2022-10-05 Factorization#

Last time#

  • FD methods in 2D

  • Cost profile

  • The need for fast algebraic solvers

Today#

  • Wave equation and Hamiltonians

  • Symplectic integrators

  • Sparse direct solvers

    • matrix orderings

    • impact on formulation

    • cost scaling

using Plots
default(linewidth=3)
using LinearAlgebra
using SparseArrays

function my_spy(A)
    cmax = norm(vec(A), Inf)
    s = max(1, ceil(120 / size(A, 1)))
    spy(A, marker=(:square, s), c=:diverging_rainbow_bgymr_45_85_c67_n256, clims=(-cmax, cmax))
end
    
function plot_stability(Rz, title; xlims=(-2, 2), ylims=(-2, 2))
    x = LinRange(xlims[1], xlims[2], 100)
    y = LinRange(ylims[1], ylims[2], 100)
    heatmap(x, y, (x, y) -> abs(Rz(x + 1im*y)), c=:bwr, clims=(0, 2), aspect_ratio=:equal, title=title)
end

struct RKTable
    A::Matrix
    b::Vector
    c::Vector
    function RKTable(A, b)
        s = length(b)
        A = reshape(A, s, s)
        c = vec(sum(A, dims=2))
        new(A, b, c)
    end
end

function rk_stability(z, rk)
    s = length(rk.b)
    1 + z * rk.b' * ((I - z*rk.A) \ ones(s))
end

rk4 = RKTable([0 0 0 0; .5 0 0 0; 0 .5 0 0; 0 0 1 0], [1, 2, 2, 1] / 6)

function ode_rk_explicit(f, u0; tfinal=1., h=0.1, table=rk4)
    u = copy(u0)
    t = 0.
    n, s = length(u), length(table.c)
    fY = zeros(n, s)
    thist = [t]
    uhist = [u0]
    while t < tfinal
        tnext = min(t+h, tfinal)
        h = tnext - t
        for i in 1:s
            ti = t + h * table.c[i]
            Yi = u + h * sum(fY[:,1:i-1] * table.A[i,1:i-1], dims=2)
            fY[:,i] = f(ti, Yi)
        end
        u += h * fY * table.b
        t = tnext
        push!(thist, t)
        push!(uhist, u)
    end
    thist, hcat(uhist...)
end
ode_rk_explicit (generic function with 1 method)

Gas equations of state#

There are many ways to describe a gas

Name

variable

units

pressure

\(p\)

force/area

density

\(\rho\)

mass/volume

temperature

\(T\)

Kelvin

(specific) internal energy

\(e\)

energy/mass

entropy

\(s\)

energy/Kelvin

Equation of state#

\[ \rho, e \mapsto p, T \]

Ideal gas#

(23)#\[\begin{align} p &= \rho R T & e &= e(T) \end{align}\]
\[ p = (\gamma - 1) \rho e \]
pressure(rho, T) = rho*T

contour(LinRange(0, 2, 30), LinRange(0, 2, 30), pressure, xlabel="\$\\rho\$", ylabel="\$T\$")
../_images/2022-10-05-factorization_6_0.svg

Conservation equations#

Mass#

Let \(\mathbf u\) be the fluid velocity. The mass flux (mass/time) moving through an area \(A\) is

\[ \int_A \rho \mathbf u \cdot \hat{\mathbf n} .\]

If mass is conserved in a volume \(V\) with surface \(A\), then the total mass inside the volume must evolve as

\[ \int_V \rho_t = \left( \int_V \rho \right)_t = - \underbrace{\int_A \rho\mathbf u \cdot \hat{\mathbf n}}_{\int_V \nabla\cdot (\rho\mathbf u)},\]

where we have applied the divergence theorem. Dropping the integrals over arbitrary volumes, we have the evolution equation for conservation of mass.

\[ \rho_t + \nabla\cdot (\rho \mathbf u) = 0 \]

Momentum#

The momentum \(\rho \mathbf u\) has a flux that includes

  • convection \(\rho \mathbf u \otimes \mathbf u\)

    • this is saying that each component of momentum is carried along in the vector velocity field

  • pressure \(p I\)

  • viscous \(-\boldsymbol\tau\)

A similar integral principle leads to the momentum equation

\[ (\rho \mathbf u)_t + \nabla\cdot\big[ \rho \mathbf u \otimes \mathbf u + p I - \boldsymbol \tau \big] = 0 \]

Simplifications#

  • Ignore viscous stress tensor \(\boldsymbol \tau\)

  • Ignore energy equation (not yet written) and assume constant temperature

    • \(p = a^2 \rho\) where \(a\) is the acoustic wave speed

\[\begin{split}\begin{pmatrix} \rho \\ \rho \mathbf u \end{pmatrix}_t + \nabla\cdot \begin{bmatrix} \rho \mathbf u \\ \rho \mathbf u \otimes \mathbf u + \rho a^2 I \end{bmatrix} = 0 \end{split}\]

Linearization#

Split each state variable into a mean state and a small fluctuation

  • \(\rho = \bar\rho + \tilde\rho\)

  • \(u = \bar u + \tilde u\)

Let \(\widetilde{\rho u} = (\bar\rho + \tilde\rho) (\bar u + \tilde u) - \bar\rho\bar u \approx \tilde \rho \bar u + \bar\rho \tilde u\), where we have dropped the second order term \(\tilde \rho\tilde u\) because both are assumed small.

We consider background state \(\bar u = 0\) and constant \(\bar\rho(x,y,t) = \bar\rho\). Then

\[\begin{split}\begin{pmatrix} \tilde \rho \\ \bar\rho \mathbf{\tilde u} \end{pmatrix}_t + \nabla\cdot \begin{bmatrix} \bar\rho \mathbf{\tilde u} \\ \tilde \rho a^2 I \end{bmatrix} = 0 \end{split}\]

Two forms of acoustic wave equation#

Divide the momentum equation through by background density and dropping the tildes yields the standard form.

\[\begin{split}\begin{pmatrix} \rho \\ \mathbf u \end{pmatrix}_t + \nabla\cdot \begin{bmatrix} \bar\rho \mathbf u \\ \rho \frac{a^2}{\bar\rho} I \end{bmatrix} = 0 .\end{split}\]

Examine second equation

\[ \frac{a^2}{\bar\rho} \nabla\cdot\big[ \rho I \big] = \frac{a^2}{\bar\rho} \nabla \rho \]
and thus $$\begin{pmatrix} \rho \ \mathbf u \end{pmatrix}_t +

(24)#\[\begin{bmatrix} & \bar\rho \nabla\cdot \\ \frac{a^2}{\bar\rho} \nabla & \\ \end{bmatrix}\]
(25)#\[\begin{pmatrix} \rho \\ \mathbf u \end{pmatrix}\]

Let’s differentiate the first equation,

\[ \rho_{tt} + \bar\rho\nabla\cdot(\mathbf u_t) = 0\]
and substitute in the second equation
\[ \rho_{tt} = a^2 \nabla\cdot(\nabla \rho)\]

  • Note: we had to assume these derivatives exist!

We can reduce this to a first order system as

\[\begin{split}\begin{pmatrix} \rho \\ \dot \rho \end{pmatrix}_t + \begin{bmatrix} & -I \\ -a^2 \nabla\cdot\nabla & \end{bmatrix} \begin{pmatrix} \rho \\ \dot\rho \end{pmatrix} = 0\end{split}\]

Question#

  • How is the problem size different?

  • What might we be concerned about in choosing the second formulation?

Laplacian in periodic domain#

function laplacian_matrix(n)
    h = 2 / n
    rows = Vector{Int64}()
    cols = Vector{Int64}()
    vals = Vector{Float64}()
    wrap(i) = (i + n - 1) % n + 1
    idx(i, j) = (wrap(i)-1)*n + wrap(j)
    stencil_diffuse = [-1, -1, 4, -1, -1] / h^2
    for i in 1:n
        for j in 1:n
            append!(rows, repeat([idx(i,j)], 5))
            append!(cols, [idx(i-1,j), idx(i,j-1), idx(i,j), idx(i+1,j), idx(i,j+1)])
            append!(vals, stencil_diffuse)
        end
    end
    sparse(rows, cols, vals)
end
cond(Matrix(laplacian_matrix(5)))
2.9959163385932148e16
L = laplacian_matrix(10)
ev = eigvals(Matrix(L))
scatter(real(ev), imag(ev))
../_images/2022-10-05-factorization_19_0.svg

Wave operator#

\[\begin{split}\begin{pmatrix} \rho \\ \dot \rho \end{pmatrix}_t = \begin{bmatrix} & I \\ a^2 \nabla\cdot\nabla & \end{bmatrix} \begin{pmatrix} \rho \\ \dot\rho \end{pmatrix}\end{split}\]
function wave_matrix(n; a=1)
    Z = spzeros(n^2, n^2)
    L = laplacian_matrix(n)
    [Z I; -a^2*L Z]
end
wave_matrix(2)
8×8 SparseMatrixCSC{Float64, Int64} with 16 stored entries:
   ⋅     ⋅     ⋅     ⋅   1.0   ⋅    ⋅    ⋅ 
   ⋅     ⋅     ⋅     ⋅    ⋅   1.0   ⋅    ⋅ 
   ⋅     ⋅     ⋅     ⋅    ⋅    ⋅   1.0   ⋅ 
   ⋅     ⋅     ⋅     ⋅    ⋅    ⋅    ⋅   1.0
 -4.0   2.0   2.0    ⋅    ⋅    ⋅    ⋅    ⋅ 
  2.0  -4.0    ⋅    2.0   ⋅    ⋅    ⋅    ⋅ 
  2.0    ⋅   -4.0   2.0   ⋅    ⋅    ⋅    ⋅ 
   ⋅    2.0   2.0  -4.0   ⋅    ⋅    ⋅    ⋅ 
A = wave_matrix(8; a=2) * .1
ev = eigvals(Matrix(A))
plot_stability(z -> rk_stability(z, rk4), "RK4", xlims=(-4, 4), ylims=(-4, 4))
scatter!(real(ev), imag(ev), color=:black)
../_images/2022-10-05-factorization_22_0.svg

Question: would forward Euler work?#

Example 2D wave solver with RK4#

n = 20
A = wave_matrix(n)
x = LinRange(-1, 1, n+1)[1:end-1]
y = x
rho0 = vec(exp.(-9*((x .+ 1e-4).^2 .+ y'.^2)))
sol0 = vcat(rho0, zero(rho0))
thist, solhist = ode_rk_explicit((t, sol) -> A * sol, sol0, h=.02)
size(solhist)
(800, 51)
@gif for tstep in 1:length(thist)
    rho = solhist[1:n^2, tstep]
    contour(x, y, reshape(rho, n, n), title="\$ t = $(thist[tstep])\$")
end
┌ Info: Saved animation to 
│   fn = /home/jed/cu/numpde/slides/tmp.gif
└ @ Plots /home/jed/.julia/packages/Plots/lW9ll/src/animation.jl:137