# Implementing the Numerov method for solving ODEs with NDSolve

I'd like to implement the Numerov scheme for solving an ODE (Scroedinger Eq time-independent) with NDSolve. I tried in analogy with the Runge Kutta example in the documentation:

Numerovv[]["Step"[rhs_, h_, t_, x_]] := Module[{},
-x[t - h] (2 (12 + 5 h^2 rhs[t - h]) )/(-12 + h^2 rhs[t]) +
x[t - 2 h] (12 - h^2 rhs[t - 2 h]) /(-12 + h^2 rhs[t])
];
Numerovv[___]["StepInput"] = {"F"["T"], "H", "T", "X"};
Numerovv[___]["StepOutput"] = "X";
Numerovv[___]["StepMode"] := Fixed;
Numerovv[___]["DifferenceOrder"] := 4;


But when I select Method -> "Numerovv", I get

The value of the option Method -> Numerovv is not a known built-in method, a symbol that could be a user-defined method, or a list with a name followed by method options. >>

Do I miss something?

• This is not the error I get when I use your code. I get an error about the step function not returning an acceptable form. This is the code I use, and the error message I get is ""The Step function for Method -> Numerovv returned {-1.\ \ {1.,0.}[-0.0472329]+2.\ {1.,0.}[-0.0236164],-1.\ \ {1.,0.}[-0.0472329]+1.99944\ {1.,0.}[-0.0236164]}, which is not an \ acceptable form."" – Jason B. Mar 3 '16 at 10:19
• In your specification of "StepInput", you define the function to take one variable, which matches the step function, but you don't define "X" to be a variable with an argument. That is one issue, but fixing it doesn't solve the problem. The documentation on user-defined NDSolve methods is scarce. If someone can find more info than that linked in the question I'd be interested to read it. – Jason B. Mar 3 '16 at 10:23
• The problem is that the built-in method framework is setup for $\mathbf y^\prime=\mathbf f(x,\mathbf y)$, while Numerov is a method specialized for second order conservative equations, $y^{\prime\prime}=f(x,y)$. An "adapter" would be needed, but I'm drawing a blank on how to write one. – J. M. will be back soon Mar 3 '16 at 11:13
• – Mariusz Iwaniuk Mar 3 '16 at 11:51
• As I said, the sticking point is not that the method is single- or multistep; it's that NDSolve[]'s framework assumes that methods plugged into it are for $\mathbf y^\prime=\mathbf f(x, \mathbf y)$. Numerov is not a method in that format, so adaptation is necessary. That you might need to figure out. – J. M. will be back soon Mar 14 '16 at 5:39