Numerical method for ODEs without frequency error











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To solve an ODE numerically,

one usually use finite difference methods.



For example,

simple harmonic oscillator, i.e.
$$y''=-omega^2y$$



can be discretized by



$$frac{y_{n+1}-2y_n+y_{n-1}}{Delta x^2}=-omega^2y_n$$



However, if we use this formula to solve the equation,

the frequency of the numerical solution is different from

true frequency $omega$.



Then my question is,

is there any discretization method without frequency error?

For example, is Runge-Kutta method similarly suffer from frequency error?



EDIT

Simple harmonic oscillator is just an example.

My equation is actually PDE (Wave equation).






numerical-methods finite-differences finite-difference-methods







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    up vote
    0
    down vote

    favorite












    To solve an ODE numerically,

    one usually use finite difference methods.



    For example,

    simple harmonic oscillator, i.e.
    $$y''=-omega^2y$$



    can be discretized by



    $$frac{y_{n+1}-2y_n+y_{n-1}}{Delta x^2}=-omega^2y_n$$



    However, if we use this formula to solve the equation,

    the frequency of the numerical solution is different from

    true frequency $omega$.



    Then my question is,

    is there any discretization method without frequency error?

    For example, is Runge-Kutta method similarly suffer from frequency error?



    EDIT

    Simple harmonic oscillator is just an example.

    My equation is actually PDE (Wave equation).






    numerical-methods finite-differences finite-difference-methods







    share|cite|improve this question


























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      To solve an ODE numerically,

      one usually use finite difference methods.



      For example,

      simple harmonic oscillator, i.e.
      $$y''=-omega^2y$$



      can be discretized by



      $$frac{y_{n+1}-2y_n+y_{n-1}}{Delta x^2}=-omega^2y_n$$



      However, if we use this formula to solve the equation,

      the frequency of the numerical solution is different from

      true frequency $omega$.



      Then my question is,

      is there any discretization method without frequency error?

      For example, is Runge-Kutta method similarly suffer from frequency error?



      EDIT

      Simple harmonic oscillator is just an example.

      My equation is actually PDE (Wave equation).






      numerical-methods finite-differences finite-difference-methods







      share|cite|improve this question















      To solve an ODE numerically,

      one usually use finite difference methods.



      For example,

      simple harmonic oscillator, i.e.
      $$y''=-omega^2y$$



      can be discretized by



      $$frac{y_{n+1}-2y_n+y_{n-1}}{Delta x^2}=-omega^2y_n$$



      However, if we use this formula to solve the equation,

      the frequency of the numerical solution is different from

      true frequency $omega$.



      Then my question is,

      is there any discretization method without frequency error?

      For example, is Runge-Kutta method similarly suffer from frequency error?



      EDIT

      Simple harmonic oscillator is just an example.

      My equation is actually PDE (Wave equation).






      numerical-methods finite-differences finite-difference-methods




      numerical-methods finite-differences finite-difference-methods






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited yesterday

























      asked yesterday









      Shu S

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      183



























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