Design of optimal Runge-Kutta methods - FEniCS Project
Design of optimal Runge-Kutta methods - FEniCS Project Design of optimal Runge-Kutta methods - FEniCS Project
Runge–Kutta methods as a convex combination of Euler Consider the two-stage method: Is ||u n+1 || ≤ ||u n ||? y 1 = u n + ∆tF (u n ) u n+1 = u n + 1 2 ∆t � F (u n ) + F (y 1 ) � D. Ketcheson (KAUST) 24 / 36
Runge–Kutta methods as a convex combination of Euler Consider the two-stage method: y 1 = u n + ∆tF (u n ) u n+1 = 1 2 un + 1 � � 1 1 y + ∆tF (y ) . 2 Take ∆t ≤ ∆tFE. Then ||y 1 || ≤ ||u n ||, so ||u n+1 || ≤ 1 2 ||un || + 1 2 ||y1 + ∆tF (y 1 )|| ≤ ||u n ||. ||u n+1 || ≤ ||u n || D. Ketcheson (KAUST) 24 / 36
- Page 1 and 2: Design of optimal Runge-Kutta metho
- Page 3 and 4: Outline 1 High order Runge-Kutta me
- Page 5 and 6: Solution of hyperbolic PDEs The fun
- Page 7 and 8: Solution of hyperbolic PDEs The fun
- Page 9 and 10: Solution of hyperbolic PDEs The fun
- Page 11 and 12: Time Integration Using a better tim
- Page 13 and 14: Time Integration Using a better tim
- Page 15 and 16: Time Integration Using a better tim
- Page 17 and 18: Runge-Kutta Methods To solve the in
- Page 19 and 20: Outline 1 High order Runge-Kutta me
- Page 21 and 22: The Stability Function For the line
- Page 23 and 24: Absolute Stability For the linear e
- Page 25 and 26: Stability optimization This leads n
- Page 27 and 28: Stability Optimization: a toy examp
- Page 29 and 30: Stability Optimization: a toy examp
- Page 31 and 32: Stability Optimization: one more ex
- Page 33 and 34: Stability Optimization: one more ex
- Page 35 and 36: Nonlinear accuracy Besides the cond
- Page 37 and 38: Strong stability preservation Desig
- Page 39 and 40: Strong stability preservation Desig
- Page 41: The Forward Euler condition Recall
- Page 45 and 46: Example: A highly oscillatory flow
- Page 47 and 48: Low storage methods 3S Algorithm S3
- Page 49 and 50: Two-step optimization process Our o
- Page 51 and 52: Two-step optimization process Our o
- Page 53 and 54: Optimizing for the SD spectrum On r
- Page 55 and 56: Optimizing for the SD spectrum Prim
- Page 57 and 58: Application: flow past a wedge Dens
<strong>Runge</strong>–<strong>Kutta</strong> <strong>methods</strong> as a convex combination <strong>of</strong> Euler<br />
Consider the two-stage method:<br />
y 1 = u n + ∆tF (u n )<br />
u n+1 = 1<br />
2 un + 1 � � 1 1<br />
y + ∆tF (y ) .<br />
2<br />
Take ∆t ≤ ∆tFE. Then ||y 1 || ≤ ||u n ||, so<br />
||u n+1 || ≤ 1<br />
2 ||un || + 1<br />
2 ||y1 + ∆tF (y 1 )|| ≤ ||u n ||.<br />
||u n+1 || ≤ ||u n ||<br />
D. Ketcheson (KAUST) 24 / 36
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