Introduction to Riemann Solvers

Transcription

1 CO 5 BOLD WORKSHOP 2, 2 4 June Introduction to Riemann Sovers Oskar Steiner

2 . Three major advancements in the numerica treatment of the hydrodynamic equations Three major progresses in computationa fuid dynamics of the past 4 years incude: the conservative formuation of the computationa scheme in terms of finite voumes, the technique of approimate Riemann-sovers for the computation of numerica fues, the fu-imiter technique for maintaining stabiity and monotonicity of higher-order accurate scheme. A these techniques are part of the numerica scheme of CO 5 BOLD.

3 2. Conservation aws finite voumes Consider the continuity equation: ρ t + (ρu) =. () Integration over a finite voume, V, and time period, T, eads to the integra form of this equation: V ρ(t,)dv V ρ(,)dv = T V (ρu) ndsdt (2) Soutions to Eq. (2) are caed weak soutions to the partia differentia equation (). Additionay to the soutions of Eq. (), the set of soutions to Eq. (2) encompasses discontinuous soutions, because no derivatives appear in Eq. (2). Discontinuous soutions to the Euer equations represent shock fronts of the rea word.

4 Conservation aws finite voumes (cont.) Consider the mass conservation in a one-dimensiona tube: A ( ρv) m ( ρv) 2 m(t + t) = m(t) + ρv A t ρv 2 A t (3) ρ (t + t) = ρ (t) t ( ρv 2 ρv ) (4) in the imit of and t ρ t = (ρv) But Eq. (3) is identica to the integra form Z V ρ(t,)dv Z V ρ(,)dv = Z T I V (ρu) n ds dt

5 Conservation aws finite voumes (cont.) The conservative, finite voume formuation has three highy desirabe properties: Conserved quantities (mass, momentum, energy) remain accuratey conserved Discontinuous soutions are incude by soving the integra form of the partia differentia equation It fufis one of two requirements of the theorem of La and Wendroff (9) that says: The approimate soution that is computed with a consistent and conservative scheme converges to a weak soution of the conservation aw.

6 Conservation aws finite voumes (cont.) Euer s equation in one dimension is given by where q t + f(q) =, q n+ i q = ρ ρu E C A f(q) = = q n i + t [f i /2 f i+/2 ], ρu ρu 2 + p u(e + p) C A In 3-D we have q t + f(q) + g(q) y + h(q) z =, with q = ρ ρu ρv ρw E C A f(q) = ρu ρu 2 + p ρuv ρuw u(e + p) C A

7 3. Riemann sovers A conservative finite voume scheme is an eact representation of the integra form of the partia differentia form of the conservation aw. The probem consists in computing the correct fu function f(q), i.e., ρv in the case of the continuity equation. It turns out that these fues can be computed eacty.

8 Riemann sovers (cont.) Idea of S.K. Godunov (959): Piecewise constant reconstruction with discontinuities at ce interfaces q q q i i+

9 Riemann sovers (cont.) The shock-tube probem t ρ ρ ρ r p p p r v v = v r =

10 Riemann sovers (cont.) The shock-tube probem t t ρ ρ ρ ρ ρ* ρ* r ρr ρ r p p p p p* p r p r v v v * v = v r = v v r

11 Riemann sovers (cont.) The shock-tube probem t t ρ ρ ρ ρ ρ* ρ* r ρr ρ r t p p p p p* t q * q * r p r p r q q r v v v * v = v r = v v r

12 Riemann sovers (cont.) The shock-tube probem t t ρ ρ ρ ρ ρ* ρ* r ρr ρ r t p p p p p* t q * q * r p r p r q q r v v v * v = v r = v v r q q = q r q = q r * q = q * q = q rf q = q

13 3.. The Riemann sover of Harten, La, and van Leer (HLL) Consider the system of one-dimensiona conservation aws q t + f(q) =, q(, ) = ( q if <, q r if >. T s t u = u a s r = q * q * r u+a q q r T s T s r r The integra form in the contro voume [, r ] [, T] is given by: Z r q(, T)d = Z r q(, )d + Z T f(q(, t))dt Z T f(q( r, t))dt

14 The HLL sover (cont.) Z r Z r q(, T) d = q(, )d + Z T f(q(, t)) dt Z T f(q( r, t)) dt = r q r q + T(f f r ), f = f(q ), f r = f(q r ) Z r q(, T) d = Z Ts q(, T) d + Z Tsr Ts q(, T) d + Z r Ts r q(, T) d = Z Tsr Ts q(, T) d + (Ts )q + ( r Ts r )q r T(s r s ) Z Tsr Ts q(, T)d := q h = s rq r s q + f f r s r s

15 The HLL sover (cont.) T u s = u a s r = T s q q * t q * r q r T u+a s r r Appying the integra form to the contro voume [, ] [, T] we obtain: Z Ts q(, T)d = Ts q + T(f f ), where f is the fu f(q) aong the t-ais. Hence, f = f s q T Z Ts q(, T)d. Doing the same for the contro voume [, r ] [, T] eads to f r = f r s r q r T Z Tsr q(, T)d. It foows that f = f r.

16 The HLL sover (cont.) Harten, La, and van Leer put forward the foowing approimation: s q t sr q h q(, t) = q r 8 >< >: q if t s, q h if s t s r, q r if t s r. f h = f + s (q h q ) or f h = f r + s r (q h q r ) f h = s rf s f r + s s r (q r q ) s r s

17 The HLL sover (cont.) The corresponding interce fu for the approimate Godunov method is then given by: f h i+/2 = f if s, s r f s f r + s s r (q r q ) s r s if s s r, f r if s r that can be used in the epicit conservative formua q n+ i = q n i + t [f i /2 f i+/2 ].

18 3.2. The HLLC sover The HLLC scheme is a modification of the HLL scheme in which the missing contact and shear waves are restored. s q t q* s* q* q r s r q(, t) = 8 >< >: q if t s, q if s t s, q r if s t s r, q r if t s r. Integrating over appropriate contro voumes, or more directy, by appying the Rankine-Hugoniot Conditions across each wave, we obtain f = f + s (q q ), f r = f + s (q r q ), f r = f r + s r (q r q r ).

19 The HLLC sover (cont.) The corresponding interce fu for the approimate Godunov method is then given by: f hc i+/2 = f if s, f + s (q q ) if s s, f r + s r (q r q r ) if s s r, f r if s r that can be used in the epicit conservative formua q n+ i = q n i + t [f i /2 f i+/2 ].

20 The HLLC sover (cont.) The intermediate states q and q r can be derived from f = f + s (q q ), f r = f + s (q r q ), f r = f r + s r (q r q r ), u = u r = u, p = p r = p, v = v, vr = v r, w = w, wr = w r, s = u, q k = ρ k sk u k s k s 2 «4 s v k w k E k ρ k + (s u k )[s + p k ρ k (s k u k ) 3 5, k =, r

21 3.3. Wave-speed estimates In order to competey determine the numerica fues in the HLL Riemann sover we need estimates for the wave speeds s and s r, and, for the HLLC sover, s. Given a positive speed s +, a simpe choice woud consist in s = s +, s r = s +. It is interesting to note that if we set s + equa to the maima speed according to the CFL-condition, i.e., s + = t, we obtain La-Friederichs numerica fu f i+/2 = 2 (f f r ) 2 which brings us back to a cassica scheme. t (q r q ),

22 Wave-speed estimates (cont.) More ingenious choices are motivated by a Roe average of the eft and right states, e.g., s = ũ ã, s r = ũ + ã, where ũ = ρ u + ρ r u r ρ + ρ r, ã =» (γ )( H 2ũ2 ) 2, with the enthapy H = (E + p)/ρ approimated as H = ρ H + ρ r H r ρ + ρ r.

23 Wave-speed estimates (cont.) In a different approach we first suppose to have estimates for p and u for the pressure and veocity in the Star Region. Then we compute the foowing wave speeds: s = u a r, s = u, s r = u r + a r r r, where r k = 8 >< >: if p p k raaction head» + γ + 2 2γ (p ) if p > p k shock p k p and u can be found from a inearization of the Riemann probem yieding p = 2 (p + p r ) 2 (u r u ) ρā, u = 2 (u + u r ) 2 where ρ = 2 (ρ + ρ r ), ā = 2 (a + a r ). (p r p ) ρā,

24 3.4. The Riemann sover of Roe Consider again the Riemann probem q t + f(q) =, 8 < q(, ) = : q if <, q r if >, where for the -spit three-dimensiona Euer equation ρ ρu ρu ρu 2 + p q = ρv, f(q) = ρuv B C ρw ρuw E u(e + p) C A.

25 The Roe sover (cont.) Using the chain rue, the conservation aw may be written as q t + f(q) = q t + A(q)q =, A(q) = f q. Roe s approach consists in repacing the Jacobian matri A(q) by a constant Jacobian à = Ã(q,q r ) resuting in the Riemann probem for the inear system q t + Ãq =, 8 < q(, ) = : q if <, q r if >, which can be soved eacty.

26 The Roe sover (cont.) Once the matri Ã(q,q r ), its eigenvaues λ i (q,q r ), and corresponding right eigenvectors k (i) (q,q r ) are avaiabe, the difference between right and eft state can be epanded in terms of the eigenvectors: mx q = q r q = α i k (i), i= from which one finds the wave strengths α i (q,q r ). ρ u L L w v L L p L t ρ * L ρ * R w v L L p* u* w v R R ρ u w v p R R R R R q i+/2 () = q + X α i k (i), λ i q i+/2 () = q r X α i k (i), λ i f i+/2 = f + X λ i α i λi k (i), f i+/2 = f r X α i λi k (i). or or λ i

27 The Roe sover (cont.) The -direction Jacobian matri for the Euer equations, A(q), is A = 2 4 ˆγH u 2 a 2 (3 γ)u ˆγv ˆγw ˆγ uv v u uw w u 2 u[(γ 3)H a2 ] H ˆγu 2 ˆγuv ˆγuw γu where ˆγ = γ and a = p γp/ρ. The eigenvaues are λ = u a, λ 2 = λ 3 = λ 4 = u, λ 5 = u + a. 3 5,

28 The Roe sover (cont.) The corresponding right eigenvaues are k () = 2 4 u a v w H ua 3 5, k (2) = 2 4 u v w 2 V 2 3 5, k (3) = 2 4 v 3 5, k (4) = 2 4 w 3 5, k (5) = 2 4 u + a v w H + ua 3 5, where H = E + p ρ, E = 2 ρv 2 + ρe, V 2 = u 2 + v 2 + w 2.

29 The Roe sover (cont.) Roe requires the constant Jacobian matri à = Ã(q,q r ) to satisfy the agebraic properties of the Jacobian A(q), i.e., λ i = λ i (q,q r ) R i, Ã(q,q) = A(q), f(q r ) f(q ) = Ã(q r q ). These conditions may be fufied with the foowing Roe averagged quantities to be used in the formuae for λ i and k (i) shown on the brevious pages: ũ = ṽ = w = ρ u + ρ r u r ρ + ρ r, ρ v + ρ r v r ρ + ρ r, ρ w + ρ r w r ρ + ρ r, H = ρ H + ρ r H r ρ + ρ r, ã = (γ )[ H 2 Ṽ 2 ] 2, Ṽ 2 = ũ 2 + ṽ 2 + w 2.

30 4. Higher order accurate methods Piece-wise inear, oca reconstruction (variabe etrapoation method) as part of the MUSCL scheme (Monotone Upstream-centred Scheme for Conservation Laws) u () i u L i u () i u () i+ q i () = q n i + i i [, ], q L i = q n i 2 i, u R i q R i = q n i + 2 i, eads to the Generaized Riemann Probem q t + f(q) =, 8 < q(, ) = : q i () if <, q i+ () if >.

31 Higher order accurate methods (cont.) a) u () i i+/2 b) t u () i+ Initia data (a) and soution structure (b) of the generaized Riemann probem. In the MUSCL-Hancock method intermediate boundary etrapoated vaues q R i obtained by and q L i+ are q L i = q L i + t 2 [f(ql i ) f(q R i )], q R i = q R i + t 2 [f(ql i ) f(q R i )], which then form the piece-wise constant data for the conventiona Riemann probem q t + f(q) =, 8 < q(, ) = : q R i if <, q L i+ if >.

32 Higher order accurate methods (cont.) An important property of the genera scaar conservation aw is monotonicity: u t + f(u) = (5) If two initia data functions v () and u () for Eq. (5) satisfy v () u (), then the corresponding soutions v(, t) and u(, t) satisfy v(, t) u(, t) t >. Correspondingy, a monotone scheme has the foowing property: if v n i u n i i then v n+ i u n+ i i. The Theorem of Godunov states that: There are no monotone, inear schemes for the soution of Eq. (5) of second or higher order of accuracy. (A inear scheme has generay the form q n+ i = k R X k= k L b k q n i+k, k L, k R N + )

33 Higher order accurate methods (cont.) In order to circumvent Godunov s theorem, higher order non-inear schemes were invented. One way of doing so consists in finding a sope imiter ξ i such that i = ξ i i, i = 2 ( + ω)(qn i q n i ) + 2 ( + ω)(qn i+ q n i ), ω [,]. ξ i depends in a non-inear way on the ratio r, where r = qn i q n i q n i+ qn i 2 ξ( r ) TVD region for sope imiters. For negative r the TVD region is the singe ine ξ =, for positive r the TVD region corresponds to the pink region. 2 r SUPERBEE, van Leer, MINMOD, etc type of sope imiters are a subset of this region.

34 Tabe of content. Three major advancements in the numerica treatment of the hydrodynamic Equations 2. Conservation aws finite voumes 3. Riemann sovers 3.. The Riemann sover of Harten, La, and van Leer (HLL) 3.2. The HLLC sover 3.3. Wave-speed estimates 3.4. The Riemann sover of Roe 4. Higher order accurate schemes References

35 References Laney, C.B.: 998, Computationa Gasdynamics, Cambridge University Press, Cambridge LeVeque, R.J.: 22, Finite Voume Methods for Hyperboic Probems, Cambridge University Press, Cambridge LeVeque, R.J.: 992, Numerica Methods for Conservation Laws, Birkhäuser Verag, Base LeVeque, R.J., Mihaas, D., Dorfi, E.A., and Müer, E.: 998, Computationa Methods for Astrophysica Fuid Fow, O. Steiner & A. Gautschy (eds.), Springer-Verag, Berin Toro, E.F.: 999, Riemann Sovers and Numerica Methods for Fuid Dynamics, Springer-Verag, Berin