Stable Semi-Discrete Schemes for the 2D Incompressible Euler Equations

Transcription

1 Stable Semi-Discrete Schemes for the D Incompressible Euler Equations Doron Levy Department of Mathematics Stanford University dlevy@math.stanford.edu dlevy Maryland, May 004 Maryland, May 004

2 Outline Problem formulation Semi-discrete central schemes Other applications Discrete incompressibility and a maximum principle Maryland, May 004

3 The Incompressible D Euler and NS Equations u t + (u )u = p + ν u, x Ω R u = 0 A vorticity formulation: ω = u, u = (u, v).. A conservative form: ω t + (uω) x + (vω) y = ν ω.. A convective form: ω t + uω x + vω y = ν ω. The divergence conditions = a streamfunction ψ: u = ψ = ( ψ y, ψ x ). A Poisson equation: ψ = ω. Maryland, May 004 3

4 Boundary conditions. No-slip boundary conditions for u: u = 0 on Ω.. No boundary conditions for ω (vorticity generation on the boundary, etc.) 3. Boundary conditions for ψ: ψ(x, t) = for the Poisson equation). Pure Streamfunction Formulation n ψ(x, t) = 0, x Ω. (over-determined ψ t + ( ψ) ( ψ) = ν ψ, x Ω ψ(x, t) = ψ(x, t) = 0, n x THM (Uniqueness): Solutions are unique in H 0(Ω). Ω. THM (Decay of solutions): ψ(x, t) L (Ω) e νλ Ωt ψ(x, 0) L (Ω), λ Ω > 0. Maryland, May 004 4

5 Numerics for the Pure Streamfunction Formulation (Ben-Artzi, Fishelov, Kupferman,...) Time-discretization: Crank-Nicolson (nd-order). ( 4 ) νk ψ n+ = ( ) νk ψ n+ = ( + 4 ) νk ψ n k[(u )ω]n, ( + ) νk ψ n k[(u )ω]n+. Spatial-discretization:. Laplacian (5-points), Bi-harmonic operator (a compact discretization due to Stephenson), The advection term, Linear solver ( α )ψ i,j = RHS.. Pure streamfunction vs. vorticity-streamfunction formulation. Maryland, May 004 5

6 Semi-Discrete Central-Schemes for Conservation Laws (Kurganov-Tadmor) u t + f(u) x = 0 n u j n+ w j n+ u j w n+ j / w n+ j u n j n+ w j+/ n+ w j+ n u j+ x n j 3/,R xj /,L n x n j /,R x x n j+/,l j x j / x j x j+/ x n j+/,r x x j+ n j+3/,l Maryland, May 004 6

7 Reconstruction: In I j, ũ(x, t n ) = u n j + (u x) n j (x x j). Nonlinear limiters: MinMod with θ ( u x = MM θ un j un j, un j+ un j, θ un j+ un j x x x ). Local speeds of propagation: a j+ (t) = max! ρ ( ) f u (u), u C u j+,u + j+ Evolution points: x n j+,r = x j+ + an j+ t. An exact evolution: w n+ j+ Projection: u j = x x j+ x j =..., w n+ j =.... w(ξ, t n+ )dξ. Maryland, May 004 7

8 In the limit lim t 0 u n+ j t u n j The semi-discrete scheme: d dt u j = H (t) H j+ j (t). x The numerical flux: H j+ = f ( ) u + j+ + f ( ) u j+ a j+ ( u + j+ u j+ ) u n j n u j+ x j / n j+/,l n j+/,r x n j /,R x x x x j x j+/ x j+ n j+3/,l Maryland, May 004 8

9 D Semi-discrete Godunov-type schemes A Reconstruction: u t + f(u) x + g(u) y = 0 ũ(x, y, t) = u j,k + (u x ) j,k (x x j ) + (u y ) j,k (y y k ). Local speeds of propagation: ( f a x j+ = max,k ρ ± u Evolution points = temporary averages ( u ± j+,k ) ), a y j,k+ = max ± ρ ( g u ( u ± j,k+ ) ). Maryland, May 004 9

10 Projection to the original grid + limit as t 0 The semi-discrete scheme: d H x dt u j+ jk(t) =,k Hx j,k x H y j,k+ H y j,k y. The numerical fluxes: H x j+,k(t) = ) ) f (u +j+,k(t) + f (u j+,k(t) a x j+ ] [u +j+ ( ) ( ) H y j,k+ (t) = g u + + g u j,k+ (t) a j,k+ (t) [ ] j,k+ (t) u + (t) u j,k+ j,k+ (t) j,k Maryland, May 004 0

11 Comments. Third-order extensions (Kurganov, DL, Noelle, Petrova,...). Different reconstructions 3. Numerical fluxes (local speeds of propagation, the projection step,...) Maryland, May 004

12 Additional terms: Parabolic, Source terms,... u t + f(u) x = Q(u, u x ) x The diffusion flux: P j+ = d dt u j = H (t) H j+ j (t) x ( [ Q u j, u j+ u j + P j+ (t) P j (t). x ) ( + Q u j+, u j+ u j )]. u t + f(u) x = S(u) The source term: Sj =? d dt u j = H (t) H j+ j (t) x + S j. Maryland, May 004

13 Example: Shallow Water Waves (Bryson + DL) h t + (hu) x + (hv) y = 0, (hu) t + (hu + ) gh + (huv) y = ghb x, (hv) t + (huv) x + (hv + ) gh = ghb y. x x Unstructured grids Source term discretization that preserves stationary steady-states Maryland, May 004 3

14 A Converging-Diverging Channel + Bottom Topography (Bryson-DL) Maryland, May 004 4

15 Example: Dynamics of the Solar Atmosphere (Bryson-Kosovichev-DL) Spicules near the solar limb. They are blue-shifted (hence dark) in this narrow-band hydrogen-alpha image due to their motion towards the observer. (National Solar Observatory, Sacramento Peak) Maryland, May 004 5

16 Phenomena:. Spicules. Perhaps cooler photospheric material shooting up into the corona.. Coronal Oscillations. Particle oscillations in the corona have been observed to have periods of about 5 minutes above quiet regions and 3 minutes about sunspots (De Moortel et al., 00). The model: Euler equations for material in a flux tube of area A(x, t) and gravity g(x): ρ ρu E t + ρu ρu + p (E + p)u x = ρua A x ρa A t ρu A A x ρua A t g(x)ρ + F (x, t)ρ (E + p)ua A x EA A t g(x)ρu F (x, t): a velocity forcing term that perturb the base of the solar atmosphere. Maryland, May 004 6

17 Trajectories of Velocity Shocks, Quiet Sun Model (Bryson-Kosovichev-DL) shock histories, quiet sun model, 000 m/sec impulse shock histories, quiet sun model, 000 m/sec impulse height ( 0 6 meters) 6 height ( 0 6 meters) time (minutes) time (minutes) Constant flux tube Expanding flux tube An initial impulse amplitude of 000 meters/second. Numerical issues:. hydrostatic equilibrium at initialization and at the boundaries. Static A = A(x).. D nonuniform grids. Maryland, May 004 7

18 Back to the Euler Equations (Vorticity) The conservative formulation: ω t + (uω) x + (vω) y = 0 Reconstruction (in I j,k ): ω(x, y) = ω j,k + (ω x ) j,k (x x j ) + (ω y ) j,k (y y k ). The local speeds of propagation (as long as the velocities are in L ): a x j+,k = u j+,k, ay j,k+ = v j,k+. (from the convective formulation (ω t + uω x + vω y = 0)). Maryland, May 004 8

19 The scheme: d H x dt ω j+ jk(t) =,k Hx j,k x H y j,k+ H y j,k y. The numerical fluxes: H x j+,k = ) a (ω x ) +j+,k + ω j+,k u j+,k j+ (ω,k +j+,k ω j+,k, H y = ( ) a y ( ) ω + + ω j,k+ v j,k+ j,k+ j,k+ j,k+ ω + ω. j,k+ j,k+ The remaining ingredient: the velocities (u j±,k, v j,k± ) =? j,k Maryland, May 004 9

20 Reconstructing the Velocities The 5-point Laplacian: ψ jk = ω jk. v u v u k Define the velocities: u j+,k = v j,k+ = y x ( ψj,k+ + ψ j+,k+ ( ψ j+,k + ψ j+,k+ j ψ ) j,k + ψ j+,k + ψ j,k + ψ j,k+ ). A discrete incompressibility relation: u j+,k u j,k x + v j,k+ v j,k y = 0. Maryland, May 004 0

21 The Incompressible NS Equations ω t + (uω) x + (vω) y = ν ω dω j,k dt = Hx j+/,k (t) Hx j /,k (t) x Hy j,k+/ (t) Hy j,k / (t) y + νq j,k (t) Q j,k (t) = ω j+,k(t) + 6ω j+,k (t) 30ω j,k (t) + 6ω j,k (t) ω j,k (t) x + ω j,k+(t) + 6ω j,k+ (t) 30ω j,k (t) + 6ω j,k (t) ω j,k (t) y. Third-order (KL, KP, KNP) - reconstructions (of ω, u, v), numerical flux, no theory. Maryland, May 004

22 A Maximum Principle Theorem (DL): Consider the fully-discrete scheme: ) = ωj,k n λ (H xj+,k Hxj,k µ ω n+ j,k ( H y j,k+ ) H y. j,k with the numerical fluxes H x j+,k = ) a (ω x ) +j+,k + ω j+,k u j+,k j+ (ω,k +j+,k ω j+,k, H y = ( ) a y ( ) ω + + ω j,k+ v j,k+ j,k+ j,k+ j,k+ ω + ω. j,k+ j,k+ Assume that the velocities u, v L, that the derivatives are reconstructed with the MinMod limiter and that the following CFL condition holds max(λ max u u, µ max v ) v 8. Then max j,k (ωn+ j,k ) max j,k (ωn j,k) Maryland, May 004

23 Proof. Writing ω n+ jk as a convex combination of ω ± j±,k, ω± j,k± : ω n+ j,k = [ λ ω + j+,k [ +ω j+,k 4 + λ (a xj+,k u j+,k ) ] ( u j+,k ax j+,k x j,k uω x j,k ω )] Use the properties of the MM limiter: max j,k ( ) ω ± j±,k, ω± j,k± max j,k (ωn j,k). 3. High-order in time: a convex combination of FE (TVD-RK). Maryland, May 004 3

24 Comments. Simpler than the fully-discrete case (DL-Tadmor). There: an exact discrete incompressibility was used to obtain the convex combination. Here - not needed.... Clear upwinding structure: [ ω n+ j,k = ω j+,k 4 λ ) (u ] j+,k + axj+,k [ λ ( ) ] +ω + a x j+,k j+,k u j+,k +... j,k Maryland, May 004 4

25 Incompressible NS, Double Shear-Layer Problem (Kurganov-DL) 3rd-order semi-discrete. T = 0. ν = 0.0. y y x x N = N = 8 8. Initial data: u(x, y, 0) = tanh ( 5 π tanh ( 5 π ( y π )), y π, ( 3π y)), y > π, v(x, y, 0) = 0.05 sin(x). Maryland, May 004 5

26 Conclusion Simple Semi-discrete Godunov-type schemes Parabolic terms, source terms,... Applications A new maximum principle Maryland, May 004 6