Tripadi

Numerov's method is a numerical method to solve ordinary differential equations of second order in which the first-order term does not appear. It is a fourth-order linear multistep method. The method is implicit, but can be made explicit if the differential equation is linear.

Numerov's method was developed by Boris Vasil'evich Numerov.

The method

The Numerov method can be used to solve differential equations of the form

$(\frac{d^2}{d{x}^2}+f(x))y(x)=0$

The function $y(x)$ is sampled in the interval [a..b] at equidistant positions $x_n$. Starting from function values at two consecutive samples $x_{n-1}$ and $x_n$ the remaining function values can be calculated as

$y_{n+1}=\frac{(2-\frac{5h^2}{6}{f}_n)y_n-(1+\frac{h^2}{12}{f}_{n-1})y_{n-1}}{1+\frac{h^2}{12}{f}_{n+1}}$

where $f_n=f(x_n)$ and $y_n=y(x_n)$ are the function values at the positions $x_n$ and $h=x_n-x_{n-1}$ is the distance between two consecutive samples.

Nonlinear equations

For nonlinear equations of the form

$\frac{d^2}{d{t}^2}y=f(t,y)$

the method is given by

$y_{n+1}=2y_n-y_{n-1}+\frac{1}{12}{h}^2(f_{n+1}+10f_n+f_{n-1}).$

This is an implicit linear multistep method, which reduces to the explicit method given above if the function f is linear in y. It achieves order 4 Template:Harv.

Application

In numerical physics the method is used to find solutions of the radial Schrödinger Equation for arbitrary potentials.

$[-\frac{{\hslash }^2}{2\mu }(\frac{1}{r}\frac{{\partial }^2}{\partial r^2}r-\frac{l(l+1)}{r^2})+V(r)]R(r)=ER(r)$

The above equation can be rewritten in the form

$[\frac{{\partial }^2}{\partial r^2}-\frac{l(l+1)}{r^2}+\frac{2\mu }{{\hslash }^2}(E-V(r))]u(r)=0$

with $u(r)=rR(r)$. If we compare this equation with the defining equation of the Numerov method we see

$f(x)=\frac{2\mu }{{\hslash }^2}(E-V(x))-\frac{l(l+1)}{x^2}$

and thus can numerically solve the radial Schrödinger equation.

Derivation

Start with the Taylor expansion of $y(x)$ about a point $x_0$:

$y(x)=y(x_0)+(x-x_0)y^{\prime }(x_0)+\frac{(x-x_0{)}^2}{2!}{y}^{″}(x_0)+\frac{(x-x_0{)}^3}{3!}{y}^{‴}(x_0)+\frac{(x-x_0{)}^4}{4!}{y}^{⁗}(x_0)+\frac{(x-x_0{)}^5}{5!}{y}^{\prime \prime \prime \prime \prime }(x_0)+𝒪(h^6)$

Denote the distance from $x$ to $x_0$ by $h=x-x_0$ and, noting that this means $x=x_0+h$, we can write the above equation as

$y(x_0+h)=y(x_0)+h{y}^{\prime }(x_0)+\frac{h^2}{2!}{y}^{″}(x_0)+\frac{h^3}{3!}{y}^{‴}(x_0)+\frac{h^4}{4!}{y}^{⁗}(x_0)+\frac{h^5}{5!}{y}^{\prime \prime \prime \prime \prime }(x_0)+𝒪(h^6)$

Computationally, this amounts taking a step forward by an amount h. If we want to take a step backwards, replace every h with -h for the equation of $y(x_0-h)$:

$y(x_0-h)=y(x_0)-h{y}^{\prime }(x_0)+\frac{h^2}{2!}{y}^{″}(x_0)-\frac{h^3}{3!}{y}^{‴}(x_0)+\frac{h^4}{4!}{y}^{⁗}(x_0)-\frac{h^5}{5!}{y}^{\prime \prime \prime \prime \prime }(x_0)+𝒪(h^6)$

Note that only the odd powers of h experienced a sign change. On an evenly spaced grid, the nth site on a computational grid corresponds to position $x_n$ if the step-size between grid points are of length $h$ (hence h should be small for the computation to be accurate). This means we have sampling points $(x_{n-1},y_{n-1})$ and $(x_{n+1},y_{n+1})$. Taking the equations for $y_{n-1}$ and $y_{n+1}$ from continuous space to discrete space, we see that

$y_{n+1}=y(x_n+h)=y(x_n)+h{y}^{\prime }(x_n)+\frac{h^2}{2!}{y}^{″}(x_n)+\frac{h^3}{3!}{y}^{‴}(x_n)+\frac{h^4}{4!}{y}^{⁗}(x_n)+\frac{h^5}{5!}{y}^{\prime \prime \prime \prime \prime }(x_n)+𝒪(h^6)$

$y_{n-1}=y(x_n-h)=y(x_n)-h{y}^{\prime }(x_n)+\frac{h^2}{2!}{y}^{″}(x_n)-\frac{h^3}{3!}{y}^{‴}(x_n)+\frac{h^4}{4!}{y}^{⁗}(x_n)-\frac{h^5}{5!}{y}^{\prime \prime \prime \prime \prime }(x_n)+𝒪(h^6)$

The sum of those two equations gives

$y_{n-1}+y_{n+1}=2y_n+h^2{y}^{\prime }{\text{'}}_n+\frac{h^4}{12}{y}^{‴}{\text{'}}_n+𝒪(h^6)$

We solve this equation for $y^{\prime }{\text{'}}_n$ and replace it by the expression $y^{\prime }{\text{'}}_n=-f_n{y}_n$ which we get from the defining differential equation.

$h^2{f}_n{y}_n=2y_n-y_{n-1}-y_{n+1}+\frac{h^4}{12}{y}^{‴}{\text{'}}_n+𝒪(h^6)$

We take the second derivative of our defining differential equation and get

$y^{⁗}(x)=-\frac{d^2}{d{x}^2}[f(x)y(x)]$

We replace the second derivative $\frac{d^2}{d{x}^2}$ with the second order difference quotient and insert this into our equation for $f_n{y}_n$ (note that we take the mixed forward and backward finite difference, not the double forward difference or the double backward difference)

$h^2{f}_n{y}_n=2y_n-y_{n-1}-y_{n+1}-\frac{h^4}{12}\frac{f_{n-1}{y}_{n-1}-2f_n{y}_n+f_{n+1}{y}_{n+1}}{h^2}+𝒪(h^6)$

We solve for $y_{n+1}$ to get

$y_{n+1}=\frac{(2-\frac{5h^2}{6}{f}_n)y_n-(1+\frac{h^2}{12}{f}_{n-1})y_{n-1}}{1+\frac{h^2}{12}{f}_{n+1}}+𝒪(h^6).$

This yields Numerov's method if we ignore the term of order $h^6$. It follows that the order of convergence (assuming stability) is 4.

References

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  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  This book includes the following references:
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

External links

• Lecture notes: Computerphysik und Numerik - by Jan Krieger
• Lecture notes of Werner Scholz - At Vienna University of Technology
• Lecture notes of Alexander Wagner