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
with the partial differential equation if for any smooth function o(t, r) P-Bk→0ask,h→0, the convergence being pointwise convergence at each grid point Example The laz- Friedrichs scheme对 Lax-Friedrichs格式: Pk.hg φm+1-是(mn+1+m-1),m 用 Taylor公式 omn+1=m±hpx+h2x±h3bx+O(h2) 我们可以得到 Pk,ho=ot+aor+okott-okh'oxr -ah'oxrr +O(h+k h+k) 这样当h,k→0,且k-1h2→0时,Pn-P→0. Definition. 4 A finite difference scheme Pk hUm=0 for a first-order equa tion is stable if there is an integer J and positive numbers ho and ko such that for any positive time T, there is a constant CT such that h∑m2≤cr∑∑mP2 jor0≤mk≤T0<h≤ho,and0<k≤ko. 我们引入记号 dh=(h∑ 所以上述稳定性等价于 ll≤c∑‖‖ 5
with the partial differential equation if for any smooth function φ(t, x) P φ − Pk,hφ → 0 as k, h → 0, the convergence being pointwise convergence at each grid point. Example The lax-Friedrichs Scheme éLax-Friedrichs ªµ Pk,hφ = φ n+1 m − 1 2 (φ n m+1 + φ n m−1 ) k + a φ n m+1 − φ n m−1 2h ^Taylorúªµ φ n m±1 = φ n m ± hφx + 1 2 h 2φxx ± 1 6 h 3φxxx + O(h 4 ) ·± Pk,hφ = φt + aφx + 1 2 kφtt − 1 2 k −1h 2φxx + 1 6 ah2φxxx + O(h 4 + k −1h 4 + k 2 ) ùh, k → 0§ k −1h 2 → 0§Pk,hφ − P φ → 0. Definition. 4 A finite difference scheme Pk,hv n m = 0 for a first-order equation is stable if there is an integer J and positive numbers h0 and k0 such that for any positive time T, there is a constant CT such that h X∞ m=−∞ |v n m| 2 ≤ CT h X J j=0 X∞ m=−∞ |v j m| 2 for 0 ≤ nk ≤ T,0 < h ≤ h0, and 0 < k ≤ k0. ·Ú\PÒ kwkh = h X∞ m=−∞ |vm| 2 !1 2 ¤±þã½5 du kv n kh ≤ C ∗ T X J j=0 kv j kh 5
Example:考虑计算格式 m+ B 这个计算格式当a|+|≤1时是稳定的。 差分方程的稳定性与PDE方程初值问题的适定性是密切相关的。 Definition. 5 The initial value problem for the first-order partial d ifferen- tial equation Pu=0 is well-posed if for any time T>0, there is a constant CT such that any solution u(t, a)safisfies 1,) d z CT/a(0,)dfor0≤t≤T 1.1.5 The Lax-Richtmyer Equivalence Theorem Theorem. 6 The Lat-Richtmyer Equivalence Theorem. A consistent finite difference scheme for a partial differential equation for which the initial value problem is well-posed is convergent if and only if it is stable 1.1.6 The Courant-Friedrichs-Lewy Condition Theorem. 7 For an explicit scheme for the hyperbolic equation +a-=0 of the form n+1 aum_1+ Bum+rr n+1 with k/h=a held constant, a necessary condition for stability is the Courant friedrichs-Lewy (CFL) condition, a<1 For systems of equations for which v is a vector and a, B and y are matrices, we must have aidl s 1 for all eigenvalues ai of the matriT A
Example: ÄOªµ v n+1 m = αvn m + β n m+1 ùOª|α| + |β| ≤ 1´½" ©§½5PDE§Ð¯K·½5´'" Definition. 5 The initial value problem for the first-order partial d ifferential equation P u = 0 is well-posed if for any time T ≥ 0, there is a constant CT such that any solution u(t, x) safisfies Z ∞ −∞ |u(t, x)| 2 dx ≤ CT Z ∞ −∞ |u(0, x)| 2 dx for 0 ≤ t ≤ T. 1.1.5 The Lax-Richtmyer Equivalence Theorem Theorem. 6 The Lax-Richtmyer Equivalence Theorem. A consistent finite difference scheme for a partial differential equation for which the initial value problem is well-posed is convergent if and only if it is stable. 1.1.6 The Courant-Friedrichs-Lewy Condition Theorem. 7 For an explicit scheme for the hyperbolic equation ∂u ∂t + a ∂u ∂x = 0 of the form v n+1 m = αvn m−1 + βvn m + γvn m+1 with k/h = λ held constant, a necessary condition for stability is the Courantfriedrichs-Lewy(CFL) condition, |aλ| < 1. For systems of equations for which v is a vector and α, β and γ are matrices, we must have |aiλ| ≤ 1 for all eigenvalues ai of the matrix A 6
Theorem. 8 There are no explicit, unconditionally stable, consistent finite difference schemes for hyperbolic systems of partial differential equations 这个结论发表在 Courant, R, K.O.Friedrichs, and H. Lewy, 1928, Uber die partiellen dif- ferenzengleichungen der mathematischen physik, Mathematische Annalen, 100: 32- 74
Theorem. 8 There are no explicit, unconditionally stable, consistent finite difference schemes for hyperbolic systems of partial differential equations. ù(ØuL3µ Courant, R., K.O.Friedrichs, and H.Lewy, 1928, Uber die partiellen dif- ¨ ferenzengleichungen der mathematischen physik, Mathematische Annalen,100:32- 74 7
Chapter 2 Analysis of Finite Difference Schemes 2.1 Fourier Analysis 当函数(x)定义在实轴R上,它的 Fourier变换(u)定义为 a(∞) Fourier反变换定义为 u(a) 类似地,如果网格函数定义在整数网格m∈Z上,它的 Fourier变化定义 为:对∈[-丌,]且(-丌)=0(丌) (5) Fourier反变换定义为
Chapter 2 Analysis of Finite Difference Schemes 2.1 Fourier Analysis ¼êu(x)½Â3¢¶Rþ,§FourierCuˆ(ω)½Â uˆ(ω) = 1 √ 2π Z ∞ −∞ u(x)e −iωxdx FourierC½Â u(x) = 1 √ 2π Z ∞ −∞ uˆ(ω)e iωxdω aq/§XJ ¼êv½Â3ê m ∈ Zþ,§FourierCz½Â µéξ ∈ [−π, π] vˆ(−π) = ˆv(π) vˆ(ξ) = 1 √ 2π X∞ m=−∞ vme −imξ FourierC½Â vm = 1 √ 2π Z π −π veˆ imξ(ξ)dξ 8
如果网格点之间的距离是h,通过变量代换,我们可以定义:对∈ h,r/] 6(E)=√2m=-∞ 反变换公式是 i(s)ende 个重要的结果是 Parseval关系:在连续情况下 Ja(a)ld.r a(u)2dx台‖l|2=‖al|2 类似的,在离散情况下 (5)2d vmPh=‖l2 Parseval等式广泛被用于稳定性分析。前面定义的稳定性估计可以改成等 价的形式: lh≤G∑0lh 2.2 Fourier Analysis and Partial Differential Equations 通过对 Fourier反变换公式求导,我们可以得到 e"wwwi(w)dw 因此我们有 D/)()=ia(u) 从这里可以看到,通过 Fourier变化,可以把求导变成乘法,这样使得我们 可以把PDE中的问题看成是代数问题 9
XJ :mål´h§ÏLCþ§·±½Âµéξ ∈ [−π/h, π/h], vˆ(ξ) = 1 √ 2π X∞ m=−∞ vme −imhξh Cúª´ vm = 1 √ 2π Z π/h −π/h vˆ(ξ)e imhξdξ (J´Parseval'Xµ3ëY¹e Z ∞ −∞ |u(x)| 2 dx = Z ∞ −∞ |uˆ(ω)| 2 dx ⇔ kuk2 = kuˆk2 aq§3lѹe kvˆk 2 h = Z π/h −π/h |vˆ(ξ)| 2 dξ = X∞ m=−∞ |vm| 2h = kvk 2 h Parseval ª2^u½5©Û"c¡½Â½5O±U¤ d/ªµ kvˆ n kh ≤ C ∗ T X J j=0 kvˆ j kh 2.2 Fourier Analysis and Partial Differential Equations ÏLéFourierCúª¦§·± ∂u ∂x(x) = 1 √ 2π Z ∞ −∞ e iωxiωuˆ(ω)dω Ïd·k ˆ∂u ∂x! (ω) = iωuˆ(ω) lùp±w§ÏLFourierCz§±r¦C¤¦{§ù¦· ±rPDE¥¯Kw¤´ê¯K" 9