I try to solve numerically the differential equation with mixed derivative. I found one possibility for the mixed derivative like y(i+1,j-1)-y(i,j-1)-y(i-1,j+1)+y(i-1,j). But may be there are another ways...
Consult Chapter 25 of the book Handbook of Mathematical Functions, edited by M. Abramowitz and I.A. Stegun (Dover Publications).
The discretisation you provided is incomplete. To compute the mixed derivative of u with respect to x and y you start from u_xy = (u_y)_x (or the other way around, your choice). Then you apply central differencing twice. The following applies to a uniform grid with spacing h. First find u_y at (i+1,j) and (i-1,j):
u_y(i+1,j) = (u(i+1,j+1)-u(i+1,j-1))/(2*h)
u_y(i-1,j) = (u(i-1,j+1)-u(i-1,j-1))/(2*h)
To find u_xy Apply central differencing again
u_xy = (u_y)_x = (u_y(i+1,j)-u_y(i-1,j))/(2*h)
The result is
u_xy = (u(i+1,j+1)-u(i+1,j-1)-u(i-1,j+1)-u(i-1,j-1))/(4*h^2)
This discretisation is second order accurate. You will have to deal with the boundary points of your domain and may need to switch from central differences to forward or backward differences, depending on the boundary.
Also note that if u is a known function (rather than discrete values from an array for example), you can achieve much higher accuracy with Lagrange or Chebyshev polynomial approximations or using Ridders method of polynomial extrapolations.
(uxy)ij =[ (uy)i+1,j - (uy)i - 1,j ] / (2hx)
= [ ( ui+1,j+1 - ui+1, j -1)/ (2hy) - ( ui-1,j+1 - ui-1, j-1)/ (2hy) ] /(2hx)
=( ui+1,j+1 - ui+1, j -1 - ui-1,j+1 + ui-1, j-1)/ (2hy)(2hx)
Oops! I forgot to change the sign of the last term in the numerator. Copying and pasting is a bad practice.
R.C. Mittal's answer is for meshes that are uniform in x and y with hx and hy spacing in the x and y directions, respectively, hence the hx and hy. If you have a non-uniform mesh use the differences in the absissae and ordinates of the grid points as needed instead of h, hx or hy.
Thanks to all! I understood the main idea.
My example was just a part of scheme. The changing of directions is used in the method. So in general the full approximation has been achived.
See my paper "Supraconvergence and supercloseness of a scheme for elliptic equations on non-uniform grids" Numerical Functional Analysis and Optimization, 27 (5-6). 539-564, 2006, where we propose a finite difference discretization using non-uniform grids and solutions with lower regularity smoothness.
Did you try high-order compact (HOC) finite difference methods? See
http://www.math.uh.edu/~hjm/june1995/p00397-p00408.pdf
- Badges
- Science topic
- Similar topics
- Mathematics
- Applied Mathematics
- Differential Equations