that good? S SM S SM ( )( ) ( )( ) Second ow ields constanc of pressure: P = P = P + ρ S v S v M = P = P + ρ S v S v M 35

36 Third and fourth rows give constanc of transverse velocit: v = v ; v = v ; v = v ; v = v z z z z Fifth row gives the total energ densit: ε = ( ) ε S v P v + P S M S S M ; ε = ( ) ε S v P v + P S M S S M ε and ε are formal entities. Their derivation is purel based on a formal jump condition. Don't ever use them to derive a "pressure". The HC iemann solver inherits all the same good properties that we catalogued for its progenitor H iemann solver. Thus it too will: 1) represent isolated shocks eactl. 2) Have a built-in entrop fi for transonic rarefactions. 3) Make an scheme that uses it positivel conservative in 1D (not 2D/3D). 4) It does its H parent one better representing isolated contact 36 discontinuities eactl. It is also almost as fast as the H iemann solver

37 HI iemann Solver (Philosoph:- Start with H & build on top of it) The HC S is a decided improvement over the H S, because it restores the contact discontinuit. But notice that we had to work ver hard to derive the HC S; and it onl restores the contact discontinuit. What if we had several waves in a iemann fan that we wanted to preserve crispl? One option would be to restore multiple discontinuities into an H S. From our stud of the HC S, we realize that this would be a lot of work. Another option, suggested b Einfeldt and Munz (hence the EM ), would be to introduce a linear profile within H. Intuitivel, the right kind of profile would make more of the waves of interest flow in a particular direction, thereb reducing the dissipation of 37 that wave famil. Ke benefit:- We can improve an set of wave families!

38 Intuitivel speaking, realize that the inearized S and HC S also reduce dissipation b introducing sub-structure in the iemann fan. We build on that idea. From the linearized S, we realize that is should be related in some form to the characteristic weight :- α p = l p ( U U ). The real questions are:- What form of linear profile is optimal? And, how much of that profile is needed to eactl cancel the ecessive dissipation from the H S? The details have been worked out in Dumbser & Balsara (2016). The formulation works for hperbolic sstems in conservative or nonconservative form. Ke idea is to eploit the self-similarit of the iemann fan. As a result, we develop the idea in self-similarit variables. 38

39 (, ) F( t, ) U t Tpicall, we start with + = 0 t For special situations where the PDE sstem is indeed evolving self-similarl, we can write the problem in terms of the self-sim ilarit variable ξ = t. We can then write :- U t, = U ξ ; F t, = F ξ. ( ) ( ) ( ) ( ) ξ ξ ξ 1 = = ; = = ξ ξ ξ ξ 2 t t t t (, ) F( t, ) U t + = 0 t U ( ξ) F ( ξ) ξ + = 0 ξ ξ ealize that " ξ" parametrizes various space-time points in the space-time diagram, going from ξ=s on the left to ξ=s on the right. S C F D ξ =S U S T T F E t U S T ξ =S U S B F A 39

40 For H S we can write U if ξ < S U U if S S with U ( ) ( ξ) ( ) S S U S U F F H = ξ = S S U if S < ξ th For HI S we can improve the sharpness of the p wave b writing the linear profile as:- U if ξ < S ( ) ( ( S S ) 2 S ξ + p p p ) UHI ( ξ ) = U + 2 δ r l ( U U) if S ξ S ( S S ) U if S < ξ ecall that U is still the H state in the above formula. If there are multiple waves that we want to improve (like contact + shear waves for Euler flow) or (like contact + Alfven waves for MHD flow) just add the contributions of those waves additivel to the linear profile. To make this work out just right (so that the dissipation is held to a minimum), we have p to specif the scalar δ appropriatel for each wave famil. We do that net. 40

41 F if 0< S ( S ) SS p p p FHI ( ξ ) = F δ r l ( U U) if S 0 S ( S S) F if S < 0 ecall that F is still the H flu in the above formula. If there are multiple waves that we want to improve waves additivel to the linear profile. just add the contributions of those The HI iemann solver is full specified b setting the coefficient δ p for the p th wave as: - p p ( λ ) ( λ ) min,0 ma,0 p δ = 1 S S 1) Just like the H S, the HI S does not require an further entrop fi. 2) Just like the H S, the HI S has good positivit enforcement properties. 3) HI improves over HC/HD because multiple intermediate waves can be steepened. 41

42 6.5.3) F iemann Solver ike before, it is based on taking the etremal speeds in the H iemann solver to get: ( 1 M λ ( ) ( ) ) s s λ S ma U,v c,v + c, U ; S = S = S Ma Ma ( S ) 1 S F F = F F U U 2 2 Ma ( + ) ( ) 42

43 6.6) Intercomparing iemann Solvers for the Euler Equations Question: What differences do ou see in the densit profiles below? a) b) oe H c) d) HC HI 43

44 6.7) Eigensstem for Non-elativistic MHD Can be written in a conservation form as: ρ v ρ Note the anisotropic pressures ρ v ρ v + P + /8 π B /4 π B v Also note the magnetic tension ρ v v B B /4 ρ π ρ v ρ v v z B B z/4π terms. z + 2 ( ε +P+ B /8π) v B ( v B) /4π ε t B 0 B ( v B v B ) B z ( v z B v Bz) ρ v ρ vz ρ v v B B /4 π ρ v v z B B z/4π ρ v + P + B /8 π B /4 π ρ v v z B B z/4π ρ v v z B B z/4 π ρ v z + P + B /8 π B z/4π + 2 ( +P+ /8 ) v B ( ) /4 + ε π π B v B 2 = 0 z ( ε +P+ B /8π) v z B z( v B) /4π ( v B v B) ( v z B v Bz) 0 ( v B z v z B) ( v B z v z B) 0 First 5 rows represent conservation of mass, momentum & energ. ast B 1 3 represent the induction eqn.: + c E = 0 ; E = v B 44 t c

45 With the constraint B = 0, the induction equation ensures that if it is satisfied at the beginning of a calculation, it will be satisfied for all time. B 0 results in unphsical plasma transport orthogonal to the field. In a later chapter we will see how the constraint is imposed at a discrete level in a numerical code. For -directional variations, it ensures that B is constant ( Γ 1) B 2 Bz v + v 2v ( Γ 1v ) ( Γ 1v ) ( Γ 1v ) z ( Γ 1) ( 2 Γ) ( 2 Γ) 2 4π 4π B vv v v π B A= vv z vz 0 v 0 0 4π δ51 δ52 δ53 δ54 δ55 δ56 δ 57 1 B B ( Bv Bv ) 0 0 v 0 ρ ρ ρ 1 Bz B ( Bv z Bv z) v 45 ρ ρ ρ

46 ( ρ z z) Using the vector of primitive variables, V v v v P B B we can write the hperbolic sstem as V + A V = 0 with : t p T A = p v ρ B Bz 0 v 0 0 ρ 4πρ 4πρ B 0 0 v πρ B v πρ 2 0 ρ cs 0 0 v B B 0 0 v 0 0 Bz 0 B 0 0 v Define the Alfvenic speeds as (The are useful for defining the wave speeds.) B B B b ; b ; b ; b b + b + b ; b b 4πρ 4πρ 4πρ z z z 2 + b z 46

47 The wave speeds form an ordered set given b: { v m, v b, v m, v, v + m, v + b, v + m } f s s f f ( ) To get m and m, solve the quartic : m c + b m + c b = f s s s Here m and m are the fast and slow magnetosonic wave speeds (m m 0) s f s It is important to learn the nomenclature of the waves, see fig. below: Questions: Compare and contrast entrop waves in MHD and Euler flow. Which waves take the place of shear waves? Are magnetosonic waves compressive? Which waves are the analogues of sound waves? As B 0, which waves go to 0? 47

48 The magnetic field breaks the degenerac of the problem. Unlike sound waves, MHD waves do not propagate isotropicall relative to the fluid s rest frame. The wave propagation diagrams show B increasing from a) through c). Question: What do ou see? Interpret the wave diagrams. 48

49 Eigenvalues become degenerate eigenvectors become indeterminate. The indeterminac is multiplicative and can be treated. This ields an eigenvector set that is orthonormal and salient in all the limits seen above (oe & Balsara 1996). Have to pa careful attention to the coefficients in all the limits. p αf ρ 0 αs ρ 1 αs ρ 0 αf ρ αf mf 0 αs ms 0 αs ms 0 αf mf αs m s β,s βz αf m f β,s 0 αf m f β,s βz αs m s β,s = αs m s βz,s β αf m f βz,s 0 αf m f βz,s β αs m s βz,s αf ρ cs 0 αs ρ cs 0 αs ρ cs 0 αf ρ c s αs 4 πρ c s β 4πρ βz,s α f 4 πρ c s β 0 α f 4 πρ c s β 4 πρ βz,s αs 4 πρ c s β αs 4 πρ c s βz 4 πρ β,s αf 4 πρ c s βz 0 αf 4 πρ c s βz 4 πρ β,s αs 4 πρ c s β z 49

50 6.8) inearized iemann Solver for the MHD Equations We define the oe-averaged magnetic fields differentl, retaining the original definitions of the velocities etc. ρ B + ρ B ρ B + ρ B B ; B ρ + ρ ρ + ρ z z z This allows us to derive (after a lot of smbolic manipulation) ( ) ρ v ( ρ v) ( B) 2 B = X ρ + B B ; X v X P = Γ 1 2 4π ( ρ ) ρ ε B B 4π 50

51 Manipulations similar to those for the Euler equations allow us to obtain a formal similarit to the actual, phsical, characteristic matri: ( Γ 1) 2 X B Bz v + v + ( 2 Γ) 2v ( Γ 1v ) ( Γ 1v ) ( Γ 1v ) z ( Γ 1) ( 2 Γ) ( 2 Γ) 2 4π 4π 4π B vv v v π B A= vv z vz 0 v 0 0 4π δ51 δ52 δ53 δ54 δ55 δ56 δ 57 1 B B ( Bv Bv ) 0 0 v 0 ρ ρ ρ 1 Bz B ( Bv z Bv z) v ρ ρ ρ ecall the Euler case. Familiar etensions to the definition of the sound speed follow: 2 ( Γ 1) v B 2 ( Γ 2) 2 Γ P X c s = ; P ρ H ρ ρ Γ 2 4πρ Γ 4π Analogous definitions for the Alfvenic speeds can also be obtained: B B 4πρ 4πρ b ; b B ; b ; b b + b + b ; b b + b 4πρ z z z z 51

52 The ordered set of analogous eigenvalues are given b: { v m f, v b, v m s, v, v + m s, v + b, v + mf} and the are obtained from the analogous quartic: ( ) m c + b m + c b = s s We can also obtain the analogous eigenvectors: p αf ρ 0 αs ρ 1 αs ρ 0 αf ρ αf mf 0 αs ms 0 αs ms 0 αf mf αs m s β,s βz αf m f β,s 0 αf m f β,s βz αs m s β,s αs m s βz,s β αf m f βz,s 0 αf m f βz,s β αs m s βz,s = 2 X 2 X X 2 X 2 X αf ρ cs 0 αs ρ cs αs ρ cs 0 αf ρ cs 4π 4π 4π 4π 4π αs 4 πρ c s β 4 πρ βz,s α f 4 πρ c s β 0 α f 4 πρ c s β 4 πρ βz,s αs 4 πρ c s β αs 4 πρ c s βz 4 πρ β,s α f 4 πρ c s βz 0 α f 4 πρ c s βz 4 πρ β,s αs 4 πρ c s β z We use the same entrop fies as before. There is some analsis to show that the effects of non-conveit are not severe. 52

53 6.9) H, F, HD and HI iemann Solvers for MHD 6.9.1) H and F iemann Solvers We keep B constant across all these iemann solvers. The H and F iemann solvers continue to have the same definitions. All that changes is the definitions of the etremal signal speeds: For H we use: 1 M ( λ ( ) ) λ ( ) ( 1 M λ ( ) ( ) ) f f λ ( ) S min U, v m,0 ; S ma U, v + m,0 f f For F we use: S ma U,v m,v + m, U Ma 53

54 6.9.2) HD iemann Solver for MHD ecall : The Euler equations have one contact discontinuit. estoring it in the H iemann solver gave us the HC iemann solver. The MHD equations have three linearl degenerate waves : One contact discontinuit and two Alfven waves. estoring these waves in the H iemann solver gives us the HD iemann solver. The D stands for these discontinuities. 54

55 The HD iemann solver uses the following wave model with 6 constant states: F if S > 0 F if S 0 S F if S 0 S F ( S ) M HD = F if SM 0 S F if S 0 S F if S < 0 S M, S and S will correspond to the speed of the contact discontinuit and the Alfven waves. We will obtain S using a strateg that is similar to the Euler case. M 1) The HD iemann solver for MHD is also provabl positivel conservative. 2) Just like the H iemann solver, it too has an in-built entrop fi. 3) ike the HC, it too will capture isolated fast magnetosonic shocks eactl. It does average over the slow magnetosonic shocks. However, the slow magnetosonic shocks can onl support a small pressure jump, so their contribution to the overall discontinuit is not too large. Hence, 55 it is ok to ignore them.

56 As with the HC iemann solver, we wish to keep the total pressure constant across the contact discontinuit and Alfven waves: 2 B π P T P + 8 As with the HC iemann solver, this is achieved b asserting that the -component of the velocit is a constant across the iemann fan: M ( ) ρ ( ) ρ ( S v ) ρ ( S v ) ρ v S v v S v + P P S = v = v = v = v = T T * * As with the HC iemann solver, the states U and U are obtained b writing jump conditions: S ρ ρ S M ρ ρ v 2 2 ρ S ρ SM + PT B 4π 2 2 M ρ v PT B 4 v ρ + π ρ v ρ S v M BB 4π ρ v ρ vv BB 4π ρ v ρ S v z M z BB z 4π = S vz vvz BB z 4 ρ ρ π ε ( ε + PT) SM B( v B) 4π ε ( PT ) v B( ) 4 ε + v B π B B SM Bv B B v Bv B z Bz SM Bv B z z Bz v Bv z 56

57 The first row gives: ρ = ρ S S v S M 2 2 ρ( S v ) B ( )( ) ( )( ) The second row gives: P = P = P = P + ρ S v S v T T T T M BB S v The third and sith rows give: v = v ; 4π ρ S v S S B 4π and B = B ρ M M 2 ( )( M ) 4π S v S S B 4π th th (Similar epressions are obtained from the 4 and 7 rows.) 2 The fifth row gives: ε = ( ) ε T + T M + ( v B v B) S v P v P S B 4π S S M As with the HC iemann solver, these epressions are based on a formal analog. 57

58 B B = = + 4 4πρ The Alfven wave speeds are written b analog: S S M ; S S M πρ The jump conditions across the contact discontinuit give the epected results: P = P ; v = v ; vz = v z ; B = B ; Bz = B z T T Asserting the formal jumps across the Alfven waves, unfortunatel, does not give us as much insight as we wish. Nor does it give us clean epressions for all the ** ** U and U variables. We can onl squeeze so much out of our analogies! ε ε ( v B v B ) ( ) We get : ρ = ρ ; P = P ; = ρ sgn B 4π T T astl, integrating over the dotted rectangle in the previous space-time diagram gives: ( ) ( ) ρ v + ρ v + B B sgn B 4π v = v = ; ρ + ρ B = B = ( ) ( ) ρ B + ρ B + ρρ v v sgn B 4π ρ + ρ 58

59 6.10) iemann Problem for the MHD Sstem The result of the iemann problem for the Euler sstem is eas to classif, not so for MHD, where over 200 outcomes are possible. The sstem is non-conve and non-strictl hperbolic wave families can intersect and form compound waves. A shock can be attached to a rarefaction wave, as we saw in Chapter 4. Fast magnetosonic shocks increase the transverse component of the magnetic field. Slow magnetosonic shocks decrease the transverse component of the magnetic field. Question: arefaction fans act oppositel, so how are transverse fields treated b rarefaction fans? This opens the door to switch-on fast magnetosonic shocks or switch-on slow magnetosonic rarefactions. ikewise, we can have switch-off fast magnetosonic rarefactions or switch-off slow magnetosonic shocks. 59

60 60

61 ( z z) ρ, P, v, v, v, B, B = ( 1, 1, 0, 0, 0, 4 π, 0) for <0 Γ =2; B = π = ( 0.125, 0.1, 0, 0, 0, 4 π, 0) for >0 The left-going slow compound wave is genuinel a consequence of the field reversal in a coplanar problem. If there were a non-coplanarit, it would not show up. 61

62 ( z z) ρ, P, v, v, v, B, B = ( 1.08, 0.95, 1.2, 0.01, 0.5, 3.6, 2.0) for <0 Γ =5/3; B = 2 = ( 1, 1, 0, 0, 0, 4, 2) for >0 62

63 63

64 ( z z) ρ, P, v, v, v, B, B = ( 1, 1, 0, 0, 0, 4 π, 0) for <0 Γ =5/3; B = 4π = ( 0.2, 0.1, 0, 0, 0, 0, 0) for >0 ight-going switch-on fast shock. 64

65 Preserving Stationar Contact Discontinuities (H v/s HI) Preserving Stationar Alfven waves (H v/s HI) 65