I'm following some free online material from MIT about differential equations and I'm trying to solve problem 1 – question 2 of this assignment https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/assignments/ps2.pdf
It asks to develop a second-order accurate finite difference approximation of:
$$ 6 u u_{x} – u_{xxx} $$
And I'm a bit confused. I know how to do a second-order accurate finite difference of the second derivative with $ \frac{u_{i-1} – 2 u_i + u_{i+1}}{h^2} $ but I'm not sure if that's even possible for a third order derivative $u_{xxx}$. How should I approach this problem?
Best Answer
Combine it with a central difference quotient, $$ u_{xxx}(x_i)\approx\frac{\frac{u_{i+2}-2u_{i+1}+u_i}{h^2}-\frac{u_i-2u_{i-1}+u_{i-2}}{h^2}}{2h} $$ As this is symmetric, the error term has no first order component, is thus second order.