Contents

2021-08-25 Finite Difference Intro

Last time

  • General shape of PDE solvers and stakeholders

  • Comparing/plotting cost and accuracy

  • Learning strategy

Today

  • Evaluating derivatives

  • Taylor series and truncation error

  • Stability

Consider the boundary value problem: find \(u\):

(3)\[\begin{gather} -\frac{d^2 u}{dx^2} = f(x) \quad x \in \Omega = (-1,1) \\ u(-1) = a \quad \frac{du}{dx}(1) = b . \end{gather}\]

We say

  • \(f(x)\) is the “forcing”

  • the left boundary condition is Dirichlet

  • the right boundary condition is Neumann

We need to choose

  • how to represent \(u(x)\), including evaluating it on the boundary,

  • how to compute derivatives of \(u\),

  • in what sense to ask for the differential equation to be satisfied,

  • where to evaluate \(f(x)\) or integrals thereof,

  • how to enforce boundary conditions.

Finite Difference/collocation approach to solve \(u\):

(4)\[\begin{gather} -\frac{d^2 u}{dx^2} = f(x) \quad x \in \Omega = (-1,1) \\ u(-1) = a \quad \frac{du}{dx}(1) = b . \end{gather}\]

  • Represent the function \(u(x)\) by its values \(u_i = u(x_i)\) at a discrete set of points

    \[ -1 = x_1 < x_2 < \dotsb < x_n = 1 . \]

    • The FD framework does not uniquely specify the solution values at other points

  • Compute derivatives at \(x_i\) via differencing formulas involving a finite number of neighbor points (independent of the total number of points \(n\)).

  • FD methods ask for the differential equation to be satisfied pointwise at each \(x_i\) in the interior of the domain.

  • Evaluate the forcing term \(f\) pointwise at \(x_i\).

  • Approximate derivatives at discrete boundary points (\(x_n = 1\) above), typically using one-sided differencing formulas.

Computing a derivative

using Plots

n = 11
h = 6 / (n - 1)
x = LinRange(-3, 3, n)
u = sin.(x)

plot(x, u, marker=:circle)
../_images/2021-08-25-finite-difference_7_0.svg
u_x = cos.(x)
fd_u_x = (u[2:end] - u[1:end-1]) / h

plot(x, u_x)
plot!(x[1:end-1], fd_u_x, marker=:circle)
../_images/2021-08-25-finite-difference_8_0.svg

How accurate is it?

Without loss of generality, we’ll approximate \(u'(x_i = 0)\), taking \(h = x_{i+1} - x_i\).

\[ u(x) = u(0) + u'(0)x + u''(0)x^2/2! + O(x^3)\]

and substitute into the differencing formula

\[\begin{split} \begin{split} u'(0) \approx \frac{u(h) - u(0)}{h} = h^{-1} \Big( u(0) + u'(0) h + u''(0)h^2/2 + O(h^3) - u(0) \Big) \\ = u'(0) + u''(0)h/2 + O(h^2) . \end{split}\end{split}\]
Evidently the error in this approximation is \(u''(0)h/2 + O(h^2)\). We say this method is first order accurate.

Activity on stability and accuracy computing derivatives