龙格库塔法
经典四阶龙格库塔法
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,经常被称为“RK4”或者就是“龙格库塔法”。
令初值问题表述如下。
对于该问题的RK4由如下方程给出:
其中:
RK4法是四阶方法,也就是说每步的误差是h5阶,而总积累误差为h4阶。
RK4计算程序如下:
//visualsan@yahoo.cn
#include <iostream>
using namespace std;
#include <vector>
#include <math.h>
using namespace std;
class RK
{
public:
class DiffFunc
{
public:
double operator()(double x,double y)
{
// y'=cos(x)
return cos(x);
// y'=x*y-1
// return x*y-1;
}
} m_df;
RK(double xend,double x0,double y0,double h=1e-3)
{
m_max_x = xend;
m_x0 = x0;
m_y0 = y0;
m_h = h;
m_half_h=h/2.0;
}
void Solve()
{
double yn=m_y0,xstart=m_x0;
while( xstart<m_max_x)
{
double y=yn+K(xstart,yn)*m_h;//y(n+1)=y(n)+h*y'
m_x.push_back(xstart);
m_y.push_back(yn);
cout<<xstart<<""<<yn<<endl;
yn=y;
xstart+=m_h;
}
}
//求解后的点
std::vector<double> m_x,m_y;
//步长h
double m_h;
//初始点x0,y0
double m_x0,m_y0;
//x范围
double m_max_x;
private:
double m_half_h;
double m_ptx,m_pty;
double K1(double x,double y)
{
double v=m_df(x,y);
m_ptx=x;
m_pty=y;
return v;
}
double K2(double _k1)
{
double v=m_df(m_ptx+m_half_h,m_pty+m_half_h*_k1);
return v;
}
double K3(double _k2)
{
double v=m_df(m_ptx+m_half_h,m_pty+m_half_h*_k2);
return v;
}
double K4(double _k3)
{
double v=m_df(m_ptx+m_h,m_pty+m_h*_k3);
return v;
}
double K(double x,double y)
{
double _k1=K1(x,y),_k2=K2(_k1),_k3=K3(_k2),_k4=K4(_k3);
return (_k1+2.0*_k2+2.0*_k3+_k4)/6.0;
}
};
main()
{
RK s1(1,0,0);
s1.Solve();
}
计算实例:
y'=cos(x) y(0)=0, -> solution y=sin(x)
y'=x*cos(x) y(0)=0
y'=1.0/sqrt(x*x+y*y),y(0)=0.1
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----------------------------------------------------visualsan@yahoo.cn--------
posted on 2012-02-09 21:46 DoJustForFun 阅读(7017) 评论(0) 编辑 收藏 举报