How will I implement Numerov method in a numerical example?

Question

I have been having difficulties implementing Numerov method of solving second order differential equation to solve numerical example. Please kindly suggest any text that discuss Numerov method and solve numerical examples.

Answer 1

Numerov method

The Numerov method applies to second order ODE of the type $y''(x)=f(x,y(x))$. It sets the second order difference quotient equal to the right side plus the first correction term expressed also as second order difference quotient, \begin{align} \frac{y_{n+1}-2y_n+y_{n-1}}{h^2} &=y''_k+\frac{h^2}{12}y^{(4)}_k = y''_k+\frac{1}{12}(y''_{k+1}-2y''_k+y''_{k-1})\\[.6em] &=\frac{1}{12}\Bigl(f(x_{n+1},y_{n+1})+10f(x_n,y_n)+f(x_{n-1},y_{n-1})\Bigr) \end{align} I will use a leapfrogged implementation as recommended in Hairer/Nørsett/Wanner: "Solving ODE I", \begin{align} y_{n+1}&=y_n+hv_{n+1/2}\\ v_{n+1/2}&=v_{n-1/2}+\frac{h}{12}\Bigl(f(x_{n+1},y_{n+1})+10f(x_n,y_n)+f(x_{n-1},y_{n-1})\Bigr) \end{align}

Predictor-corrector and first step

As the method is implicit, one needs to solve the implicit equation via some kind of iteration, fixed-point on the system as it is or some kind of Newton method for faster convergence, especially for stiff equations.

In the following proof-of-concept implementation, I use the most simple building blocks that satisfy the specifications.

• Use as predictor the Verlet-Størmer step $y_{n+1}^{[0]}=y_n+hv_{n-1/2}+h^2f(x_n,y_n)=2y_n-y_{n-1}+h^2f(x_n,y_n)$.
• Use fixed-point iteration as corrector.
• To compute the first step, use the 4th order Kutta method aka classical RK4. For optimal accuracy one has to apply this with sub-steps with a reduced step size.

The predictor has error $O(h^4)$, both to the exact value if $y_{n-1}$ and $y_n$ were values of an exact solution, as also to the solution of the implicit step equation. The fixed-point corrector after eliminating $v$ has contraction factor $\frac{Lh^2}{12}$, so that the error towards the exact solution of the implicit step after the first correction step should be $O(h^6)$ and after the second one $O(h^8)$. This should be sufficient to get the global error to $O(h^4)$, however with only one correction the error of the iteration is in the same magnitude as the method discretization error and thus influences the global error accumulation. 3 and more correction steps are needed if $L$ is large so that even higher order error terms can be of the size of the discretization error.

Test problem

Use the idea of manufactured solutions, $F[y]=F[p]$, where $F[y]$ is some expression with derivatives of $y$, to generate a test problem with known exact solution $p(x)=\sin(x)$. For instance, $$ F[y]=y''+4\sin(y) = F[\sin(x)] = -\sin(x)+4\sin(\sin(x)), ~~ y(0)=\sin(0)=0,~ y'(0)=\cos(0)=1 $$

The following graphs depict the (leading) error coefficient, that is, the difference of the numerical and exact solution divided by $h^4$.

That the graphs for the different step sizes are of the same scale and visually approaching some limit confirms that the method is of order $4$. Note that the first graph for 1 corrector step has a larger scale for the values than the next two, as predicted. One can see that there is some slight improvement from the second to the third corrector step in the larger step sizes. In this example it is not large enough to justify the third corrector step, but for other problems the difference might be more significant.

The last graph shows the influence of the error of the first step. Having a simple 5th order error is not enough, to get to the optimal error behavior of the whole method, the first-step error coefficient needs to be reduced (by a factor 16 or 81) by using 2 or 3 RK4 sub-steps. Using 2 steps and Richardson extrapolation could also be a useful variant

Code

method

def firststep(f,x,y, v, dx, nRK4):
    '''Apply nRK4 Kutta4 steps with step width dx/nRK4 to the first order system
    [y', v'] = [v, f(x,y)]'''
    y,v = (np.asarray(c) for c in (y,v) )
    dy = np.zeros_like(y)
    v0h = np.zeros_like(y); 
    h = dx/nRK4;
    for j in range(nRK4):
        # in principle, ky could be eliminated, at the cost of introducing unusual terms
        k1y = v;
        k1v = f(x,y);
        k2y = v + 0.5*h*k1v;
        k2v = f(x+0.5*h, y+0.5*h*k1y);
        k3y = v + 0.5*h*k2v;
        k3v = f(x+0.5*h, y+0.5*h*k2y);
        k4y = v + h*k3v;
        k4v = f(x+h, y+h*k3y);
        dy = (k1y+2*(k2y+k3y)+k4y)/6;
        v0h +=  dy
        y += h*dy;
        v += h*(k1v+2*(k2v+k3v)+k4v)/6;
        x += h;
    return y, v0h/nRK4


def Numerov_integrate(f, x0, xf, dx, y0, v0, nPECE=1, nRK4=1):
    x = np.arange(x0, xf+0.5*dx, dx);
    y = np.asarray(len(x)*[y0]);
    # use v0h for v[k-1/2], v1h for v[k+1/2] where the current step if from k to k+1
    y[1], v0h = firststep(f,x0,y0, v0, dx, nRK4);
    fkm1, fk = f(x[0],y[0]), f(x[1],y[1]);
    for k in range(1,len(x)-1):
        # predictor, use simple Verlet
        y[k+1] = y[k] + dx*(v0h+ dx*fk);
        for j in range(nPECE):
            v1h = v0h + dx/12*(f(x[k+1],y[k+1])+10*fk+fkm1);
            y[k+1] = y[k] + dx*v1h;
        fkm1, fk = fk, f(x[k+1], y[k+1]);
        v0h = v1h;

    return x,y;

test example

from numpy import sin
def mms_ode(x,y): return -sin(x)+4*(sin(sin(x))-sin(y))

# test the correctness of the implementation
m=10;
x,y = Numerov_integrate(mms_ode, 0, 4, 0.1/m, [0.0], [1.0], nPECE=2, nRK4=3)
for xk,yk in zip(x,y)[::2*m]: print "%15.10f: %15.10f,  %15.10f | %15.10e"%(xk,yk,sin(xk), (yk-sin(xk))*(10*m)**4)