Design of optimal Runge-Kutta methods - FEniCS Project
Design of optimal Runge-Kutta methods - FEniCS Project
Design of optimal Runge-Kutta methods - FEniCS Project
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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 />
D. Ketcheson (KAUST) 10 / 36
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