MATH761

MATH761 Lesson02: Finite difference calculus

Problem. Study the accuracy and stability of finite difference discretizations of the IBVP for the heat equation

q_{t} = q_{x x} in (0, 1), q (t = 0, x) = f (x), q (t, 0) = 0, q (t, 1) = 0

Approach:

Obtain analytical solution

Use finite difference calculus to define approximations of $𝒪 (k^{r}, h^{p})$ of differentiation operations

Generate Fortran code implementations to carry out numerical experiments

Analyze stability

Solution.

By separation of variables:

q (t, x) = \sum_{n = 1}^{\infty} B_{n} \sin (n π x) e^{- {(n π)}^{2} t} ≅ \sum_{n = 1}^{P} B_{n} \sin (n π x) e^{- {(n π)}^{2} t}, B_{n} = ⟨ f, b_{n} ⟩ = 2 \int_{0}^{1} f (x) \sin (n π x) d x .

Example. $f (x) = \sin (π x), P = 4, 𝑩 = {1, 0, 0, 0}$

q (t, x) = e^{- π^{2} t} \sin (π x)

Figure 1. Heat IBVP solution at $t = {0 ., 0.1, 0.2, 0.3, 0.4, 0.5}$ .