Theorem. 12 A consistent one-step scheme for the eqution +bu=0 is stable if and only if it is stable for this equation when b is equal to 0. More over, when k= Ah and A is a constant, the stability condition on g(hs, k, h) g(6,0.0)
Theorem. 12 A consistent one-step scheme for the eqution ∂u ∂t + a ∂u ∂x + bu = 0 is stable if and only if it is stable for this equation when b is equal to 0. Moreover, when k = λh and λ is a constant, the stability condition on g(hξ, k, h) is |g(θ, 0, 0)| ≤ 1 15
Chapter 3 Order of accuracy of Finite Difference Schemes 3.1 Order of Accuracy 计算PDE方程Pu=f的格式一般可以写成PRh=Rkhf Definition. 13 A scheme Pk. h=Rk.hf that is consistent with the differen- tial equation Pu=f is accurate of order p in time and order g in space if for any smooth function o(t, r) Pk o-RuhPo=O(k)+O(h9) We will say that this scheme is accurate of order(p, g) 容易知道Rk要求是恒等算子的逼近,而Pk逼近P。我们把Pn-Rk 称为计算格式的截断误差。 注意到在Lax- edrichs格式对Pkhp的 Taylor展开式中,包含项k-1h2x 我们假设时间步长的选取是空间步长的函数,ie.k=M(h),这里A是关 于h的光滑函数且A(0)=0。我们有如下定义
Chapter 3 Order of Accuracy of Finite Difference Schemes 3.1 Order of Accuracy OPDE§P u = fª±¤Pk,h = Rk,hf. Definition. 13 A scheme Pk,hv = Rk,hf that is consistent with the differential equation P u = f is accurate of order p in time and order q in space if for any smooth function φ(t, x) Pk,hφ − Rk,hP φ = O(k p ) + O(h q ) We will say that this scheme is accurate of order (p, q) N´Rk,h¦´ð f%C§ Pk,h%CP"·rPk,hφ − Rk,hφ ¡Oª äØ " 5¿3Lax-FriedrichsªéPk,hφTaylorÐmª¥§¹k −1h 2φxx" ·bmÚÀ´mÚ¼ê§i.e. k = Λ(h)§ùpΛ´' uh1w¼ê Λ(0) = 0"·kXe½Â" 16
Definition. 14 A scheme P, hU= Rk, hf with k= A(h)that is consistent with the differential equation Pu=f is accurate of order r if for any smooth functin (t, r) PR, ho- Rk, hPo=O(h) 3.1.1 Symbols of Difference Schemes Definition. 15 The symbol pk, h(s, s)of a difference operator Pk, h is defined Pi.h(eskmeimh5)=Pk.n(s, s)eskne imho That is, the symbol is the quantity multiplying the grid function eskneamhs after operating on this function with the difference operator 考虑Lax- Wendroff算子,我们有 Pk. h(s, E k 和 iak Tk, h(s, s) +1) sin he Definition. 16 The symbol p(s, s)of the differential operator P is defined P(ete)=p(s, s)estesa That is, the symbol is the quantity multiplying the function estelas after op- erating on this function with the differential operator Theorem. 17 A scheme PkhU= Rthf that is consistent with Pu= f is accurate of order (p, g) if and only if for each value of s and s 7A(,)-p(s,)=O(k)+O(h°) 17
Definition. 14 A scheme Pk,hv = Rk,hf with k = Λ(h) that is consistent with the differential equation P u = f is accurate of order r if for any smooth functin φ(t, x) Pk,hφ − Rk,hP φ = O(h r ) 3.1.1 Symbols of Difference Schemes Definition. 15 The symbol pk,h(s, ξ) of a difference operator Pk,h is defined by Pk,h(e skne imhξ) = pk,h(s, ξ)e skne imhξ That is, the symbol is the quantity multiplying the grid function e skne imhξ after operating on this function with the difference operator. ÄLax-Wendroff f§·k pk,h(s, ξ) = e sk − 1 k + ia h sin hξ + 2 a 2k h 2 sin2 1 2 hξ Ú rk,h(s, ξ) = 1 2 (e sk + 1) − iak 2h sin hξ Definition. 16 The symbol p(s, ξ) of the differential operator P is defined by P(e ste iξx) = p(s, ξ)e ste iξx That is, the symbol is the quantity multiplying the function e ste ixξ after operating on this function with the differential operator. Theorem. 17 A scheme Pk,hv = Rk,hf that is consistent with P u = f is accurate of order (p, q) if and only if for each value of s and ξ pk,h(s, ξ) rk,h(s, ξ) − p(s, ξ) = O(k p ) + O(h q ) 17
Corollary. 18 A scheme P*, hU=R, hf with k= A(h) that is consistent with Pu=f is accurate of order r if and only if for each value of s and s Pk.h(S kA(S分)p(s,)=O(h) Example.19我们可以证明 Crank- Nicolson格式的精度是(2,2)阶的。 Crank Nicolson格式为 m-1 fm++fm 4h 容易知道 Pk, h(s, 5)= +1 sin he k 和 rk,h(s, E) 2 所以 n he Pk, h(s, S)-rk, h(s, S)P(s, s) 2 h 2-(s+i)=O(k)+O2) Theorem. 20 an explicit one-step scheme for hyperbolic equations that has the form l=-∞o for homogeneous problems can be at most first-order accurate if all the co effients a are nonnegative, except for the two trivial schemes for the one-way wave equation with aA= l given b Um=um- if a>0 n+1 um+1 if a<o 18
Corollary. 18 A scheme Pk,hv = Rk,hf with k = Λ(h) that is consistent with P u = f is accurate of order r if and only if for each value of s and ξ pk,h(s, ξ) rk,h(s, ξ) − p(s, ξ) = O(h r ) Example. 19 ·±y²Crank-Nicolsonª°Ý´(2, 2) "CrankNicolsonª v n+1 m − v n m k + a v n+1 m+1 − v n+1 m−1 + v n m+1 − v n m−1 4h = f n+1 m + f n m 2 N´ pk,h(s, ξ) = e sk − 1 k + ia e sk + 1 2 sin hξ h Ú rk,h(s, ξ) = e sk + 1 2 ¤± pk,h(s, ξ)−rk,h(s, ξ)p(s, ξ) = e sk − 1 k +ia e sk+1 2 sin hξ h − e sk + 1 2 (s+iaξ) = O(k 2 )+O(h 2 ) Theorem. 20 an explicit one-step scheme for hyperbolic equations that has the form v n+1 m = X∞ l=−∞ αlv n m+l for homogeneous problems can be at most first-order accurate if all the coeffients αl are nonnegative, except for the two trivial schemes for the one-way wave equation with |aλ| = 1 given by v n+1 m = v n m−1 if a > 0 and v n+1 m = v n m+1 if a < 0 18