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: a philosophical aside An RK method builds up information about the solution derivatives through the computation of intermediate stages At the end of a step all of this information is thrown away! Use more stages =⇒ keep information around longer D. Ketcheson (KAUST) 8 / 36
Outline 1 High order Runge-Kutta methods 2 Linear properties of Runge-Kutta methods 3 Nonlinear properties of Runge-Kutta methods 4 Putting it all together: some optimal methods and applications D. Ketcheson (KAUST) 9 / 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: Runge-Kutta Methods To solve the in
- 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 and 42: The Forward Euler condition Recall
- Page 43 and 44: Runge-Kutta methods as a convex com
- 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> Methods: a philosophical aside<br />
An RK method builds up information about the solution derivatives<br />
through the computation <strong>of</strong> intermediate stages<br />
At the end <strong>of</strong> a step all <strong>of</strong> this information is thrown away!<br />
Use more stages =⇒ keep information around longer<br />
D. Ketcheson (KAUST) 8 / 36
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