MATLAB Euler方法

Question

我试图用Euler方法逼近Friedmann方程，但我的代码似乎不起作用。我得到ap和slope的值。

    N=12;
    rho0=1;
    h=0.001;
    j=N*h;
    t=0:h:j;
    a0=1;
    G=6.67*10^-11;

    rhop=-3*(sqrt((8*pi*G)/3));
    rho=zeros(size(N+1));
    rho(1)=rho0;
    a=zeros(size(N+1));
    ap=zeros(size(N+1));
    slope=zeros(size(N+1));
    a(1)=1;


    for i= 1:(N);
     slope(i)=rhop*((rho(i))^(3/2));
     ap(i)=sqrt((8*pi*G)/3)*((rho(i))^(1/2))*a(i);
     rho(i+1)=rho(i)+slope(i)*h;
     a(i+1)=a(i)+h*ap(i);

   end

   plot(t,a)
   plot(t,rho)

我做错了什么？

Answer 1

你检查了数学吗？看看你的代码，你的循环似乎编码不正确（免责声明：我不是天体物理学家）。

我还发现这个code snippet可以满足您的需求，它似乎比您正在做的更复杂，这让我相信您可能会过度简化。

function friedmann

%%%% Actual Hubble value
H0=71000/(3*10^(22))*(3600*24*365*10^(9));

%%%% Time arrays
t_begin(1)=13.7;
t_begin(2)=13.7;
t_final(1)=0;
t_final(2)=40;

%%%% Solving in two parts : backwards and forwards
for i=1:2

%%%% Initial Omega parameters    
Omega1_m=0.3;
Omega1_red=0;
Omega1_k=1-Omega1_m-Omega1_red;
Omega2_m=0.3;
Omega2_red=0.7;
Omega2_k=1-Omega2_m-Omega2_red;
Omega3_m=5;
Omega3_red=0;
Omega3_k=1-Omega3_m-Omega3_red;
Omega4_m=1;
Omega4_red=0;
Omega4_k=1-Omega4_m-Omega4_red;

%%%% Initial scale factor and derivated scale factor
a1_0=1;
a1_prim_0=H0*(Omega1_m/a1_0+Omega1_k)^(1/2);
init1=[ a1_0 ; a1_prim_0 ];
a2_0=1;
a2_prim_0=H0*(Omega2_m/a2_0+Omega2_k+Omega2_red*a2_0)^(1/2);
init2=[ a2_0 ; a2_prim_0 ];
a3_0=1;
a3_prim_0=H0*(Omega3_m/a3_0+Omega3_k)^(1/2);
init3=[ a3_0 ; a3_prim_0 ];
a4_0=1;
a4_prim_0=H0*(Omega4_m/a4_0+Omega4_k)^(1/2);
init4=[ a4_0 ; a4_prim_0 ];

%%%% options for solver
options = odeset('Events',@events,'RelTol',10^(-10),'AbsTol',10^(-10));

%%%% Differential systems solving
[t1,y1]=ode45(@(t1,y1) syseq(t1,y1,Omega1_m,Omega1_red,Omega1_k),[t_begin(i) t_final(i)],init1,options);
[t2,y2]=ode45(@(t2,y2) syseq(t2,y2,Omega2_m,Omega2_red,Omega2_k),[t_begin(i) t_final(i)],init2,options);
[t3,y3]=ode45(@(t3,y3) syseq(t3,y3,Omega3_m,Omega3_red,Omega3_k),[t_begin(i) t_final(i)],init3,options);
[t4,y4]=ode45(@(t4,y4) syseq(t4,y4,Omega4_m,Omega4_red,Omega4_k),[t_begin(i) t_final(i)],init4,options);

figure(1);
hold on;
%%%% Plot solutions
plot(t1,y1(:,1),'b',t2,y2(:,1),'r',t3,y3(:,1),'k',t4,y4(:,1),'g'); 
box on;
grid on;
xlabel('Cosmic Time in Gyr');
ylabel('Relative size of the universe');
ylim([ 0 4]);
%%%% Set label for each curve 
posX = 5;
posX1 = posX;
posX2 = posX;
posX3 = posX;
posX4 = posX;
posY1 = 3.4;
posY2 = 3.0;
posY3 = 2.6;
posY4 = 2.2;
strMax = ['\Omega_{m} = 0.3 , \Omega_{\Lambda} = 0']; 
text(posX1, posY1, strMax, 'FontSize', 15, 'Color', 'blue', 'VerticalAlignment', 'top');
strMax = ['\Omega_{m} = 0.3 , \Omega_{\Lambda} = 0.7'];
text(posX2, posY2, strMax, 'FontSize', 15, 'Color', 'red', 'VerticalAlignment', 'top');
strMax = ['\Omega_{m} = 5 , \Omega_{\Lambda} = 0'];
text(posX3, posY3, strMax, 'FontSize', 15, 'Color', 'black', 'VerticalAlignment', 'top');
strMax = ['\Omega_{m} = 1 , \Omega_{\Lambda} = 0'];
text(posX4, posY4, strMax, 'FontSize', 15, 'Color', 'green', 'VerticalAlignment', 'top');
title('Relative size of the universe vs Cosmic Time');

end
end

%%%% Differential system function
function dydt = syseq(t,y,Omega_m,Omega_red,Omega_k) 

H0=71000/(3*10^(22))*(3600*24*365*10^(9));

dydt=[ y(2) ;                        
 (-(H0^(2)/2)*(Omega_m/y(1)^(3)-2*Omega_red))*Omega_m*(1/(H0^(2))*y(2)^2-Omega_k-Omega_red*y(1)^(2))^(-1);                
       ];
end

%%%% Integration stopping function
function [value,isterminal,direction] = events(t,y)
% Locate the time when height passes through zero in a 
% decreasing direction and stop integration.
value = y(1);     % Detect height = 0
isterminal = 1;   % Stop the integration
direction = -1;   % Negative direction only
end