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Welcome! MiniSymposium on Algorithms and Software to Compute Conservation Laws of Nonlinear PDEs Speakers: Hereman, Bluman, Nivala, Wolf Symbolic Computation of Conservation Laws of Nonlinear PDEs in Multi-dimensions Willy Hereman Department of Mathematical and Computer Sciences Colorado School of Mines Golden, Colorado, U.

S.A. [email protected] http://www.mines.edu/fs home/whereman/ Minisymposium MS29

2006 SIAM Conference on Nonlinear Waves and Coherent Structures University of Washington, Seattle, WA Sunday, September 10, 2006, 3:30p.m. Acknowledgements Collaborators: Mark Hickman (Canterbury, NZ), Bernard Deconinck (UW), Michael Colagrosso (CSM). Graduate students Douglas Poole (CSM), Michael Nivala (UW). Several undergraduate and graduate students. Research supported in part by NSF under Grant CCR-9901929 This presentation was made in TeXpower Outline ? Conservation Laws of PDEs in multi-dimensions ? Example: shallow water wave equations (Dellar) ? Algorithmic Methods for conservation laws ? Computer Demonstration ? Tools: ? Euler operators (testing exactness) ? Calculus-based formulas for homotopy operator symbolic integration by parts inversion of the total divergence operator ? Application: shallow water wave equations ? Conclusions: future work, software and publications Notations C Computations on the Jet Space ? Independent variables x = (x, y, z) ? Dependent variables u = (u(1), u(2)u(j)u(N)) In examples: u = (u, v, θ, h, . . .) ? Partial derivatives ukx = ?ku ?xk , ukx ly = ?k+lu ?xkyl , etc. ? Di?erential functions Example: f = uvvx + x2u3 xvx + uxvxx for u(x), v(x) ? Total derivative (with respect to x) Dx = ? ?x + M (1) x k=0 u(k+1)x ? ?ukx + M (2) x k=0 v(k+1)x ? ?vkx M (1) x is the order of f in u (with respect to x), etc. Conservation Laws ? Conservation law in (1 + 1) dimensions Dtρ + DxJ =

0 (on PDE) conserved density ρ and ?ux J ? Example: Korteweg-de Vries (KdV) equation ut + uux + u3x =

0 ? Sample conservation law Dt u3 ? 3ux

2 + Dx

3 4 u4 ? 6uux

2 + 3u2 u2x + 3u2x

2 ? 6uxu3x =

0 ? Key property: Dilation invariance ? Example: KdV equation and its density-?ux pairs are invariant under the scaling symmetry (x, t, u) → ( x λ , t λ3 , λ2 u), λ is arbitrary parameter. ? Some density-?ux pairs for the KdV equation: ρ(1) = u J(1) = u2

2 + u2x ρ(2) = u2 J(2) = 2u3

3 + 2uu2x ? ux

2 ρ(3) = u3 ? 3ux

2 J(3) =

3 4 u4 ? 6uux

2 + 3u2 u2x + 3u2x

2 ? 6uxu3x . . . ρ(6) = u6 ?

60 u3 ux

2 ?

30 ux

4 +

108 u2 u2x

2 +

720 7 u2x

3 ?

648 7 uu3x

2 +

216 7 u4x

2 . . . ? Conservation law in (3 + 1) dimensions Dtρ + ? ・ J = Dtρ + DxJ1 + DyJ2 + DzJ3 =

0 (on PDE) conserved density ρ and ?ux J = (J1, J2, J3) ? Example: Shallow water wave (SWW) equations [P. Dellar, Phys. Fluids

15 (2003) 292-297] ut + (u・?)u +

2 ? * u + ?(θh) ?

1 2 h?θ =

0 θt + u・(?θ) =

0 ht + ?・(uh) =

0 where u(x, y, t), θ(x, y, t) and h(x, y, t). ? In components: ut + uux + vuy ?

2 ?v +

1 2 hθx + θhx =

0 vt + uvx + vvy +

2 ?u +

1 2 hθy + θhy =

0 θt + uθx + vθy =

0 ht + hux + uhx + hvy + vhy =

0 ? SWW equations are invariant under (x, y, t, u, v, h, θ, ?) → (λ?1 x, λ?1 y, λ?b t, λb?1 u, λb?1 v, λa h, λ2b?a?2 θ, λb ?) where W(h) = a and W(?) = b (a, b ∈ Q). ? First few densities-?ux pairs of SWW system: ρ(1) = h J(1) = ? ? uh vh ? ? ρ(2) = hθ J(2) = ? ? uhθ vhθ ? ? ρ(3) = hθ2 J(3) = ? ? uhθ2 vhθ2 ? ? ρ(4) = (u2 + v2)h + h2θ J(4) = ? ? u3h + uv2h + 2uh2θ v3h + u2vh + 2vh2θ ? ? ρ(5) = vxθ ? uyθ + 2?θ J(5) =