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1 2 ? ? 4?uθ ? 2uuyθ + 2uvxθ ? hθθy 4?vθ + 2vvxθ ? 2vuyθ + hθθx ? ? . Computer Demonstration Algorithmic Methods for Conservation Laws ? Use Noether'

s Theorem (Lagrangian formulation). ? Direct methods (Anderson, Bluman, Anco, Wolf, etc.) based on solving ODEs (or PDEs). ? Strategy (linear algebra and variational calculus). ? Density is linear combination of scaling invariant terms with undetermined coe?cients. ? Use variational derivative (Euler operator) to compute the undetermined coe?cients. ? Use the homotopy operator to compute the ?ux (invert Dx or Div) (Deconinck and Nivala). ? Work with linearly independent pieces in ?nite dimensional spaces. Review of Vector Calculus ? The curl annihilates gradients! ? The divergence annihilates curls! ? The Euler operator annihilates divergences! Formula for Euler operator (variational derivative) in 1D: L (0) u(j)(x) = M (j) x k=0 (?Dx)k ? ?u (j) kx = ? ?u(j) ? Dx ? ?u (j) x + D2 x ? ?u (j) 2x ? D3 x ? ?u (j) 3x + ・ ・ ・ Formula for Euler operator in 2D: L (0,0) u(j)(x,y) = M (j) x kx=0 M (j) y ky=0 (?Dx)kx (?Dy)ky ? ?u (j) kxx kyy = ? ?u(j) ? Dx ? ?u (j) x ? Dy ? ?u (j) y + D2 x ? ?u (j) 2x +DxDy ? ?u (j) xy +D2 y ? ?u (j) 2y ?D3 x ? ?u (j) 3x ・ ・ ・ Inverting Dx and Div Problem Statement ? In 1D: Example: For u(x) and v(x) f =3uxv2 sin u?u3 x sin u?6vvx cos u+2uxu2x cos u+8vxv2x ? Find F = f dx Thus, f = DxF. Inverting Dx and Div Problem Statement ? In 1D: Example: For u(x) and v(x) f =3uxv2 sin u?u3 x sin u?6vvx cos u+2uxu2x cos u+8vxv2x ? Find F = f dx Thus, f = DxF. ? Result (by hand): F =

4 v2 x + u2 x cos u ?

3 v2 cos u Inverting Dx and Div Problem Statement ? In 1D: Example: For u(x) and v(x) f =3uxv2 sin u?u3 x sin u?6vvx cos u+2uxu2x cos u+8vxv2x ? Find F = f dx Thus, f = DxF. ? Result (by hand): F =

4 v2 x + u2 x cos u ?

3 v2 cos u Mathematica cannot compute this integral! ? In 2D or 3D: Example: For u(x, y) and v(x, y) f = uxvy ? u2xvy ? uyvx + uxyvx ? Find F = Div?1 f Thus, f = Div F. ? In 2D or 3D: Example: For u(x, y) and v(x, y) f = uxvy ? u2xvy ? uyvx + uxyvx ? Find F = Div?1 f Thus, f = Div F. ? Result (by hand): ? F = (uvy ? uxvy, ?uvx + uxvx) ? In 2D or 3D: Example: For u(x, y) and v(x, y) f = uxvy ? u2xvy ? uyvx + uxyvx ? Find F = Div?1 f Thus, f = Div F. ? Result (by hand): ? F = (uvy ? uxvy, ?uvx + uxvx) Mathematica cannot do this! ? In 2D or 3D: Example: For u(x, y) and v(x, y) f = uxvy ? u2xvy ? uyvx + uxyvx ? Find F = Div?1 f Thus, f = Div F. ? Result (by hand): ? F = (uvy ? uxvy, ?uvx + uxvx) Mathematica cannot do this! Can this be done without integration by parts? ? In 2D or 3D: Example: For u(x, y) and v(x, y) f = uxvy ? u2xvy ? uyvx + uxyvx ? Find........