上机实验2：五点差分格式法偏微分方程（Matlab ）实验报告——五点差分格式法 一、 实验题目设G 是形如下图的十字形域，由五个相等的单位正方形组成，用五点差分格式求下列边值问题的数值解：22221,u uG x y ∂∂+=-∂∂∂于u=0,于G二、 实验原理取定沿X 轴和Y 轴方向的步长1h 和2h ，()122212h h h =+,作两族与坐标轴平行的直线：x=i 1h ,y=j 2h ,,0,1,2,i j =±±若（,i j x y ）为正则内点，沿x,y 方向分别用二阶中心差商代替xx yy u u 和则得1,1,,1,1221222[]i j ij i ji j ij i j ij u u u u u u f h h +-+--+-+-+=特别取正方形网格：12h h h ==，则原差分方程可简化为21,,11,,11()44ij i j i j i j i j ij h u u u u u f --++-+++=三、 实验程序1）function uxy = EllIni2Uxl(x,y)format long ;uxy = 0;2）function uxy = EllIni2Uxr(x,y)format long;uxy = y*(2-y);3）function uxy = EllIni2Uyl(x,y)format long;uxy = 0;4）function uxy = EllIni2Uyr(x,y)format long;if x < 1uxy = x;elseuxy = 2 - x;end5）function u = peEllip5(nx,minx,maxx,ny,miny,maxy) format long;hx = (maxx-minx)/(nx-1);hy = (maxy-miny)/(ny-1);u0 = zeros(nx,ny);for j=1:nyu0(j,1) = EllIni2Uxl(minx,miny+(j-1)*hy);u0(j,nx) = EllIni2Uxr(maxx,miny+(j-1)*hy);endfor j=1:nxu0(1,j) = EllIni2Uyl(minx+(j-1)*hx,miny);u0(ny,j) = EllIni2Uyr(minx+(j-1)*hx,maxy);endA = -4*eye((nx-2)*(ny-2),(nx-2)*(ny-2));b = ones((nx-2)*(ny-2),1).*(-1);for i=1:(nx-2)*(ny-2)if mod(i,nx-2) == 1if i==1A(1,2) = 1;A(1,nx-1) = 1;b(1) = - u0(1,2) - u0(2,1);elseif i == (ny-3)*(nx-2)+1A(i,i+1) = 1;A(i,i-nx+2) = 1;b(i) = - u0(ny-1,1) - u0(ny,2);elseA(i,i+1) = 1;A(i,i-nx+2) = 1;A(i,i+nx-2) = 1;b(i) = - u0(floor(i/(nx-2))+2,1);endendelseif mod(i,nx-2) == 0if i == nx-2A(i,i-1) = 1;A(i,i+nx-2) = 1;b(i) = - u0(1,nx-1) - u0(2,nx);elseif i == (ny-2)*(nx-2)A(i,i-1) = 1;A(i,i-nx+2) = 1;b(i) = - u0(ny-1,nx) - u0(ny,nx-1);elseA(i,i-1) = 1;A(i,i-nx+2) = 1;A(i,i+nx-2) = 1;b(i) = - u0(floor(i/(nx-2))+1,nx);endendelseif i>1 && i< nx-2A(i,i-1) = 1;A(i,i+nx-2) = 1;A(i,i+1) = 1;b(i) = - u0(1,i+1);elseif i > (ny-3)*(nx-2) && i < (ny-2)*(nx-2) A(i,i-1) = 1;A(i,i-nx+2) = 1;A(i,i+1) = 1;b(i) = - u0(ny,mod(i,(nx-2))+1);elseA(i,i-1) = 1;A(i,i+1) = 1;A(i,i+nx-2) = 1;A(i,i-nx+2) = 1;endendendendendul = A\b;for i=1:(ny-2)for j=1:(nx-2)u(i,j) = ul((i-1)*(nx-2)+j);endendformat short;四、实验结果>> u=peEllip5(25,0,3,25,0,3)u =Columns 1 through 61.14482.28963.26714.0750 4.73725.27632.2896 4.7466 6.7036 8.2957 9.5977 10.65833.2670 6.7035 9.5050 11.8066 13.6994 15.24744.0749 8.2955 11.8064 14.7262 17.1460 19.13514.7370 9.5973 13.6990 17.1457 20.0233 22.40115.2759 10.6576 15.2465 19.1343 22.4005 25.11205.7089 11.5108 16.4951 20.7444 24.3324 27.32286.0490 12.1816 17.4787 22.0159 25.8619 29.07756.3055 12.6880 18.2221 22.9785 27.0220 30.41066.4851 13.0426 18.7432 23.6540 27.8369 31.34816.5922 13.2542 19.0543 24.0573 28.3237 31.90826.6294 13.3278 19.1623 24.1973 28.4924 32.10196.5977 13.2651 19.0700 24.0771 28.3467 31.93326.4964 13.0648 18.7754 23.6946 27.8841 31.39956.3231 12.7225 18.2720 23.0417 27.0957 30.49116.0736 12.2298 17.5486 22.1047 25.9657 29.19135.7415 11.5747 16.5879 20.8626 24.4710 27.47565.3175 10.7395 15.3659 19.2868 22.5803 25.31134.7891 9.7000 13.8492 17.3385 20.2519 22.65624.1389 8.4221 11.9923 14.9662 17.4326 19.45773.3445 6.8572 9.7318 12.1014 14.0546 15.65152.3820 4.9303 6.9762 8.6530 10.0330 11.16071.25322.50583.58964.50165.2637 5.8950 Columns 7 through 125.70956.0499 6.3068 6.4869 6.5946 6.6327 11.5119 12.1833 12.6905 13.0461 13.2590 13.3341 16.4966 17.4810 18.2256 18.7482 19.0610 19.1713 20.7461 22.0186 22.9826 23.6600 24.0656 24.2085 24.3339 25.8647 27.0264 27.8435 28.3330 28.5051 27.3238 29.0798 30.4148 31.3547 31.9178 32.1153 29.7693 31.7160 33.1982 34.2426 34.8684 35.0871 31.7146 33.8168 35.4193 36.5492 37.2260 37.461533.1947 35.4172 37.1129 38.3090 39.0249 39.272834.2366 36.5445 38.3062 39.5489 40.2920 40.547534.8591 37.2178 39.0186 40.2884 41.0465 41.305135.0737 37.4491 39.2621 40.5395 41.3007 41.5577 34.8848 37.2427 39.0411 40.3067 41.0590 41.3099 34.2896 36.5959 38.3529 39.5873 40.3187 40.5588 33.2780 35.4985 37.1872 38.3711 39.0695 39.2944 31.8328 33.9329 35.5265 36.6403 37.2938 37.4988 29.9289 31.8739 33.3454 34.3698 34.9665 35.1472 27.5331 29.2885 30.6112 31.5272 32.0552 32.2070 24.6039 26.1358 27.2838 28.0725 28.5202 28.6384 21.0904 22.3671 23.3156 23.9590 24.3148 24.3945 16.9329 17.9266 18.6526 19.1329 19.3854 19.4216 12.0633 12.7537 13.2351 13.5347 13.6723 13.6596 6.4058 6.7898 6.9996 7.0985 7.1096 7.0427Columns 13 through 186.6020 6.5019 6.3302 6.0826 5.7530 5.3325 13.2733 13.0755 12.7361 12.2473 11.5970 10.7682 19.0818 18.7906 18.2915 17.5734 16.6195 15.4061 24.0918 23.7135 23.0660 22.1354 20.9014 19.3358 28.3635 27.9059 27.1234 26.0007 24.5150 22.6350 31.9512 31.4230 30.5211 29.2291 27.5227 25.3694 34.9031 34.3137 33.3090 31.8719 29.9775 27.5926 37.2603 36.6196 35.5294 33.9721 31.9226 29.348039.0571 38.3751 37.2167 35.5644 33.3930 30.669840.3201 39.6070 38.3981 36.6757 34.4151 31.583741.0688 40.3347 39.0928 37.3254 35.0078 32.1078 41.3152 40.5704 39.3129 37.5254 35.1829 32.2534 41.0640 40.3185 39.0632 37.2803 34.9450 32.0249 40.3124 39.5766 38.3409 36.5877 34.2917 31.4203 39.0503 38.3344 37.1363 35.4380 33.2139 30.4309 37.2600 36.5745 35.4318 33.8140 31.6951 29.0414 34.9165 34.2716 33.2024 31.6909 29.7112 27.228831.9871 31.3931 30.4153 29.0362 27.2300 24.962328.4318 27.8985 27.0295 25.8084 24.2105 22.201824.2032 23.7396 22.9959 21.9575 20.6016 18.896619.2468 18.8607 18.2571 17.4239 16.3420 14.983813.5018 13.1993 12.7480 12.1390 11.3586 10.38596.9016 6.6868 6.3964 6.0257 5.5674 5.0103 Columns 19 through 234.8087 4.1651 3.3801 2.4318 1.32549.7373 8.4714 6.9236 5.0218 2.635213.9009 12.0597 9.8209 7.0967 3.756117.4005 15.0454 12.2038 8.7880 4.683220.3200 17.5175 14.1611 10.1681 5.438822.7271 19.5436 15.7548 11.2845 6.044524.6754 21.1749 17.0301 12.1707 6.517326.2071 22.4504 18.0201 12.8507 6.869727.3546 23.3995 18.7492 13.3421 7.110928.1421 24.0436 19.2352 13.6577 7.247528.5864 24.3978 19.4901 13.8060 7.284028.6980 24.4711 19.5214 13.7923 7.223028.4809 24.2673 19.3322 13.6187 7.065727.9336 23.7849 18.9214 13.2846 6.811827.0481 23.0174 18.2838 12.7864 6.459525.8106 21.9528 17.4100 12.1176 6.005524.2003 20.5732 16.2859 11.2686 5.444922.1885 18.8537 14.8917 10.2260 4.771119.7377 16.7613 13.2014 8.9724 3.976216.7993 14.2523 11.1802 7.4861 3.051813.3107 11.2685 8.7810 5.7401 1.99499.1911 7.7299 5.9352 3.6985 0.82824.3378 3.5247 2.5313 1.2905 -0.3180>>。