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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 - Burgers equation Instabilities in Burgers equation example appear associated with formation of shock Shock not required for nonlinear instability Nonlinear instability can develop in smooth flow Example : viscous Burgers equation there are values of 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 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 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 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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This note was uploaded on 10/06/2009 for the course ATMOSPHERI 502 taught by Professor Jewett during the Fall '09 term at University of Illinois at Urbana–Champaign.
- Fall '09