Hi Rajesh,

There are forward, backward and central approximations for finite differences. If you want to use central differences up to the boundary for sake of simplicity then you would construct an artificial "ghost points" lying outside of your computational domain. However, you can also derive single sided difference approximations (forward, backward) for high order derivatives from Taylor series, which should guarantee also higher order accuracy at the boundary. Some good scripts would show how to derive stencils from Taylor series which should make things obvious.

@ Christian, thanks for the info.

The suggested methods will increase the precision of an estimate at any given point. You can include Markov chain methods and other successive approximations to both finite difference and finite volume. The increase in accuracy is an illusion. The uncertainty remains the same and can even increase when errors are fully propagated. There is no magic spell that works for finite volume or finite difference except more measurements.

@ Joseph, thanks for the info.

The trick is, suppose you are given B C dy/dx at the boundary x=a as alpha that is  dy/dx |x=a = alpha. We replace dy/dx by central difference at a, that is, y(a+h) - y (a-h) = 2h* alpha . If a is left boundary then a - h will be outside the domain and a is right boundary then a+h will be outside boundary. In the first case, we replace y(a-h) by        y (a+h ) - 2h * alpha and in second case replace y (a+h) by y(a-h) + 2h*alpha. These are second order approximations.

You can get any order you wish.

@ Mittal, thanks this makes  sense.

@ Raymond, thanks for your time.

If you use the reflection boundary condition i.e. the "ghost points" lying outside the domain of computation it is possible that in most cases it will not make physical sense and it will be just as or more inaccurate than a first-order forward difference.

There is an alternative approach to the Taylor series analysis to construct the finite-difference quotients, that is, the polynomial approach. It consist in assuming that at the boundary, for example, the x component of velocity u can be expressed as the polynomial u=a+b*y+c*y2 if you evaluate this equation at the boundary (grid point 1), where y=0, it yields u1=a and at grid point 2, where y=deltay deltay=y2-y1 u2=a+b*deltay+c*(deltay)2 and at grid point 3, where y=2*deltay u3=a+2*b*deltay+4*c*(deltay)2 with the three equations above you can form a system of equations and then solving for b you obtain 

b=(-3*u1+4*u2+u3)/(2*deltay) now if you differentiate the first equation u with respect to y you obtain du/dy=b+2*c*y evaluating this last equation at the boundary where y=0, yields (du/dy)1=b substituting b into this last equation you obtain (du/dy)1=(-3*u1+4*u2-u3)/(2*deltay) this last equation is a one-sided finite-difference expression for the derivative of u at the boundary. It is easy to show that this expression is of second-order-accuracy.

@ Nestor, thanks that makes sense. Thanks for your time.

I agree with Raymond - "You can get any order you wish"

@Ashish, thanks for the info.

Of course you can!! ...even fourth order if you need...

But in general just apply the same order for the scheme at the boundary that the ones you used in the domain for the discretizations of the PDE.

@ Philippe, thanks for the info.

You already got the answer to your question. However, I will add the following. If you are working with elliptic  equations then higher approximation on the boundary will produce smaller errors, but sometimes the convergence rate remains invariant. So, many people think that the effort in programming higher order approximations at the boundary is no well justify. In the case of parabolic and hyperbolic equations higher approximation at the boundary could produce stability problems. To avoid such problems with stability many people try the opposite approach, namely, higher order at inner grid points and lower order approximations at the boundary.

@ Guevara-Jordan, thanks for the update, it is highly helpful.

Dear Mr. Alayil,

Higher orders means larger stencils of the difference operators, or more complex processes if you are using compact schemes. There is a lot of literature about compact schemes available.

Anyway, higher orders will give you more oscillations which, for some problems, will be worse than a bigger localized error. Imagine a polynomial interpolation: two points, a line, three points, a parabola (one extreme), four points, a cubic (two extremes), and so forth.

There is the option of rational interpolation, which has been being observed as more stable, but without a formal proof of the "whys and hows" (up to my knowledge). I don't know what you are going into, but maybe it deserves a try.

By the way, do you recall where have you read that statement?

Best regards and good luck.

it is  possible to got the second order accuracy at the boundaries using finite difference method.stability problems can be avoided by correct bc.

You should look at the characteristic equation of the interior scheme and the resolvant equation of the boundary operator (assuming the "linearized" PDE is well posed and the interior scheme is stable in the sense of fixing the final time and refining the grid spacing and time stepping) to ensure the global stability or boundness of the linear scheme (GKS theory).

In general, one can employ a boundary operator which is one order less than the interior scheme.  This will ensure the global order of the scheme will be the same as  the interior scheme.  Of course, there is a lot of little things didn't mentioned here.  For example, nonlinear scheme or nonlinear PDE does not fall into the consideration here.

I hope this answer won't get cutoff again....

@ Helio, thanks for the info. Sorry I don't recall, but I am searching. Will let you know.

thanks!

@Kalidasan & Wai, thanks for the update.

You should also analyze the PDE and understand the physics of the problem.  For example, if it is a supersonic gas inflow, then the free stream condition must be applied and all variables must be imposed, while for a subsonic  inflow of an external flow or wall boundary, it is not clear what is the best way to totally avoid the wave reflection due to the incoming characteristics of the Euler equations (for example). 

Blindly specifying boundary condition of any order does not always give one a "correct" solution, despite the scheme is of high order.

Boundary condition for complex system is notorious in its design and implementation except in simple baby problems. 

Hello Mr. Alayil,  You can get second and higher order convergence but concentrate on stability of your proposed scheme thats important. When you get higher order but loose stability and vice versa. The challenge is to construct the numerical scheme to the problem whose physical characteristic should be resolved.

@ Wai, thanks for the info.

@Ramesh, thanks for the info.

In my opinion you may trade off between time step size and order.whenever you use a 1st order you mean you have just seen the slope at apoint and you havent seen curve feature.the mien of function needs to be considered.

using a first order and combining it with adaptive steps (short step at high slope variation points) may be best choice

@ Mahdi, thanks for the info.

Do you have any reference for the same, I would like to know more about this. Do you have any links or references on the same, it would be a great help.

thanks!

If you use Mimetic Discretizations you can achieve the same order of accuracy at the boundary that you have in the interior of the domain. Here is a reference

http://www.crcpress.com/product/isbn/9781466513433

For Poisson's equation with LINEAR boundary conditions, it is well known that the large truncation error at the boundary does not stop you from getting second-order error in the solutions: the reference is here:

J.H. Bramble and B.E. Hubbard 1962 On the formulation of finite-difference analogues of the Dirichlet problem for Poisson's equation. Numer. Math. 4: 313-327.

The same authors also analyzed Neumann and mixed BCs in their followup work.

For the heat equation and the advection equation, I discussed the truncation error and solution error in this JCP paper:

https://www.researchgate.net/publication/220206475_Handling_solid-fluid_interfaces_for_viscous_flows_Explicit_jump_approximation_vs._ghost_cell_approaches

I am also working on a nonlinear Robin boundary condition, and numerical experiments show that you lose at least one order of accuracy at the boundary if the nonlinearity is high.

Article Handling solid-fluid interfaces for viscous flows: Explicit ...

It may be important to derive your boundary condition from the internal discretization.  For example, if you have a symmetry-preserving numerical method (preserves spherical or cylindrical or planar symmetry on symmetric meshes), a symmetry/reflecting boundary condition implementation must be consistent with the discrete form internal to the mesh.  Otherwise, the boundary condition will break symmetry.

@ Jose, thanks for the info and the link. It would help me.

@Qinghai, thanks for the update and the link. I will go through it.

@Mack, thanks for the info.

I recommend publication: E.Majchrzak, Ł.Turchan, The finite difference method for transient convection-diffusion problems, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology, 1(11), 2012, 63-72.

Full text available: http://srimcs.im.pcz.pl/2012_1/art_07.pdf

http://srimcs.im.pcz.pl/2012_1/art_07.pdf

@ Lukasz, thanks for the info and link. I will go through the paper.

It is necessary  to applied the Taylor series to obtain schemes of different orders at a boundary point. For example,  to evaluate the derivative du/dx, we

consider the boundary point "b" and the points "o", "a" are interior points in the domain in axis x, according to the simple figure

                           ----a-----o-----b--------->x

the x step is h. The derivative at the boundary point "b" could be evaluated as follows

du/dx= (-3*u(a)+4*u(o)-u(b) )/ (2* h)

@ Carlos, thanks for the update.

You can use one sided approximations to the derivatives.

It is possible but it requires extra work, using mimetic methods you can have the same order of approximation at the boundary as well as the interior not just for second order but higher order as well and the boundary approximation is part of the scheme.