半线性抛物方程组的有限差分格式.pdf

∗ (210096) E-mail:ken bear@ 2 L∞ O(h2 + τ 2). 1. ut = uxx + f (u), (x, t) ∈ ΩT = (0, l) × (0, T], (1) ∂u ∂2u u(x, t) = (u1 (x, t), · · · , um (x, t)) m ut = ∂t , uxx = ∂x2 f (u) = (f (u), · · · , f (u)) m 1 m u(0, t) = α(t), u(l, t) = β (t), t ∈ (0, T], (2) u(x, 0) = u (x), x ∈ [0, l], (3) 0 (1) [1] Crank-Niclson [2] (1) Euler [3] 2 [4] (1) Æ θ- Runge-Kutta - [5, 6, 7] L- A ∗ 1 Crank-Niclson (1)-(3) 2 L∞ O(h2 + τ 2) (H1) (1)-(3) u ∈ C4,3 (ΩT ) C0 0 u C0 (H2) f (u) C1 0 0 N 1, |s| ≤ C0 + N f (s) C1 (H3) (1)-(3) u u (x). 0 2. x = x (0 ≤ j ≤ J), t = t (0 ≤ k ≤ K ) Ω , x = jh, t = kτ , j k T j k Jh = l, Kτ = T. J K Ωτ = {(x , t )|0 ≤ j ≤ J, 0 ≤ k ≤ K } m Æ {vk |0 ≤ j ≤ J, 0 ≤ h j k j k ≤ K }, Æ k+ 1 1 k k+1 k+ 1 1 k+1 k v 2 = (v + v ), δ v 2 = (v − v ), j 2 j j t j τ j