Day17-slides - Two dimensions stability Advection equation...

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1 Atms 502 Numerical Fluid Dynamics Thu., Oct. 19, 2006 10/23/06 Atms 502 - Fall 2006 - Jewett 2 Two dimensions - stability Advection equation now looks like: To determine stability, assume that coefficients u and v are constant Let max u(x,y)=U, max v(x,y)=V This is called local stability As we will see, stability restrictions in 2-D are greater than in 1-D. " t + u ( x , y ) " x + v ( x , y ) " y = 0 10/23/06 Atms 502 - Fall 2006 - Jewett 3 Two dimensions - stability Problem : 2d advection - stability Approach : discrete dispersion relation Method : look at 2-d leapfrog Result: " 2 t # + U " 2 x # + V " 2 y # = 0, or # i , j n + 1 $ # i , j n $ 1 2 % t + U # i + 1, j n $ # i $ 1, j n 2 % x + V # i , j + 1 n $ # i , j $ 1 n 2 % y = 0 i = 1: nx , j = 1: ny sin( " # t ) = μ sin( k # x ) + $ sin( l # y ) 10/23/06 Atms 502 - Fall 2006 - Jewett 4 Two dimensions - stability Require ω is real. ω real requires | μ |+| ν |<1 sin( ω t ) can be > 1 if ω is complex… sin( a ) = sin( x + iy ) = sin( x )cos( iy ) + cos( x )sin( iy ) = sin( x )cosh( y ) + i cos( x )sinh( y ) Wikipedia
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2 10/23/06 Atms 502 - Fall 2006 - Jewett 5 Two dimensions - stability Another approach: (apologies for multiple uses of “i”) " 2 t # + U " 2 x # + V " 2 y # = 0 Let " x = " y = " s : # i , j n + 1 = # i , j n $ 1 $ U " t " s # i + 1, j n $ # i $ 1, j n ( ) $ V " t " s # i , j + 1 n $ # i , j $ 1 n ( ) ˜ " n + 1 = ˜ " n # 1 # U $ t $ s e ik $ x # e # ik $ x ( ) # V $ t $ s e il $ y # e # il $ y ( ) = ˜ " n # 1 # 2 i $ t $ s U sin k $ x + V sin l $ y ( ) ˜ " n Mistake on notes Von Neumann method: 10/23/06 Atms 502 - Fall 2006 - Jewett 6 Two dimensions - stability Continuing Von Neumann: ˜ " n + 1 = ˜ " n # 1 # 2 i $ t $ s U sin k $ s + V sin l $ s ( ) ˜ " n ˜ % n + 1 = ˜ " n ˜ " n + 1 ˜ % n + 1 & ( ) * + = # 2 i $ t / $ s ( U sin k $ s + V sin l $ s ) 1 1 0 & ( ) * + ˜ " n ˜ % n & ( ) * + # 2 i $ t / $ s ( U sin k $ s + V sin l $ s ) # , 1 1 0 # , = 0 10/23/06 Atms 502 - Fall 2006 - Jewett 7 Two dimensions - stability Continuing Von Neumann: " 2 i # t / # s ( U sin k # s + V sin l # s ) " $ 1 1 0 " $ = 0 $ 2 " " 2 i # t # s U sin k # s + V sin l # s ( ) % & ( ) * $ " 1 = 0 $ = " 2 i # t # s U sin k # s + V sin l # s ( ) ± " 4 # t 2 # s 2 ()
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