5. Partial Differential Equations

Finite Difference — Quiz

Test your understanding of finite difference with 5 practice questions.

Practice Questions

Question 1

What is the order of truncation error of the standard three-point central difference formula for the second derivative $u''(x)$ on a uniform grid with spacing $\Delta x$?

Question 2

In the five-point finite difference stencil for the two-dimensional Laplace equation $u_{xx}+u_{yy}=0$ on a uniform grid, the value at node $(i,j)$ satisfies which relation?

Question 3

For the explicit FTCS scheme applied to the one-dimensional heat equation $u_t = \alpha\,u_{xx}$, stability requires that $\lambda = \alpha\,\Delta t/\Delta x^2$ satisfy which condition?

Question 4

What is a key characteristic of implicit finite difference schemes compared to explicit schemes when solving time-dependent PDEs?

Question 5

Which finite difference scheme for the linear advection equation $u_t + a\,u_x = 0$ introduces numerical diffusion through averaging of neighboring points?