微分方程数值解 发展方程的差分方法 陈文斌 复旦大学数学系 Na diffe3.tex-微分方程数值解一陈文斌-21/4/2003-17:00-p.1/29
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Diffusing equations 扩散方程初边值问题:设a>0,在区 域D={(x,t)0<x<1,0≤t≤们}找(x,t满足 au a2, 0<<1.0<t<T (x,0)=0(x),0<x<1初值条件 (0,+)=(1,t)=0,0≤t≤T边值条件 这里边值条件也可以有多种提法 Na diffe3.tex-微分方程数值解一陈文斌-21/4/2003-17:00-p.2/29
Diffusing equations * Ñ §Ð > ¯ Kµa > 0 § 3 « D ∞ = { (x, t )|0 < x < 1, 0 ≤ t ≤ T } é u (x, t ) ÷ v ∂u ∂t = a ∂ 2 u ∂x 2 , 0 < x < 1, 0 < t ≤ T u (x, 0) = φ ( x ), 0 < x < 1 Ð ^ u(0, t) = u(1, t) = 0, 0 ≤ t ≤ T > ^ ù p > ^ ± k õ « J { " NA˙diffc3.tex – © § ê ) – © R – 21/4/2003 – 17:00 – p.2/29
Diffusing equations 区域的离散:这里需要空间方向x和时间方向t的 离散,假设空间方向N等分,空间步长 为h=,x;=2*h;时间方向为步长 为 T 方程离散 边界条件的处理 Na diffe3.tex-微分方程数值解一陈文斌-21/4/2003-17:00-p.3/29
Diffusing equations « l Ñ:ù p I m x Ú m t l Ñ § b m N © § m Ú h = 1 N ,x i = i ∗ h¶ m Ú τ § t j = jτ " § l Ñ > . ^ ? n NA˙diffc3.tex – © § ê ) – © R – 21/4/2003 – 17:00 – p.3/29
Diffusing equations 区域的离散 c∞OoμAa,n 4/29
Diffusing equations « l Ñ ÇøÓòµÄÍø¸ñ x t T (x i,t j) x i t j NA˙diffc3.tex – © § ê ) – © R – 21/4/2003 – 17:00 – p.4/29
Diffusing equations 古典显格式取=0 r+1+(1-2r)r+r 0 截断误差是 R=O(T +h Na diffe3.tex-微分方程数值解一陈文斌-21/4/2003-17:00-p.5/29
Diffusing equations ; w ª θ = 0 u j+1 i = ru j i+1 + (1 − 2 r ) u j i + ru j i − 1 u 0 i = φ i u j 0 = u j N = 0 ä Ø ´ R j i = O ( τ + h 2 ) NA˙diffc3.tex – © § ê ) – © R – 21/4/2003 – 17:00 – p.5/29