# Finite Differences - Higher Order Approximations

In order to achieve better approximations using finite difference method,higher order accuracy differences can be employed. However, the treatment of boundaries is more complicated for higher order approximations because more grid points are required at the boundaries to evaluate the corresponding approximation. Higher order approximations of first order derivative with finite differences are given by (e.g. backward, forward, and centered schemes respectively):
$$\left(\frac{\partial q}{\partial x}\right)_i = \frac{2q_{i+1}+3q_i-6q_{i-1}+q_{i-2}}{6\Delta x} + \mathcal{O}(\Delta x)^3$$ $$\left(\frac{\partial q}{\partial x}\right)_i = \frac{-q_{i+2}+6q_{i+1}-3q_i+2q_{i-1}}{6\Delta x} + \mathcal{O}(\Delta x)^3$$ $$\left(\frac{\partial q}{\partial x}\right)_i = \frac{-q_{i+2}+8q_{i+1}-8q_{i-1}+q_{i-2}}{12\Delta x} + \mathcal{O}(\Delta x)^4$$ And an example for the second order derivative is:
$$\left(\frac{\partial^2 q}{\partial x^2}\right)_i = \frac{-q_{i+2}+16q_{i+1}+30q_i-16q_{i-1}+q_{i-2}}{(12\Delta x)^2} + \mathcal{O}(\Delta x)^4$$

#### 1 comment:

1. Hi,
You people are doing great job for such intense Numerical Methods. I am working on Higher Order Compact Schemes. I need help from you people, sixth order Compact Scheme with Dirichlet boundary condition. I have to use on 3D Nonlinear Coupled System. Help is highly appreciated.
Thanks you for help in advance.