于有限差分法的二维边值问题的数值分析

ureРplot((0:gridsize(i):a),v2(nodey/2,:),'k')Рxlabel('长度(m)')Рylabel('电压(V)')Рlegend(['网格尺寸=',num2str(gridsize(i))])Рtitle('沿一半宽度处电压随长度的变化') РfigureРcontour((0:gridsize(i):a),(0:gridsize(i):b),v2,30)Рcaxis([0,100])Рcolorbar('vert')%horizРxlabel('长度(m)')Рylabel('宽度(m)')Рtitle(['超松弛迭代;精度=',num2str(10^-10),'网格尺寸=',num2str(gridsize(i))])РfigureРimagesc((0:gridsize(i):a),(0:gridsize(i):b),v2)Рset(gca,'ydir','normal')Рcolorbar('vert')%horizРcaxis([0,100])Рxlabel('长度(m)')Рylabel('宽度(m)')Рtitle(['超松弛迭代;精度=',num2str(10^-10),'网格尺寸=',num2str(gridsize(i))])РkРendРendР实验结果:Р网格尺寸=0.1:Рalpha =Р 1.9391 1.8391 1.9891Рk =Р 491Рk =Р 1941Рk =Р 2503Р网格尺寸=1:Рalpha =Р 1.5279 1.4279 1.5779Рk =Р 51Рk =Р 90Рk = Р 58Р可以看出超松弛迭代法迭代次数比简单迭代法迭代次数少,当松弛因子等于最佳松弛因子时,迭代次数最少。Р最佳松弛因子可用下式计算:Рalphaop=2/(1+sin(pi/(a/gridsize(i))))