我正在使用RK4解决一个ODE系统。我生成了一条直线图，这似乎是因为k3_1的上限为-3.1445e + 24。我不明白为什么会受到限制。
function RK4system_MNModel()
parsec = 3.08*10^18;
r_1 = 8.5*1000.0*parsec; % in cm
z_1 = 0.0; % in cm also
theta_1 = 0.0;
grav = 6.6720*10^-8;
amsun = 1.989*10^33; % in grams
amg = 1.5d11*amsun; % in grams
gm = grav*amg; % constant
q = 0.9; % axial ratio
u_1 = 130.0; % in cm/sec
w_1 = 95*10^4.0; % in cm/sec
v = 180*10^4.0; % in cm/sec
vcirc = sqrt(gm/r_1); % circular speed (constant)
nsteps = 50000;
deltat = 5.0*10^11; % in seconds
angmom = r_1*v; % these are the same
angmom2 = angmom^2.0;
e = -gm/r_1+u_1*u_1/2.0+angmom2/(2.0*r_1*r_1);
time=0.0;
for i=1:nsteps
k3_1 = deltat*u_1 %%%%% THIS LINE
k4_1 = deltat*(-gm*r_1/((r_1^2.0+(1+sqrt(1+z_1^2.0))^2.0)^1.5) + angmom2/(r_1^3.0)); % u'=-dphi_dr+lz^2/(r^3.0) with lz=vi*ri this gives deltau
k5_1 = deltat*(angmom/(r_1^2.0)); % theta'=lz/r^2 this gives deltatheta
k6_1 = deltat*w_1;
k7_1 = deltat*(-gm*z_1*(1+sqrt(1+z_1^2.0))/(sqrt(1+z_1^2.0)*(r_1^2.0+(1+sqrt(1+z_1^2.0))^2.0)^1.5));
r_2 = r_1+k3_1/2.0;
u_2 = u_1+k4_1/2.0;
theta_2 = theta_1+k5_1/2.0;
z_2 = z_1 + k6_1/2.0;
w_2 = w_1 + k7_1/2.0;
k3_2 = deltat*u_2;
k4_2 = deltat*(-gm*r_2/((r_2^2.0+(1+sqrt(1+z_2^2.0))^2.0)^1.5)+angmom2/(r_2^3.0));
k5_2 = deltat*(angmom/(r_2^2.0)); % theta'=lz/r^2 =====> deltatheta
k6_2 = deltat*w_2;
k7_2 = deltat*(-gm*z_2*(1+sqrt(1+z_2^2.0))/(sqrt(1+z_2^2.0)*(r_2^2.0+(1+sqrt(1+z_2^2.0))^2.0)^1.5));
r_3 = r_1+k3_2/2.0;
u_3 = u_1+k4_2/2.0;
theta_3 = theta_1+k5_2/2.0;
z_3 = z_1 + k6_2/2.0;
w_3 = w_1 + k7_2/2.0;
k3_3 = deltat*u_3; % r'=u
k4_3 = deltat*(-gm*r_3/((r_3^2.0+(1+sqrt(1+z_3^2.0))^2.0)^1.5)+angmom2/(r_3^3.0));% u'=-dphi_dr+lz^2/(r^3.0)
k5_3 = deltat*(angmom/(r_3^2.0)); % theta'=lz/r^2
k6_3 = deltat*w_3;
k7_3 = deltat*(-gm*z_3*(1+sqrt(1+z_3^2.0))/(sqrt(1+z_3^2.0)*(r_3^2.0+(1+sqrt(1+z_3^2.0))^2.0)^1.5));
r_4 = r_1+k3_2;
u_4 = u_1+k4_2;
theta_4 = theta_1+k5_2;
z_4 = z_1 + k6_2;
w_4 = w_1 + k7_2;
k3_4 = deltat*u_4; % r'=u
k4_4 = deltat*(-gm*r_4/((r_4^2.0+(1+sqrt(1+z_4^2.0))^2.0)^1.5)+angmom2/(r_4^3.0)); % u'=-dphi_dr+lz^2/(r^3.0)
k5_4 = deltat*(angmom/(r_4^2.0)); % theta'=lz/r^2
k6_4 = deltat*w_4;
k7_4 = deltat*(-gm*z_4*(1+sqrt(1+z_4^2.0))/(sqrt(1+z_4^2.0)*(r_4^2.0+(1+sqrt(1+z_4^2.0))^2.0)^1.5));
r_1 = r_1+(k3_1+2.0*k3_2+2.0*k3_3+k3_4)/6.0; % New value of R for next step
u_1 = u_1+(k4_1+2.0*k4_2+2.0*k4_3+k4_4)/6.0; % New value of U for next step
theta_1 = theta_1+(k5_1+2.0*k5_2+2.0*k5_3+k5_4)/6.0; % New value of theta
z_1 = z_1+(k6_1+2.0*k6_2+2.0*k6_3+k6_4)/6.0;
w_1 = w_1+(k7_1+2.0*k7_2+2.0*k7_3+k7_4)/6.0;
e = -gm/r_1+u_1*u_1/2.0+angmom2/(2.0*r_1*r_1); % energy
ecc = (1.0+(2.0*e*angmom2)/(gm^2.0))^0.5; % eccentricity
x(i) = r_1*cos(theta_1)/(1000.0*parsec); % X for plotting orbit
y(i) = r_1*sin(theta_1)/(1000.0*parsec); % Y for plotting orbit
time = time+deltat;
r(i) = r_1;
z(i) = z_1;
time1(i)= time;
end
请注意，异常发生在指示的行上。
答案 0 :(得分：0)
k3_1没有上限，u_1的计算结果返回-3.1445e+24 / deltat deltat的值u_1是不变的。
u_1 = u_1+(k4_1+2.0*k4_2+2.0*k4_3+k4_4)/6.0;
在行中计算：
u_1(1) = 6.500e+13 % Hard coded before the loop
u_1(2) = -1.432966614767040e+04 % Calculated using the equation above
u_1(3) = -2.878934017859105e+04 % Calculated using the equation above
u_1(4) = -4.324903004768405e+04
第一次迭代后，返回：
u_1(n+1) = u_1(n) + du根据等式du，k4_1 = deltat*(-gm*r_1/((r_1^2.0+(1+sqrt(1+z_1^2.0))^2.0)^1.5)+angmom2/(r_1^3.0)); % u'=-dphi_dr+lz^2/(r^3.0) with lz=vi*ri this gives delta
k4_2 = deltat*(-gm*r_2/((r_2^2.0+(1+sqrt(1+z_2^2.0))^2.0)^1.5)+angmom2/(r_2^3.0));
k4_3 = deltat*(-gm*r_3/((r_3^2.0+(1+sqrt(1+z_3^2.0))^2.0)^1.5)+angmom2/(r_3^3.0));% u'=-dphi_dr+lz^2/(r^3.0)
k4_4 = deltat*(-gm*r_4/((r_4^2.0+(1+sqrt(1+z_4^2.0))^2.0)^1.5)+angmom2/(r_4^3.0)); % u'=-dphi_dr+lz^2/(r^3.0)
代表相对较小的差异。两个第一个值之间的差异是非常大，所以我假设这个计算不正确。
如果您发现该计算是正确的，那么您的错误就在以下一行中：
$arrayData = DB::select(DB::raw("SELECT RIGHT('abc3',1) ")):