龙格库塔法推导

数值分析中，龙格－库塔法（Runge-Kutta）是用于模拟常微分方程的解的重要的一类隐式或显式迭代法。

这些技术由数学家卡尔·龙格和马丁·威尔海姆·库塔于1900年左右发明。

经典四阶龙格库塔法龙格库塔法的家族中的一个成员如此常用，以至于经常被称为“RK4”或者就是“龙格库塔法”。

令初值问题表述如下。

则，对于该问题的RK4由如下方程给出：其中这样，下一个值(y n+1)由现在的值(y n)加上时间间隔(h)和一个估算的斜率的乘积决定。

该斜率是以下斜率的加权平均：∙k1是时间段开始时的斜率；∙k2是时间段中点的斜率，通过欧拉法采用斜率k1来决定y在点t n + h/2的值；∙k3也是中点的斜率，但是这次采用斜率k2决定y值；∙k4是时间段终点的斜率，其y值用k3决定。

当四个斜率取平均时，中点的斜率有更大的权值：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。

注意上述公式对于标量或者向量函数(y可以是向量)都适用。

显式龙格库塔法显示龙格－库塔法是上述RK4法的一个推广。

它由下式给出其中如果要求方法有精度p则还有相应的条件，也就是要求舍入误差为O(h p+1)时的条件。

这些可以从舍入误差本身的定义中导出。

例如，一个2阶精度的2段方法要求b1 + b2 = 1, b2c2 = 1/2, 以及b2a21 = 1/2。

在Matlab下输入：edit，然后将下面两行百分号之间的内容，复制进去，保存%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%function dxdt=ode_Miss_ghost(t,x)%分别用x(1),x(2),x(3),x(4)代替N1,P1,N2,P2N1=x(1);P1=x(2);N2=x(3);P2=x(4);K=2;tau_c=3e-9;tan_p=6e-12;beta =5e-5;delta=0.692;eta =0.0001;fm =8e6;Ith =26e-3;Ib =1.5*Ith;Im =0.3*Ith;I1=Ib+Im*sin(2*pi*fm*t)+K*P2;I2=Ib+Im*sin(2*pi*fm*t)+K*P1;dxdt=[(I1/Ith-N1-(N1-delta)/(1-delta)*P1)/tau_e;((N1-delta)/(1-delta)*(1-eta*P1)*P1-P1+beta*N1)/tau_p;(I2/Ith-N2-(N2-delta)/(1-delta)*P2)/tau_e;((N2-delta)/(1-delta)*(1-eta*P2)*P2-P2+beta*N2)/tau_p;]; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%在Matlab下面输入：t_start=0;t_end=2e-9;y0=[1e-3;1e-4;0;0]; %初值[x,y]=ode15s('ode_Miss_ghost',[0,t_end],y0);plot(x,y);legend('N1','P1','N2','P2');xlabel('x');Mathlab定步长龙格－库塔法[345阶]、自适应步长rkf45经常看到很多朋友问定步长的龙格库塔法设置问题，下面吧定步长三阶、四阶、五阶龙格库塔程序贴出来，有需要的可以看看ODE3 三阶龙格－库塔法CODE:function Y = ode3(odefun,tspan,y0,varargin)%ODE3 Solve differential equations with a non-adaptive method of order 3.% Y = ODE3(ODEFUN,TSPAN,Y0) with TSPAN = [T1, T2, T3, ... TN] integrates% the system of differential equations y' = f(t,y) by stepping from T0 to% T1 to TN. Function ODEFUN(T,Y) must return f(t,y) in a column vector.% The vector Y0 is the initial conditions at T0. Each row in the solution% array Y corresponds to a time specified in TSPAN.%% Y = ODE3(ODEFUN,TSPAN,Y0,P1,P2...) passes the additional parameters% P1,P2... to the derivative function as ODEFUN(T,Y,P1,P2...).%% This is a non-adaptive solver. The step sequence is determined by TSPAN% but the derivative function ODEFUN is evaluated multiple times per step.% The solver implements the Bogacki-Shampine Runge-Kutta method of order 3.%% Example% tspan = 0:0.1:20;% y = ode3(@vdp1,tspan,[2 0]);% plot(tspan,y(:,1));% solves the system y' = vdp1(t,y) with a constant step size of 0.1, % and plots the first component of the solution.%if ~isnumeric(tspan)error('TSPAN should be a vector of integration steps.');endif ~isnumeric(y0)error('Y0 should be a vector of initial conditions.');endh = diff(tspan);if any(sign(h(1))*h <= 0)error('Entries of TSPAN are not in order.')endtryf0 = feval(odefun,tspan(1),y0,varargin{:});catchmsg = ['Unable to evaluate the ODEFUN at t0,y0. ',lasterr];error(msg);endy0 = y0(:); % Make a column vector.if ~isequal(size(y0),size(f0))error('Inconsistent sizes of Y0 and f(t0,y0).');endneq = length(y0);N = length(tspan);Y = zeros(neq,N);F = zeros(neq,3);Y(:,1) = y0;for i = 2:Nti = tspan(i-1);hi = h(i-1);yi = Y(:,i-1);F(:,1) = feval(odefun,ti,yi,varargin{:});F(:,2) = feval(odefun,ti+0.5*hi,yi+0.5*hi*F(:,1),varargin{:});F(:,3) = feval(odefun,ti+0.75*hi,yi+0.75*hi*F(:,2),varargin{:});Y(:,i) = yi + (hi/9)*(2*F(:,1) + 3*F(:,2) + 4*F(:,3));endY = Y.';ODE4 四阶龙格－库塔法CODE:function Y = ode4(odefun,tspan,y0,varargin)%ODE4 Solve differential equations with a non-adaptive method of order 4.% Y = ODE4(ODEFUN,TSPAN,Y0) with TSPAN = [T1, T2, T3, ... TN] integrates % the system of differential equations y' = f(t,y) by stepping from T0 to% T1 to TN. Function ODEFUN(T,Y) must return f(t,y) in a column vector.% The vector Y0 is the initial conditions at T0. Each row in the solution% array Y corresponds to a time specified in TSPAN.%% Y = ODE4(ODEFUN,TSPAN,Y0,P1,P2...) passes the additional parameters % P1,P2... to the derivative function as ODEFUN(T,Y,P1,P2...).%% This is a non-adaptive solver. The step sequence is determined by TSPAN % but the derivative function ODEFUN is evaluated multiple times per step.% The solver implements the classical Runge-Kutta method of order 4.%% Example% tspan = 0:0.1:20;% y = ode4(@vdp1,tspan,[2 0]);% plot(tspan,y(:,1));% solves the system y' = vdp1(t,y) with a constant step size of 0.1,% and plots the first component of the solution.%if ~isnumeric(tspan)error('TSPAN should be a vector of integration steps.');endif ~isnumeric(y0)error('Y0 should be a vector of initial conditions.');endh = diff(tspan);if any(sign(h(1))*h <= 0)error('Entries of TSPAN are not in order.')endtryf0 = feval(odefun,tspan(1),y0,varargin{:});catchmsg = ['Unable to evaluate the ODEFUN at t0,y0. ',lasterr]; error(msg);endy0 = y0(:); % Make a column vector.if ~isequal(size(y0),size(f0))error('Inconsistent sizes of Y0 and f(t0,y0).');endneq = length(y0);N = length(tspan);Y = zeros(neq,N);F = zeros(neq,4);Y(:,1) = y0;for i = 2:Nti = tspan(i-1);hi = h(i-1);yi = Y(:,i-1);F(:,1) = feval(odefun,ti,yi,varargin{:});F(:,2) = feval(odefun,ti+0.5*hi,yi+0.5*hi*F(:,1),varargin{:}); F(:,3) = feval(odefun,ti+0.5*hi,yi+0.5*hi*F(:,2),varargin{:}); F(:,4) = feval(odefun,tspan(i),yi+hi*F(:,3),varargin{:});Y(:,i) = yi + (hi/6)*(F(:,1) + 2*F(:,2) + 2*F(:,3) + F(:,4));endY = Y.';定步长RK4(自编):CODE:function Y=RungeKutta4(f,xn,y0)% xn=0:.1:1;% y0=1;% y_n=[];% f=@(X1,Y1) Y1-2*X1/Y1;y_n=[];h=diff(xn(1:2));for i=1:length(xn)-1K1=f(xn(i),y0);K2=f(xn(i)+h/2,y0+h*K1/2);K3=f(xn(i)+h/2,y0+h*K2/2);K4=f(xn(i)+h,y0+h*K3);y_n=[y_n;y0+h/6*(K1+2*K2+2*K3+K4)];y0=y_n(end);endY=y_n;ODE5 五阶龙格－库塔法CODE:function Y = ode5(odefun,tspan,y0,varargin)%ODE5 Solve differential equations with a non-adaptive method of order 5.% Y = ODE5(ODEFUN,TSPAN,Y0) with TSPAN = [T1, T2, T3, ... TN] integrates % the system of differential equations y' = f(t,y) by stepping from T0 to% T1 to TN. Function ODEFUN(T,Y) must return f(t,y) in a column vector.% The vector Y0 is the initial conditions at T0. Each row in the solution% array Y corresponds to a time specified in TSPAN.%% Y = ODE5(ODEFUN,TSPAN,Y0,P1,P2...) passes the additional parameters % P1,P2... to the derivative function as ODEFUN(T,Y,P1,P2...).%% This is a non-adaptive solver. The step sequence is determined by TSPAN % but the derivative function ODEFUN is evaluated multiple times per step.% The solver implements the Dormand-Prince method of order 5 in a general % framework of explicit Runge-Kutta methods.%% Example% tspan = 0:0.1:20;% y = ode5(@vdp1,tspan,[2 0]);% plot(tspan,y(:,1));% solves the system y' = vdp1(t,y) with a constant step size of 0.1,% and plots the first component of the solution.if ~isnumeric(tspan)error('TSPAN should be a vector of integration steps.');endif ~isnumeric(y0)error('Y0 should be a vector of initial conditions.');endh = diff(tspan);if any(sign(h(1))*h <= 0)error('Entries of TSPAN are not in order.')endtryf0 = feval(odefun,tspan(1),y0,varargin{:});catchmsg = ['Unable to evaluate the ODEFUN at t0,y0. ',lasterr];error(msg);endy0 = y0(:); % Make a column vector.if ~isequal(size(y0),size(f0))error('Inconsistent sizes of Y0 and f(t0,y0).');endneq = length(y0);N = length(tspan);Y = zeros(neq,N);% Method coefficients -- Butcher's tableau%% C | A% --+---% | BC = [1/5; 3/10; 4/5; 8/9; 1];A = [ 1/5, 0, 0, 0, 03/40, 9/40, 0, 0, 044/45 -56/15, 32/9, 0, 019372/6561, -25360/2187, 64448/6561, -212/729, 0 9017/3168, -355/33, 46732/5247, 49/176, -5103/18656];B = [35/384, 0, 500/1113, 125/192, -2187/6784, 11/84]; % More convenient storageA = A.';B = B(:);nstages = length(B);F = zeros(neq,nstages);Y(:,1) = y0;for i = 2:Nti = tspan(i-1);hi = h(i-1);yi = Y(:,i-1);% General explicit Runge-Kutta frameworkF(:,1) = feval(odefun,ti,yi,varargin{:});for stage = 2:nstageststage = ti + C(stage-1)*hi;ystage = yi + F(:,1:stage-1)*(hi*A(1:stage-1,stage-1));F(:,stage) = feval(odefun,tstage,ystage,varargin{:});endY(:,i) = yi + F*(hi*B);endY = Y.';-------------------------------------------------------------------------------------------------------------------自适应步长RKF45(相当于ode45)ODE45 是4阶方法提供候选解，5阶方法控制误差。

四阶龙格－库塔法求解常微分方程的初值问题1.算法原理对于一阶常微分方程组的初值问题⎪⎪⎪⎩⎪⎪⎪⎨⎧=⋯⋯==⋯⋯=⋯⋯⋯⋯=⋯⋯=0020********'212'2211'1)(,,)(,)())(,),(),(,()())(,),(),(,()())(,),(),(,()(n n n n n n n y x y y x y y x y x y x y x y x f x y x y x y x y x f x y x y x y x y x f x y ， 其中b x a ≤≤。

若记Tn Tn Tn y x f y x f y x f y x f y y y y x y x y x y y x y )),(,),,(),,((),(),,,())(),(),(()(2102010021⋯⋯=⋯⋯=⋯⋯=，，则可将微分方程组写成向量形式⎩⎨⎧=≤≤=0')()),(,()(y a y b x a x y x f x y微分方程组初值问题在形式上和单个微分方程处置问题完全相同，只是数量函数在此变成了向量函数。

因此建立的单个一阶微分方程初值问题的数值解法，可以完全平移到求解一阶微分方程组的初值问题中，只不过是将单个方程中的函数转向向量函数即可。

标准4阶R-K 法的向量形式如下：⎪⎪⎪⎪⎩⎪⎪⎪⎪⎨⎧++=++=++==++++=+),()21,2()21,2(),()22(61342312143211K y h x hf K K y h x hf K K y h x hf K y x hf K K K K K y y n n n n n n n n n n 其分量形式为n j K y K y K y h x hf K K y K y K y h x hf K K y K y K y h x hf K y y y x hf K K K K K y y n ni i i i j j n nii i i j j n nii i i j j ni i i i j j j j j j i j i j ,,2,1).,,,;(),2,2,2;2(),2,2,2;2(),,,,;(),22(6132321314222212131212111221143211,1,⋯⋯=⎪⎪⎪⎪⎩⎪⎪⎪⎪⎨⎧+⋯⋯+++=+⋯⋯+++=+⋯⋯+++=⋯⋯=++++=++，，2.程序框图3.源代码%该函数为四阶龙格-库塔法function [x,y]=method(df,xspan,y0,h)%df为常微分方程，xspan为取值区间，y0为初值向量，h为步长x=xspan(1):h:xspan(2);m=length(y0);n=length(x);y=zeros(m,n);y(:,1)=y0(:);for i=1:n-1k1=feval(df,x(i),y(:,i));k2=feval(df,x(i)+h/2,y(:,i)+h*k1/2);k3=feval(df,x(i)+h/2,y(:,i)+h*k2/2);k4=feval(df,x(i)+h,y(:,i)+h*k3);y(:,i+1)=y(:,i)+h*(k1+2*k2+2*k3+k4)/6;end%习题9.2clear;xspan=[0,1];%取值区间h=0.05;%步长y0=[-1,3,2];%初值df=@(x,y)[y(2);y(3);y(3)+y(2)-y(1)+2*x-3];[xt,y]=method(df,xspan,y0,h)syms t;yp=t*exp(t)+2*t-1;%微分方程的解析解yp1=xt.*exp(xt)+2*xt-1%计算区间内取值点上的精确解[xt',y(1,:)',yp1']%y(1,:)为数值解，yp1为精确解ezplot(yp,[0,1]);%画出解析解的图像hold on;plot(xt,y(1,:),'r');%画出数值解的图像4.计算结果。

龙格库塔方程1.介绍龙格-库塔（RK）方法是求解常微分方程（ODE）最常见的数值方法之一。

对于大多数非线性ODE问题，解析解并不存在或难以获得，因此需要使用数值方法来近似计算解。

RK方法通过迭代逼近ODE的解来得到精确性可控、收敛性好、易实现的数值解。

RK方法的基本思想是将ODE中的一阶导数转化为一组计算步骤，以得到相邻时间点之间的函数值和一阶导数的近似值，然后将其结合起来得到一个更精确的解。

2.RK方法的推导RK方法的推导过程是基于欧拉方法的，欧拉方法是RK方法的一阶近似。

假设有ODE$\frac{dx}{dt}=f(x,t)$，欧拉方法的迭代公式为$$x_{n+1}=x_n+hf(x_n,t_n)$$其中$h$是时间步长，$t_n=n*h$。

这个公式的意思是，从$x_n$开始，用一阶导数$f(x_n,t_n)$来列出切线，然后沿着切线向前移动$h$个单位，得到$x_{n+1}$。

更高阶的RK方法则基于更精细的近似。

例如，经典的四阶RK方法（RK4）迭代公式为：\begin{align*}k_1&=f(x_n,t_n)\\k_2&=f(x_n+\frac{h}{2}k_1,t_n+\frac{h}{2})\\k_3&=f(x_n+\frac{h}{2}k_2,t_n+\frac{h}{2})\\k_4&=f(x_n+h k_3,t_n+h)\\x_{n+1}&=x_n+\frac{h}{6}(k_1+2k_2+2k_3+k_4)\end{align*}其中，$k_1$是欧拉方法的一阶导数解，依次计算得到更高阶的导数近似值$k_2-k_4$。

3.RK方法的优势RK方法与其他数值方法相比具有众多优点。

首先，RK方法的精度可控。

通过增加迭代次数或者近似阶次，RK 方法可以获得任意高的精度。

这个特性非常适用于涉及长时间尺度和小尺度特征的问题，例如天气预报，需要同时精确地处理地球的自转和大气的扰动。