Dear Mohamed
Thank you for your explanation. I agrre and understand if derivative BC s are specified, then may be in PDE s we can't use central diff scheme without some assumptions.But what about in ODES? here by introducing ghost points and using the diff eqn at the boundary node can't we solve the problem? I am not sure!!
However my origianl question was what is the use of first order FDM formulae if that is not computationally cheaper!!
I have done a mathematical simulation model using FDM for water balance study and minimized the error by improvisation.
I have upload my PhD thesis related to above in RG in the following link
https://www.researchgate.net/publication/255909976_CONJUNCTIVE_USE_OF_SURFACE_AND_GROUNDWATER_TO_IMPROVE_FOOD_PRODUCTIVITY_IN_A_RESTRICTED_AREA_Degree_of_Doctor_of_Philosophy?ev=prf_pub
Thesis CONJUNCTIVE USE OF SURFACE AND GROUNDWATER TO IMPROVE FOOD P...
If a,b,c and d are constants, then central difference will give better results. If Neumann's BC is given then one exterior point coming at each boundary x=a1 and x=a2, can be reflected to interior point by the condition.
Yes if a, b, c and d are functions then it is different case if singularity is there.
If you solve the problem by an implicit method through matrix inversion, only the first-order BC can be internally consistent. For certain type of partial differential systems, this formulation can also yield a globally second-order solution. In most circumstance, the second-order BC must be lagged and to be treated explicitly. However, the Dirichlet-type BC is the only exception.
The given equation is a simple two point boundary value problem. It is an ODE. So no concept of impilicit or explicit method is there.You will get a tri diagonal matrix which can easily be solved by Thomas algorithm. One should not invert the matrix.
Thank you Sir
So can I conclude that first order FDM should be used only if central diff scheme cannot be applied ? Since the first order FDM is of lower accuracy yet computationally as expensive as central diff !
Having said this...
The part I need you to kindly help me is give an example of a situation where central difference scheme cannot be employed and only first order FDM can be used!!
Thank you
Consider the case when a, b , c and d are function of x in general.By taking central differences, if the corresponding coefficients matrix is not diagonally dominant(when advection term bu' is very large in comparison to au'') then it may give you oscillations.
In such cases upwinding or downwinding(first order approximation) is applied.
For more detail, you may section 3.5 of the link
http://www4.ncsu.edu/~zhilin/TEACHING/MA402/chapter3.pdf
@ R Mittal
Thank you very much sir, I went through the section you mentioned, its very useful.
In fact I am trying to solve an eqn of the form u''+p*cos(u)=0, u(0)=0,u'(0)=0;and p>0.Do you know if an analytical solution exist for such an eqn ?
Another thing to mention is why are they calling such an ODE as elliptic !! I thought such classifications are meant for only PDEs.
This is an initial value problem . What we discussed was related to two point boundary value problems. This can be solved numerically by Runge Kutta type of methods.I think elliptic in the sense that it is related to elliptic integrals.
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