Diffusing equations Crank-Nicholson格式取 1+(1+7) +1 T-20 l+1+(1-7)2+ 0 截断误差是B=O(72+12)与显格式不同的是:每 求一层(下一个时间点)的解,都需要求解一个代 数方程。 NA diffe2tex-微分方程数值解-陈文斌-17/4/2003-7:49-p.1446
Diffusing equations Crank-Nicholson ª θ = 1 2 − r 2 u j+1 i+1 + (1 + r ) u j+1 i − r 2 u j+1 i − 1 = r 2 u j i+1 + (1 − r ) u j i + r 2 u j i − 1 u 0 i = φ i u j 0 = u j N = 0 ä Ø ´ R j i = O ( τ 2 + h 2 ) w ª Ø Ó ´µz ¦ £ e m : ¤ ) § Ñ I ¦ ) ê § " NA˙diffc2.tex – © § ê ) – © R – 17/4/2003 – 7:49 – p.14/46
Diffusing equations 在两层格式中,如果取=号,则截断误差估计是 (O)=O(72)+O(7h2)+O(h4 可以有各种各样的计算格式,但是有一点我们不能 记 稳定性! NA diffe. tex-微分方程数值解-陈文斌-17/4/2003-7:49-p.1546
Diffusing equations 3 ü ª ¥ § X J θ = 1 2 − 1 12 r ,K ä Ø O ´ R j i ( θ) = O ( τ 2) + O (τh 2) + O ( h 4 ) ± k « O ª § ´ k : · Ø U # Pµ ½ 5 NA˙diffc2.tex – © § ê ) – © R – 17/4/2003 – 7:49 – p.15/46
Diffusing equations 影响区域类似于初边值问题的处理,只是我们用显 式格式 =72+1+(1-2n7)2+r 影响区域 Matlab: meshdemo NA diffe2tex-微分方程数值解-陈文斌-17/4/2003-7:49-p.1646
Diffusing equations K « a q u Ð > ¯ K ? n § ´ · ^ w ª ª ( u j+1 i = ru j i+1 + (1 − 2 r ) u j i + ru j i − 1 u 0 i = φ i K « Matlab:meshdemo NA˙diffc2.tex – © § ê ) – © R – 17/4/2003 – 7:49 – p.16/46
Convection Equations 对流方程 +a at ax 显式左偏格式 +1 +a f i, R=O(T+h NA diffe2tex-微分方程数值解-陈文斌-17/4/2003-7:49-p.1746
Convection Equations é 6 § ∂u ∂t + a ∂u ∂x = f w ª ª u j+1 i τ + a u j i − u j i − 1 h = f j i , R = O ( τ + h ) NA˙diffc2.tex – © § ê ) – © R – 17/4/2003 – 7:49 – p.17/46
Convection Equations 显式右偏格式 +1 a +1 B=0(+ T 显式中心差分格式 +1 +a f, R=O(T +h NA diffe2tex-微分方程数值解-陈文斌-17/4/2003-7:49-p.1846
Convection Equations w ª m ª u j+1 i τ + a u j i+1 − u j i h = f j i , R = O ( τ + h ) w ª ¥ % © ª u j+1 i τ + a u j i+1 − u j i − 1 h = f j i , R = O ( τ + h ) NA˙diffc2.tex – © § ê ) – © R – 17/4/2003 – 7:49 – p.18/46