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Unformatted text preview: 1 Atms 502 Numerical Fluid Dynamics Tue., Oct. 17, 2006 10/18/06 Atms 502  Fall 2006  Jewett 2 Multistage methods (1) • Earlier schemes were singlestage The terms on the right side  the spatial derivatives  were evaluated once Thus 2timelevel and 3timelevel schemes can be singlestage • Multistage methods Try to improve accuracy by evaluating right handside F terms at additional times between n ∆ t and (n+1) ∆ t. 10/18/06 Atms 502  Fall 2006  Jewett 3 • 2stage schemes (amplitude behavior) Multistage methods (2) " 2 = 1 + (1 # 2 $% )( & ’ t ) 2 + $ 2 % 2 ( & ’ t ) 4 Strong damping β =1 α =1 Matsuno Weakly unstable β =1 α =1/2 Midpoint Weakly unstable β =1/2 α =1 Heun RK RK RK: weak instability/2 nd order αβ >1/2: less accuracy but damped … damping can be useful! 10/18/06 Atms 502  Fall 2006  Jewett 4 AdamsBashforth • 2ndorder: Weakly unstable (P), but with damped computational mode • 3rdorder: Highly efficient 1 comp. mode limits ∆ t Higherorder methods (1) " n + 1 = " n + ( h /2) 3 F ( " n ) # F ( " n # 1 ) ( ) " n + 1 = " n + ( h /12) 23 F ( " n ) # 16 F ( " n # 1 ) + 5 F ( " n # 2 ) $ % & & & ’ ( ) ) ) 2 10/18/06 Atms 502  Fall 2006  Jewett 5 RungeKutta • 3rdorder: • RK3 / RK4 methods have two free parameters; can choose to minimize storage Higherorder methods (2) q 1 = hF ( " n ) q 2 = hF ( " 1 ) # (5/9) q 1 q 3 = hF ( " 2 ) # (153/128) q 2 " 1 = " n + (1/3) q 1 " 2 = " 1 + (15/16) q 2 " n + 1 = " 2 + (8/15) q 3 RK3 and RK4 RK2 10/18/06 Atms 502  Fall 2006  Jewett 6 Time differencing: summary • Lists: Order of scheme Formula Storage factor Number of full arrays needed (if explicit) • Lists: Efficiency factor Largest stable step ÷ by # evaluations of F Amplitude, phase error Max stable ω ∆ t Durran Table 2.1, pp 6869 10/18/06 Atms 502  Fall 2006  Jewett 7 Method discussion (1) • For smooth solutions:...
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 Spring '09
 JEWETT
 Convection, Velocity, Jewett, Richard Courant

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