[soes]

For instance in a Laplace equation is it generally possible to difference one component central difference in space and the other 1st order accurate forward/backward difference?

Generally, if an equation is sum of derivative components should all components be differenced with similar method (i.e. central, or backward ..) or should all be differenced with same order of accuracy (1st order, 2nd order ...), or there is no specific limitation on the schemes?

Finite difference

[FMDenaro]

Originally Posted by Soheil.esmaeilzadeh

For instance in a Laplace equation is it generally possible to difference one component central difference in space and the other 1st order accurate forward/backward difference?

Generally, if an equation is sum of derivative components should all components be differenced with similar method (i.e. central, or backward ..) or should all be differenced with same order of accuracy (1st order, 2nd order ...), or there is no specific limitation on the schemes?

Finite difference

there is no theoretical reason to create a non-homogeneous discretization for elliptical equation

[soes]

Consider a 2d rectangular domain, for example in the far right edge I do backward in x and central in y, is this allowed? or on the left edge forward in x and central in y for instance.

[FMDenaro]

Originally Posted by Soheil.esmaeilzadeh

Consider a 2d rectangular domain, for example in the far right edge I do backward in x and central in y, is this allowed? or on the left edge forward in x and central in y for instance.

I do not see a reason for doing that...

[soes]

I do not want to use ghost cells to treat the boundary for instance, so I prefer to write a backward in x at the far right edge for example. I mean I am just asking about the possibility of this to do from numerical aspect.

[FMDenaro]

Originally Posted by Soheil.esmaeilzadeh

I do not want to use ghost cells to treat the boundary for instance, so I prefer to write a backward in x at the far right edge for example. I mean I am just asking about the possibility of this to do from numerical aspect.

but the treatment of the BC.s for the Laplace equation is a different issue from the discretization of the operator....

If you have Neumann BC.s you can work in several way, for example using the Div Grad operators so that at a boundary you can directly set n.Grad.
Alternatively, you can discretize the d/dn derivative with backward/forward formula (but at second order of accuracy) and insert the expression in the central discretization of the Lap operator.

[soes]

Thanks for you reply.

Actually, in reality I have a boundary condition at the bottom as:
w=(p_x/p_xx)*w_x+w_z*(2*p_x/p_xx-p)-p_x*w_zx
assuming that p is like a profile function such as p=sin(k*x) and w is velocity in vertical direction (let's call it z)

Actually when I want to discretize this equation at the bottom grid line, I think of using central in x and forward in z(vertical direction) for the points inside domain; and forward and backward at bottom left and bottom right respectively for both x and z.

That's how I came up with my question.

[FMDenaro]

Originally Posted by Soheil.esmaeilzadeh

Thanks for you reply.

Actually, in reality I have a boundary condition at the bottom as:
w=(p_x/p_xx)*w_x+w_z*(2*p_x/p_xx-p)-p_x*w_zx
assuming that p is like a profile function such as p=sin(k*x) and w is velocity in vertical direction (let's call it z)

Actually when I want to discretize this equation at the bottom grid line, I think of using central in x and forward in z(vertical direction) for the points inside domain; and forward and backward at bottom left and bottom right respectively for both x and z.

That's how I came up with my question.

well, that's not clear ... you write a BC.s for w in terms of a (mixed) second derivative of w ??? That sounds strange...

of course I imagine the the function p is known and does not need a discretization of its derivatives.

[soes]

Yes yes, p is known like a sin function so it's derivative is also known at each x location, no problem with that.

Yes, basically the no penetration condition at a corrugated bottom I change it to this complex function. But that's how it is but my question is regarding the discretisation mainly

[FMDenaro]

Originally Posted by Soheil.esmaeilzadeh

Yes yes, p is known like a sin function so it's derivative is also known at each x location, no problem with that.

Yes, basically the no penetration condition at a corrugated bottom I change it to this complex function. But that's how it is but my question is regarding the discretisation mainly

Honestly, I don't know if your mathematical problem would be properly closed this way, you should check for elliptic problems with oblique/degenerate oblique BC.s.

I am supposing some compatibility constraint must be satisfied

[soes]

I will write a description of my formulation in a few hours