I am trying to solve a system of equations and have a question regarding the validity of my approach when implementing a fifth-order Cash-Karp Runge-Kutta (CKRK) embedded method with the method of lines. To give the questions some context, let me state the problem I am attempting to solve:(adsbygoogle = window.adsbygoogle || []).push({});

$$ \frac {\partial E}{\partial z} = - \frac {1}{c^2}\frac {\partial E}{\partial t} - \frac{1}{k} \frac {\partial ^2 E}{\partial z^2} - \frac{1}{kc^2} \frac {\partial^2 E}{\partial t^2} + iP \tag{1}

$$ $$

\\

\frac {\partial P}{\partial t} = iNE^* \tag{2}\\

$$ $$

\frac {\partial N}{\partial t} = iPE \tag{3}

$$

$$ E(z=0) = \frac{\partial E}{\partial z}(z=0) = E(t=0) = \frac{\partial E}{\partial t}(t=0) = 0,\\ P(t=0) = P_0e^{z/c}, N(t=0) = N_0e^{z/c} $$

where ## c = 3 \times 10^8, k = 1000, P_0 ## and ##N_0## are constants##, i=\sqrt{-1}####; 0 \leq t \leq 1000, 0 \leq z \leq 1000 ##

I am implementing CKRK on the above, and even though the first spatial derivative of ##E## depends on the second spatial derivative of ##E##, the numerical method appears to work when solving (1)-(3) when I use the scheme of approximating the time and spatial derivatives of ##E## on the right hand side of (1) by a backward difference approximation (I am using an accuracy of 5).

To switch (1) above to a system of first order spatial derivatives in ##z##, I could make the substitution:

$$ U = \frac {\partial E}{\partial z} $$

and solve the following equations for ##E## instead:

$$

U = \frac {\partial E}{\partial z} \tag{4}\\

$$ $$\frac {\partial U}{\partial z} = - \frac {k}{c^2}\frac {\partial E}{\partial t} - kU - \frac{1}{c^2}\frac {\partial^2 E}{\partial t^2} + kP \tag{5}

$$

But when testing these same initial/boundary conditions using the same numerical method on the coupled equations (2) - (5), the code takes too long to finish (the step sizes required become extremely small). I believe it is due to the fact that the coefficients on the right hand side of (5) are very large and cause stability issues. I have tried to rescale the values for ##z,t,P,N,## and ##E##, but doing so causes one of the other coupled equations to become unstable or has no effect (e.g. scaling ##z## does nothing to the value ## E = U\Delta z ## since both ##U## and ##\Delta z## would scale reciprocally and cancel any effect). It is due to similar reasons I am solving ##E## in the ##z##-direction as opposed to doing the substitution ##U = \frac {\partial E}{\partial t}## and solving it in ##t## which is the standard method of lines approach (when I tried this method, the ##\Delta t## given by CKRK becomes very small).

So ultimately, instead of using equations (2) - (5), I was wondering if applying CKRK to (1) - (3) is still a valid approach where I approximate the derivatives of ##E## on the right-side of (1) by backward finite differences? It seems very odd to apply CKRK to a first order spatial derivative that depends on an approximation of the second order spatial derivative, but is this wrong? (I would be using stored intermediate values of ##E## to ensure the backward finite difference approximations are also following the Runge-Kutta method.)

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A Backward finite differences on higher order derivative

Have something to add?

Draft saved
Draft deleted

Loading...

Similar Threads - Backward finite differences | Date |
---|---|

Backward euler method for heat equation with neumann b.c. | Mar 17, 2013 |

MATLAB solution to system of ODEs with forward and backward propagation | Feb 10, 2012 |

Numerical differentiation using forward, backward and central finite difference | Feb 2, 2010 |

System of ODE Boundary Value Problem with 2nd Order Backward Difference | Dec 5, 2009 |

Solving Backward Euler with Newton's Method | Dec 3, 2009 |

**Physics Forums - The Fusion of Science and Community**