Well-Balanced Schemes for the Euler Equations with Gravity

Transcription

1 Well-Balanced Schemes for the Euler Equations with Gravity Alina Chertock North Carolina State University joint work with S. Cui, A. Kurganov, S.N. Özcan and E. Tadmor supported in part by

2 Euler Equations with Gravity U t + f(u) x + g(u) y = S(U) U := ρ ρu ρu ρv f(u) = ρu + p ρuv E u(e + p) g(u) := 0 S(U) = ρφ x ρφ y ρ(uφ x vφ y ) ρv ρuv ρv + p v(e + p) ρ is the density u, v are the x- and y-velocities E is the total energy p is the pressure; E = p γ + ρ (u + v ) φ is the time-independent linear gravitational potential; φ x = 0 and φ y = g

3 Euler Equations with Gravity Plays an important role in modeling model astrophysical and atmospheric phenomena in many fields including supernova explosions, (solar) climate modeling and weather forecasting; This is a system of balance laws: in many physical applications, solutions of the system are small perturbations of the steady states. Capturing such solutions numerically is a challenging task since the size of these perturbations may be smaller than the size of the truncation error on a coarse grid; To overcome this difficulty, one can use very fine grid, but in many physically relevant situations, this may be unaffordable; It is important to design a well-balanced numerical method, that is, the method which is capable of exactly preserving some steady state solutions. Then, perturbations of these solutions will be resolved on a coarse grid in a non-oscillatory way. A type of solutions of interest are steady state ones.

4 Steady States ρ t + (ρu) x + (ρv) y = 0 (ρu) t + (ρu + p) x + (ρuv) y = 0 (ρv) t + (ρuv) x + (ρv + p) y = ρg E t + (u(e + p)) x + (v(e + p)) y = ρvg Steady state solution: u 0, v 0, w Const Here, w := p + R, R(x, y, t) := g y ρ(x, ξ, t) dξ Numerical Challenges : Well-balanced scheme should exactly balance the flux and source terms so that the steady states are preserved. 3

5 Well-Balanced Methods Some References Developed mainly in the context of shallow water models. Euler equations with gravitational fields: R. Leveque and D. Bale (998) quasi-steady wave-propagation methods for models with a static gravitational field N. Botta, R. Klein, S. Langenberg, and S. Lützenkirchen (004) well-balanced finite-volume methods, which preserve a certain class of steady states for nearly hydrostatic flows C. T. Tian, K. Xu, K. L. Chan, and L. C. Deng (007), K. Xu, J. Luo, and S. Chen (00), J. Luo, K. Xu, and N. Liu (0) gaskinetic schemes for multi-d gas dynamic equations and well-balanced numerical methods for problems, in which the gravitational potential was modeled by a piecewise step function Y. Xing and C.-W. Shu (03) higher order finite-difference methods for the gas dynamics with gravitation M. Zenk, C. Berthon and C. Klingenberg (04), P. Chandrashekar, C. Klingenberg (05) FV methods... 4

6 -D System For Simplicity ρ t + (ρv) y = 0 (ρv) t + (ρv + p) y = ρg E t + (v(e + p)) y = ρvg Steady state solution: v 0, w Const Here, w := p + R, R(y, t) := g y ρ(ξ, t) dξ How to design a well-balanced scheme? 5

7 Finite-Volume Methods U t + g(u) y = S U n k y C k U(y, t n ) dy : cell averages over C k := (y k, y k+ ) Semi-discrete FV method: d dt U k(t) = G (t) G k+ k (t) y + S k G k+ (t): numerical fluxes S k : quadrature approximating the corresponding source terms Central-Upwind (CU) Scheme: [Kurganov, Lin, Noelle, Petrova, Tadmor, et al.; ] 6

8 {U k (t)} Ũ(, t) { U N,S k } (t) { } G k+ (t) {U k (t + t)} (Discontinuous) piecewise-linear reconstruction: Ũ(y, t) := U k (t) + (U y ) k (y y k ), y C k It is conservative, second-order accurate, and non-oscillatory provided the slopes, {(U y ) j }, are computed by a nonlinear limiter Example Generalized Minmod Limiter ( (U y ) k = minmod θ U k U k y, U k+ U k, θ U ) k+ U j y y where minmod(z, z,...) := min k {z k }, if z k > 0 k, max k {z k }, if z k < 0 k, 0, otherwise, and θ [, ] is a constant 7

9 {U k (t)} Ũ(, t) { U N,S k } (t) { } G k+ (t) {U k (t + t)} U N k and U S k are the point values at y k+ and y k : Ũ(y, t) = U k + (U y ) k (y y k ), y C k U N k := U k + x (U y) k U S k := U k x (U y) k k+/ k k / j / j j+/ 8

10 {U k (t)} Ũ(, t) { U N,S k } (t) { } G k+ (t) {U k (t + t)} where d dt U k = G G k+ k y + S k G k+ = b + g(u k+ k N) b k+ g(u S k+ ) b + k+ b k+ + β k+ ( U S k+ U N k ) β k+ = b + b k+ k+ b + k+ b k+ b + k+ b k+ = max ( v N k + c N k, v S k+ + c S k+, 0 ) = min ( v N k c N k, v S k+ c S k+, 0 ), c = γp ρ and S k = (0, gρ k, g(ρv) k ) T 9

11 No Well-Balanced Property For steady-state solution: v 0, w Const, we have dρ k dt = β (ρ S k+ k+ ρn k ) β (ρ S k k ρn k ) y d(ρv) k dt de k dt = (ps k+ + pn k ) (ps k + pn k ) y = β k+ (ps k+ pn k ) β k (ps k pn k ) (γ ) y The RHS does not necessarily vanish and hence the steady state would not be preserved at the discrete level; This would also true for the first-order version of the scheme; For smooth solutions, the balance error is expected to be of order ( y), but a coarse grid solution may contain large spurious waves; The lack of balance between the numerical flux and source terms is a fundamental problem of the scheme. 0

12 Well-Balanced Scheme -D system in the y-direction: ρ ρv E t + ρv ρv + p v(e + p) y = 0 ρg ρvg Define R(y) := g y ρ(ξ)dξ R y = ρg Then (ρv + p + R) }{{} y = 0 w w := p + R

13 Well-Balanced Scheme Incorporating the source term ρg into the flux: w := p + R, R(y, t) := g y ρ(ξ, t) dξ ρ ρv E t + ρv ρv + w v(e + p) y = 0 0 ρvg Reconstruct equilibrium variables W = (ρ, ρv, w) T Compute the point values of conservative variables U = (ρ, ρv, E) T : {U k (t)} Ũ(, t) { W N,S k } (t) { U N,S k } (t) { } G k+ (t) {U k (t+ t)}

14 -D Reconstruction Compute the point values of ρ and ρv at y k : ρ(y) = k ρv(y) = k [ ρk + (ρ y ) k (y y k ) ] χ Ck (y) ρ N,S k [ (ρv)k + ((ρv) y ) k (y y k ) ] χ Ck (y) (ρv) N,S k Integrate ρ(y) to obtain an approximation of R = gρ dy: R(y) = g y ρ(ξ) dξ = g k y kl [ y k ρ i + ρ k (y y k ) + (ρ ] y) k (y y k )(y y k+ ) i=k L χ Ck (y) 3

15 -D Reconstruction The point values of R at the cell interfaces and cell centers are R k+ = g y k i=k L ρ i and R k = g y k ρ i + g y ρ k g( y) (ρ y ) k 8 i=k L or recursively R kl = 0, R k+ = R k + g yρ k, R k = R k + g y ρ k g( y) (ρ y ) k, 8 The values of w at the cell centers are set as k = k L,..., k R w k = p k + R k where and p k = (γ ) ( E k ρ ) k v k v k = (ρv) k /ρ k 4

16 -D Reconstruction Equipped with w k = p k + R k, we apply the minmod reconstruction procedure to {w k } and obtain the point values of w at the cell interfaces: where w N k = w k + y (w y) k, w S k+ = w k+ y (w y) k+ ( (w y ) k = minmod θ w k+ w k y Finally, the point values of p and E are, w k+ w k, θ w ) k w k y y p N k = w N k R k+, p S k = w S k R k and E N k = pn k γ + ( (ρv) N k ) ρ N k, E S k = ps k γ + ( (ρv) S k ) ρ S k 5

17 Well-Balanced Evolution d dt U k = G G k+ k y + S k. Here, the second and third components of the numerical fluxes G are computed as before G () k+ b + ( ) ( k+ ρ N = k (vk N) + wk N b k+ ρ S k+ (vk+ S ) + wk+) S b + b k+ k+ ( + β k+ (ρv) S k+ (ρv) N ) k G (3) k+ = b + v k+ k N(EN k + pn k ) b k+ v S k+ (ES k+ + ps k+ ) b + k+ b k+ + β k+ ( E S k+ E N k ) 6

18 Well-Balanced Evolution However, to ensure a well-balanced property of the scheme, the first component of the flux and the source term have to be modified: G () k+ = b + (ρv) N k+ k b k+ (ρv) S k+ b + k+ b k+ + β k+ B ( wk+ w k y yk R + y k L max k {w k } ) (ρ S k+ ρ N ) k S k = (0, 0, g(ρv) k ) T B ψ 7

19 Proof of the Well-Balanced Property Theorem. The semi-discrete scheme coupled with the well-balanced reconstruction and evolution is well-balanced in the sense that it preserves the steady state v 0, w Const. Proof: Assume that at certain time level, we have v N k v k v S k 0 and w N k w k w S k ŵ Const To show that the proposed scheme is well-balanced, we need to show that d dt U k = G G k+ k y + S k 0 S k = (0, 0, gv k ρ k ) T = 0 8

20 Proof of the Well-Balanced Property Theorem. The semi-discrete scheme coupled with the well-balanced reconstruction and evolution is well-balanced in the sense that it preserves the steady state v 0, w Const. Proof: Assume that at certain time level, we have v N k v k v S k 0 and w N k w k w S k ŵ Const To show that the proposed scheme is well-balanced, we need to show that d dt U k = G G k+ k y + S k 0 G () k+ = b + (ρv) N k+ k b k+ (ρv) S k+ b + k+ b k+ + β k+ B ( wk+ w k y yk R + y k L max k {w k } ) (ρ S k+ ρ N k ) = 0 9

21 Proof of the Well-Balanced Property Theorem. The semi-discrete scheme coupled with the well-balanced reconstruction and evolution is well-balanced in the sense that it preserves the steady state v 0, w Const. Proof: Assume that at certain time level, we have v N k v k v S k 0 and w N k w k w S k ŵ Const To show that the proposed scheme is well-balanced, we need to show that d dt U k = G G k+ k y + S k 0 G () k+ = b + k+ ( ) ( ρ N k (vk N) + wk N b k+ ρ S k+ (vk+ S ) + wk+) S b + b k+ k+ = 0 0

22 Proof of the Well-Balanced Property Theorem. The semi-discrete scheme coupled with the well-balanced reconstruction and evolution is well-balanced in the sense that it preserves the steady state v 0, w Const. Proof: Assume that at certain time level, we have v N k v k v S k 0 and w N k w k w S k ŵ Const To show that the proposed scheme is well-balanced, we need to show that d dt U k = G G k+ k y + S k 0 G (3) k+ = β k+ ( E S k+ E N k = β k+ γ ) = β k+ γ ( p S k+ p N k ) [ (w S k+ R k+ ) (wn k R k+ ) ] = 0

23 -D Well-Balanced Scheme ρ ρu ρv E t + ρu ρu + p ρuv u(e + p) x + ρv ρuv ρv + p v(e + p) y 0 = 0 gρ gρv Incorporating the source term ρg into the flux: w := p + R, R(x, y, t) := g y ρ(x, ξ, t) dξ ρ ρu ρv E t + ρu ρu + p ρuv u(e + p) x + ρv ρuv ρv + w v(e + p) y 0 = 0 0 gρv

24 -D Reconstruction In the x-direction: Reconstruct conservative variables {U j,k (t)} Ũ(,, t) { U E,W j,k } In the reconstruction step, we compute { } F j+,k {U j,k (t + t)} { } ρ j,k ρ(,, t) ρ E,W j,k (t) { } (ρu) j,k ρu(,, t) (ρu) E,W j,k (t) { } (ρv) j,k ρv(,, t) (ρv) E,W j,k (t) From the EOS: ( p j,k = (γ ) { p E,W j,k (t) } E j,k (ρu) j,k + (ρv) j,k { E E,W j,k (t) } ρ j,k ) p(,, t) { p E,W j,k (t) } 3

25 -D Reconstruction In the y-direction: Reconstruct equilibrium variables W = (ρ, ρu, ρv, w) T : {U j,k (t)} Ũ(,, t) { W N,S j,k } In the reconstruction step, we compute { U N,S j,k } {G j,k+ } { } ρ j,k ρ(,, t) ρ N,S j,k (t) { } (ρu) j,k ρu(,, t) (ρu) N,S j,k (t) { } (ρv) j,k ρv(,, t) (ρv) N,S j,k (t) Compute w j,k = p j,k + R j,k, p j,k = (γ ) R j,kl = 0, R j,k+ = R j,k + g yρ j,k, {U j,k (t+ t)} ( ) E j,k (ρu) j,k +(ρv) j,k ρ j,k R j,k = R j,k + g y ρ j,k g( y) (ρ y ) j,k 8 4

26 -D Reconstruction In the y-direction: Reconstruct equilibrium variables W = (ρ, ρu, ρv, w) T : {U j,k (t)} Ũ(,, t) { W N,S j,k } { U N,S j,k } {G j,k+ } {U j,k (t+ t)} Reconstruct w w j,k w(,, t) { w N,S j,k (t) } Finally, obtain the point values: and from the EOS: E N j,k = pn j,k γ + p N j,k = w N j,k R j,k+, p S j,k = w S j,k R j,k ( ) ( ) (ρu) N j,k + (ρv) N j,k ρ N j,k, E S j,k = ps j,k γ + ( ) ( ) (ρu) S j,k + (ρv) S j,k ρ S j,k 5

27 Well-Balanced Evolution d dt U j,k = F j+,k F j,k x G j,k+ G j,k y + S j,k Similar to the -D case... Theorem. The -D semi-discrete central-upwind scheme scheme described with the described well-balanced reconstruction and evolution is well-balanced in the sense that it preserves the steady state exactly. 6

28 Example -D Isothermal Equilibrium Solution [Xing, Shu; 03] The ideal gas with γ =.4; domain [0, ] The gravitational force is φ y = g = The steady-state initial conditions are ρ(y, 0) = e y, p(y, 0) = e y, v(y, 0) = 0 A zero-order extrapolation at the boundaries is used Well-Balanced Test L -errors at T = : N ρ ρv E 40.33E-06.5E-06.7E E E E E-06.48E-05.3E E E-06.4E-05 7

29 η = 0 η = Perturbation A small initial pressure perturbation: ρ(y, 0) = e y, v(y, 0) 0, p(y, 0) = ge y + ηe 00(y 0.5), x initial state WB, N=00 WB, N=000 Non WB, N=00 x initial state WB, N=00 WB, N=000 Non WB, N=

30 Example -D Isothermal Equilibrium Solution [Xing, Shu; 03] The ideal gas with γ =.4; domain [0, ] [0, ] The gravitational force is φ y = g = The steady-state initial conditions are ρ(x, y, 0) =.e.y, p(x, y, 0) = e.y, u(x, y, 0) v(x, y, 0) 0 Solid wall boundary conditions imposed at the edges of the unit square E E+00.43E E E E E E E E+00.85E E-06 N N ρ ρu ρv E E E E E E E+00.07E E E E+00 7.E-06.57E-05 9

31 Perturbation A small initial pressure perturbation: p(x, y, 0) = e.y + ηe ((x 0.3) +(y 0.3) ), η =

32 W B : 50 50, NW B : 50 50,

33 Example -D Explosion The ideal gas with γ =.4; domain domain [0, 3] [0, 3] The gravitational force is φ y = g = The steady-state initial conditions are ρ(x, y, 0), u(x, y, 0) = v(x, y, 0) 0 p(x, y, 0) = gy + { 0.005, (x.5) + (y.5) < 0.0, 0, otherwise. A zero-order extrapolation at the boundaries is used 3

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36 THANK YOU! 35