2022-08-24 Finite Differences#

Last time#

  • General shape of PDE solvers and stakeholders

  • Comparing/plotting cost and accuracy

  • Learning strategy

Today#

  • Discussion and scoping

  • Evaluating derivatives

  • Taylor series and truncation error

  • Stability

Examples of PDE#

  • Navier-Stokes (viscous fluids)

    • nonlinear

    • incompressible or compressible

  • Elasticity

    • linear elasticity

    • hyperelasticity (geometric nonlinear + material nonlinearity)

    • time dependent (dynamics) or steady state

  • Hamilton-Jacobi-Bellman

    • optimal control

  • Wave equations

    • acoustics

    • elasticity

    • electromagnetics

    • frequency domain

Choices in scoping the class#

Theory#

Analysis first, confirm using numerics. Limited to simpler models.

Applied#

Numerics first, pointers to useful theory.

Build from scratch#

Limited to simpler models, but you’ll understand everything under the hood.

Build using libraries#

More installation and software layers, but can solve more interesting problems.

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
default(linewidth=3)

n = 41
h = 6 / (n - 1)
x = LinRange(-3, 3, n)
u = sin.(x)
plot(x, u, marker=:circle)
../_images/2022-08-24-finite-differences_14_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/2022-08-24-finite-differences_15_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#