Question

Using the Taylor series expansion, What is the first term of the truncation error of the finite difference equation ((frac{partial u}{partial x})_{i,j}=frac{u_{i+1,j}-u_{i,j}}{Delta y})?

(a) (-(frac{partial^2 u}{partial x^2})_{i,j}frac{Delta x}{2})

(b) ((frac{partial^2 u}{partial x^2})_{i,j}frac{Delta x}{3})

(c) (-(frac{partial^2 u}{partial x^2})_{i,j}frac{Delta x}{3})

(d) ((frac{partial^2 u}{partial x^2})_{i,j}frac{Delta x}{2})

Answer 1

The correct choice is (a) (-(frac{partial^2 u}{partial x^2})_{i,j}frac{Delta x}{2})

Explanation: The Taylor series expansion of ui+1,j is

(u_{i+1,j}=u_{i,j}+(frac{partial u}{partial x})_{i,j} Delta x+(frac{partial ^2 u}{partial x^2})_{i,j}  frac{(Delta x)^2}{2}+⋯)

(frac{u_{i+1,j}-u_{i,j}}{Delta x}=(frac{partial u}{partial x})_{i,j}+(frac{partial^2 u}{partial x^2})_{i,j}frac{Delta x}{2}+⋯)

((frac{partial u}{partial x})_{i,j}=frac{u_{i+1,j}-u_{i,j}}{Delta x}-(frac{partial^2 u}{partial x^2})_{i,j}frac{Delta x}{2}-…)

The term –((frac{partial^2 u}{partial x^2})_{i,j}frac{Delta x}{2}-… )is truncated. So, the first term of truncation error is –((frac{partial^2 u}{partial x^2})_{i,j}frac{Delta x}{2}).