微分方程数值解课件 陈文斌 May5,2003
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Chapter 1 Hyperbolic Partial Differential Equations 1.1 Overiew Hyperbolic Partial Differential Equations 方程的解为 当α>0时,解往右移动,当α<0时,解向左移动。在(t,x)位置的解 只依赖于ξ=x-at的的初始值。我们把x-at= const的线称为特征 线( characteristics)。a称为特征线的传播速度。 考虑 况+aDx2+bu=f(,x) a(0 定义变换 t=7,5=x-at
Chapter 1 Hyperbolic Partial Differential Equations 1.1 Overiew Hyperbolic Partial Differential Equations: ∂u ∂t + a ∂x ∂x = 0 §) u(t, x) = u0(x − at) a > 0§) m£Ä§a < 0§)£Ä"3(t, x) ) 6uξ = x − atЩ"·rx − at = const¡A (characteristics)"a¡ADÂÝ" Ä ∂u ∂t + a ∂u ∂x + bu = f(t, x) u(0, x) = u0(x) ½ÂC t = τ, ξ = x − at 1
我们定义(r,)=u(t,x)。 =-b+f(r,5+ar) 这个方程有解 i(r,.5)=uo(S)e-br+/f(o,5+ao)e-b(r-o)do u(t, r)=uo(a-at)e f(s, I-a(t-s) Definition. 1 A system of the form au au Ao+ Bu= F(t, a is hyperbolic if the matric A is diagonalizable with real eigenvalues 1.1.1 Equations with variable Coefficients at 0 0,i(0,5)=uo(5) d 1.1.2 Boundary Condtions 考虑初边值问题 dur 00≤x≤1,t≥0 如果a>0,则波往右方移动。初始数据u(0,x)=u0(x)和边界数据u(t,0)= 9(t) (a-at) g(t-aac) if -at<0 2
. ·½Âu˜(τ, ξ) = u(t, x)" ∂u˜ ∂τ = −bu˜ + f(τ, ξ + aτ ) ù§k) u˜(τ, ξ) = u0(ξ)e −bτ + Z τ 0 f(σ, ξ + aσ)e −b(τ−σ) dσ. u(t, x) = u0(x − at)e −bt + Z t 0 f(s, x − a(t − s))e −b(t−s) ds. Definition. 1 A system of the form ∂u ∂t + A ∂u ∂x + Bu = F(t, x) is hyperbolic if the matrix A is diagonalizable with real eigenvalues. 1.1.1 Equations with variable Coefficients ∂u ∂t + a(t, x) ∂u ∂x = 0 du˜ dτ = 0, u˜(0, ξ) = u0(ξ) dx˜ dτ = a(τ, x), x(0) = ξ 1.1.2 Boundary Condtions ÄÐ>¯Kµ ∂u ∂t + a ∂u ∂x = 0 0 ≤ x ≤ 1, t ≥ 0 XJa > 0§KÅ m£Ä"Щêâu(0, x) = u0(x)Ú>.êâu(t, 0) = g(t) u(t, x) = u0(x − at), if x = at > 0 g(t − a −1x) if x − at < 0 2
1.1.3 Introduction to finite difference schemes k +a 0 h leapfrog sche tm+1-(vm+1+m-1),vm+1-om 0 Lax-Friedrichs scheme k Example考虑初边值问题 at dr 2≤x≤3,0<t 初始条件 I if rl≤1 if|r|≥1 当x=-2时,我们指定a=0 0 02 3
1.1.3 Introduction to Finite difference schemes v n+1 m − v n m k + a v n m+1 − v n m h = 0 v n+1 m − v n m k + a v n m − v n m−1 h = 0 v n+1 m − v n m k + a v n m+1 − v n m−1 h = 0 v n+1 m − v n−1 m 2k + a v n m+1 − v n m−1 2h = 0 leapfrog scheme v n+1 m − 1 2 (v n m+1 + v n m−1 ) k + a v n m+1 − v n m−1 2h = 0 Lax-Friedrichs scheme Example ÄÐ>¯K ∂u ∂t + ∂u ∂x = 0 − 2 ≤ x ≤ 3, 0 ≤ t Щ^ u0(x) = 1 − |x| if |x| ≤ 1 0 if |x| ≥ 1 x = −2§·½u = 0 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t=1.60 3
在上面的图中,我们假设λ=0.8,h=0.1,t=1.6时刻的解和计算解, 用Lax- Friedrichs格式计算: + 在上面的图中,我们假设λ=1.6,h=0.1,t=0.8时刻的解和计算解,用 上述Lax- Friedrichs格式计算。 1.1. 4 Convergnce and Consistency Definition. 2 A one-step finite difference scheme approximating a partial differential equation is a convergent scheme if for any solution to the partial differential equation, u(t, r), and solutions to the finite difference scheme, um such that um converges to o(a)as mh convergences to T, then um converges to u(t, r)as(nk, mh) converges to(t, )as h, k converge to 0 Definition. 3 Given a partial differential equaiton Pu=f and a finite difference scheme, Pk hU=f, we say the finite difference scheme is consistent
3þ¡ã¥§·bλ = 0.8,h = 0.1,t = 1.6)ÚO)§ ^Lax-Friedrichs ªOµ v n+1 m = 1 2 (v n m+1 + v n m−1 ) − 1 2 λ(v n m+1 − v n m−1 ) −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 t=0.80 3þ¡ã¥§·bλ = 1.6,h = 0.1,t = 0.8)ÚO)§^ þãLax-Friedrichs ªO" 1.1.4 Convergnce and Consistency Definition. 2 A one-step finite difference scheme approximating a partial differential equation is a convergent scheme if for any solution to the partial differential equation, u(t, x), and solutions to the finite difference scheme, v n m, such that v 0 m converges to u0(x) as mh convergences to x, then v n m converges to u(t, x) as (nk, mh) converges to (t, x) as h, k converge to 0. Definition. 3 Given a partial differential equaiton P u = f and a finite difference scheme, Pk,hv = f, we say the finite difference scheme is consistent 4