课程设计报告-四阶runge-kutta方法

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1、《计算机数值方法》课程设计报告题目四阶Runge-Kutta方法学生姓名班级计科12学号成绩指导教师延安大学计算机学院2014年9月1日12目录一、摘要·······································································5二、问题重述··································································5三、方法原理及实现··························

2、································5四、计算公式或算法··························································5五、Matlab程序·······························································6六、测试数据及结果··························································6七、12结果分析·····

3、····························································10八、方法改进··································································10九、心得体会··································································10十、参考文献········································

4、··························10一、摘要本课程设计主要内容是用四阶Runge-Kutta方法解决常微分方程组初值问题的数值解法，12首先分析题目内容和要求，然后使用Matlab编写程序计算结果并绘图，最后对计算结果进行分析并得出结论。一、问题描述在计算机上实现用四阶Runge-Kutta求一阶常微分方程初值问题的数值解，并利用最后绘制的图形直观分析近似解与准确解之间的比较。三、方法原理及实现龙格-库塔(Runge-Kutta)方法是一种在工程上应用广泛的高精度单步算法。由于此算法精度

5、高，采取措施对误差进行抑制，所以其实现原理也较复杂。该算法是构建在数学支持的基础之上的。龙格库塔方法的理论基础来源于泰勒公式和使用斜率近似表达微分，它在积分区间多预计算出几个点的斜率，然后进行加权平均，用做下一点的依据，从而构造出了精度更高的数值积分计算方法。如果预先求两个点的斜率就是二阶龙格库塔法，如果预先取四个点就是四阶龙格库塔法。经典的方法是一个四阶的方法，它的计算公式是：四、计算公式或算法1．输入（编写或调用计算的函数文件），2．3．For12End4．输出五、Matlab程序x=[a:h:b];y

6、(1)=y1;n=(b-a)/h+1;fori=2:nfk1=f(x(i-1),y(i-1));fk2=f(x(i-1)+h/2,y(i-1)+fk1*h/2);fk3=f(x(i-1)+h/2,y(i-1)+fk2*h/2);fk4=f(x(i-1)+h,y(i-1)+fk3*h);y(i)=y(i-1)+h*(fk1+2*fk2+2*fk3+fk4)/6;endy六、测试数据及结果用调试好的程序解决如下问题：应用经典的四阶Runge-Kutta方法解初值问题取（1）步骤一：编写函数具体程序.121.求解

7、解析解程序：dsolve('Dy=(y^2+y)/t','y(1)=-2','t')结果：2.综合编写程序如下：a=1;b=3;h=0.5;y(1)=-2;x(1)=a;n=(b-a)/h+1;yy(1)=-2;fori=2:nk1=(y(i-1)^2+y(i-1))/x(i-1);k2=((y(i-1)+h*k1/2)^2+(y(i-1)+h*k1/2))/(x(i-1)+h/2);k3=((y(i-1)+h*k2/2)^2+(y(i-1)+h*k2/2))/(x(i-1)+h/2);k4=((y(i-1

8、)+h*k3)^2+(y(i-1)+h*k3))/(x(i-1)+h);y(i)=y(i-1)+h*(k1+2*k2+2*k3+k4)/6;%四阶Runge-Kutta公式解x(i)=x(i-1)+h;%有解区间的值yy(i)=-x(i)/(x(i)-1/2);%解析解s(i)=abs(y(i)-yy(i));%误差项end[x'y'yy's']（2）步骤二：执行上述Runge-Kutta算法，计算结果为1.00