Design of optimal Runge-Kutta methods - FEniCS Project
Design of optimal Runge-Kutta methods - FEniCS Project Design of optimal Runge-Kutta methods - FEniCS Project
The Stability Function For the linear equation u ′ = λu, a Runge-Kutta method yields a solution u n+1 = φ(λ∆t)u n , where φ is called the stability function of the method: Example: Euler’s Method φ(z) = det(I − z(A − ebT ) det(I − zA) u n+1 = u n + ∆tF (u); φ(z) = 1 + z. For explicit methods of order p: p� 1 φ(z) = j! zj + j=0 s� j=p+1 αjz j . D. Ketcheson (KAUST) 10 / 36
Absolute Stability For the linear equation u ′ (t) = Lu we say the solution is absolutely stable if |φ(λ∆t)| ≤ 1 for all λ ∈ σ(L). D. Ketcheson (KAUST) 11 / 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: The Stability Function For the line
- 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
The Stability Function<br />
For the linear equation<br />
u ′ = λu,<br />
a <strong>Runge</strong>-<strong>Kutta</strong> method yields a solution<br />
u n+1 = φ(λ∆t)u n ,<br />
where φ is called the stability function <strong>of</strong> the method:<br />
Example: Euler’s Method<br />
φ(z) = det(I − z(A − ebT )<br />
det(I − zA)<br />
u n+1 = u n + ∆tF (u); φ(z) = 1 + z.<br />
For explicit <strong>methods</strong> <strong>of</strong> order p:<br />
p� 1<br />
φ(z) =<br />
j! zj +<br />
j=0<br />
s�<br />
j=p+1<br />
αjz j .<br />
D. Ketcheson (KAUST) 10 / 36
Change language