Milo Viviani (joint work with prof. Michele Benzi)
Scuola Normale Superiore - Pisa
Due Giorni di Algebra Lineare Numerica e Applicazioni
Napoli, 14-15 Febbraio 2022
Thm. Given Y∈M(N,C), equation (*) has a unique solution for sufficiently small h>0 in some neighbourhood of Y.
Furthermore, when equation (*) takes place in su(N), there exists such a neighbourhood for any h≤3∥L∥op∥Y∥1.
G(X):=X−h[LX,X]−h2(LX)X(LX)−Y
DG(X)=I−h(B1+hB2)
B2=(L⋅)X(LX)+(LX)⋅(LX)+(LX)X(L⋅)
B1=[L⋅,X]+[LX,⋅]
Need an approximation for DG(X)−1
Rmk. the second one is the best compromise between exactness and computational cost
Spatial semidiscretization of the Euler equations on a compact orientable surface S, ωt∈C∞(S)
ω˙=∇ω⋅∇⊥Δ−1ω
Give the Hamiltonian isospectral flow on su(N), for N≥1
W˙=[W,ΔN−1W]
Rmk. ΔN is tridiagonal, the evaluation of ΔN−1W (via LU-factorization done once) requires O(N2) operations
Time-step h=0.5, normalized initial data
Rmk. Explicit fixed point performs the best in terms of CPU time
Spatial semidiscretization of the Landau–Lifshitz–Gilbert Hamiltonian PDE, σt:S1→S2
σ˙=σ×∂xxσ
Give the Hamiltonian isospectral flow on su(2)⊕N, for N≥1
S˙i=[Si,Si−1+Si+1]
for i=1,...,N with S1≡SN
Time-step h=0.5, normalized initial data
Rmk. Explicit fixed point performs the best in terms of stabilty and CPU time.