时间步长与空间长对稳定性影响.doc

对下面的一阶对流扩散方程 (1.1) 对求解区间 Matlab程序 function u = yingfeng(a,dt,n,minx,maxx,M) %方程中的常数：a %时间步长：dt %空间节点个数：n %求解区间的左端：minx%求解区间的右端：naxx %时间步的个数：M %求解区间上的数值解：u format long; h = (maxx-minx)/(n-1); if a0 for j=1:(n+M) u0(j) = IniU(minx+(j-M-1)*h); end else for j=1:(n+M) u0(j) = IniU(minx+(j-1)*h); end end u1 = u0; for k=1:M if a0 for i=(k+1):n+M u1(i) = -dt*a*(u0(i)-u0(i-1))/h+u0(i); end else for i=1:n+M-k u1(i) = -dt*a*(u0(i+1)-u0(i))/h+u0(i); end end u0 = u1; end if a0 u = u1((M+1):M+n); else u = u1(1:n); end format long; function ux=IniU(x) format long; if x=0 if x=-0.1 ux=10*(x+0.1); else ux=0; end else if x=0.1 ux=-10*(x-0.1); else ux=0; end end end 在保持其他量不变的情况下，通过改变dt的值来改变来分析时间步长dt和空间步长h的取值关系 事实上令则有 ; u=yingfeng(1,0.00001,101,0,1,100); plot(u) u=yingfeng(1,0.001,101,0,1,100); plot(u) u=yingfeng(1,0.003,101,0,1,100); plot(u) u=yingfeng(1,0.005,101,0,1,100); plot(u) u=yingfeng(1,0.006,101,0,1,100); plot(u) u=yingfeng(1,0.007,101,0,1,100); plot(u) u=yingfeng(1,0.008,101,0,1,100); plot(u) u=yingfeng(1,0.009,101,0,1,100); plot(u) u=yingfeng(1,0.01,101,0,1,100); plot(u) u=yingfeng(1,0.011,101,0,1,100); plot(u) u=yingfeng(1,0.012,101,0,1,100); plot(u) u=yingfeng(1,0.013,101,0,1,100); plot(u) u=yingfeng(1,0.014,101,0,1,100); plot(u) u=yingfeng(1,0.017,101,0,1,100); plot(u) u=yingfeng(1,0.020,101,0,1,100); plot(u) u=yingfeng(1,1,101,0,1,100); plot(u) 由以上得到的图像我们可以知道当，既时波是稳定的，且随着时间的增长向右传播；时，波开始不稳定，且向左不断的扩散。 特别的当方程的解不是平滑的曲线