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Unformatted text preview: 1 Atms 502 Numerical Fluid Dynamics Tue., Dec. 05, 2006 12/6/06 Atms 502 - Fall 2006 - Jewett 2 Nonlinear instability • Durran - Burger’s equation Instabilities in Burger’s equation example appear associated with formation of shock… Shock not required for nonlinear instability Nonlinear instability can develop in smooth flow Example : viscous Burger’s equation there are values of ν for which true solution does not form shock, but for which instability develops in the numerical solution with advective form of differencing "# " t + # "# " x = $ " 2 # " x 2 12/6/06 Atms 502 - Fall 2006 - Jewett 3 Conservation • Recall: advective form: flux form: • If periodic domain (in x and z) non-divergent flow ψ is conserved: " t + u " x + w " z = " t + u " ( ) x + w " ( ) z = 0 if u x + w z = " dxdz = c # where c is independent of time 12/6/06 Atms 502 - Fall 2006 - Jewett 4 Conservation • Periodic domain (in x and z), nondivergent ψ is conserved: • More generally … (Durran p.244) Starting with: a scalar conservation law Integrating over [x1,x2] and [t1,t2]: " dxdz = c # * following wilhelmson "# " t + " " x f # ( ) = 0 for $% < x < % " ( x , t 2 ) dx = x 1 x 2 # " ( x , t 1 ) dx + x 1 x 2 # f " ( x 1 , t ) [ ] dt $ t 1 t 2 # f " ( x 2 , t ) [ ] dt t 1 t 2 # % & ’ ( ’ ) * ’ + ’ What you are left with What you started with Time-integrated Fuxes through boundaries at x 1 , x 2 = + The ψ which satisfies this expression for every subdomain [x1,x2]•[t1,t2] is the weak solution to the conservation law. 2 12/6/06 Atms 502 - Fall 2006 - Jewett 5 Finite Volume methods • Overview: Fields with shocks are problematic - derivative is not defined Computing “ discontinuous solutions to PDEs …” Solution might converge to result that is not a weak solution of the conservation law, or converge to a genuine weak solution but not to physically relevant one. Numerical approximations to equations with discontinuous solutions must satisfy additional conditions beyond stability and consistency to guarantee that the numerical solution converges to the correct solution as ∆ x, ∆ t approach zero. Durran: Essential to use a numerical scheme that conserves domain integral ; easy to do if flux form used Following Schopf, Durran 12/6/06 Atms 502 - Fall 2006 - Jewett 6 Finite Volume methods • And: Following Schopf, Durran Schopf 2005 Durran : “advective form does not converge to correct solution as...
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- Fall '09
- Numerical Analysis, Computational fluid dynamics, conservation law, ATMs, Jewett, MUSCL scheme