Contents

2021-10-15 Sparse Linear Algebra

Last time

  • Note on p-Laplacian activity

  • Weak forms and jump conditions

  • 2D wave equations via gas dynamics

  • Appropriate time integrators

  • Hamiltonian structure and energy drift

Today

  • Recap and questions

  • Sparse direct solvers

    • matrix orderings

    • impact on formulation

    • cost scaling

  • Why iterative solvers

  • Discussion with Ioana Fleming

using Plots
using LinearAlgebra
using SparseArrays

default(linewidth=4)

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

function laplacian_matrix(n)
    "Laplacian in n×n periodic domain"
    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

laplacian_matrix (generic function with 1 method)

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?

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/2021-10-15-sparse-linalg_10_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/1RWWg/src/animation.jl:114

Accuracy, conservation of mass with RK4

thist, solhist = ode_rk_explicit((t, sol) -> A * sol, sol0, h=.05,
    tfinal=1)

tfinal = thist[end]
M = exp(Matrix(A*tfinal))
sol_exact = M * sol0
sol_final = solhist[:, end]
norm(sol_final - sol_exact)

0.02015111748435016

mass = vec(sum(solhist[1:n^2, :], dims=1))
plot(thist[2:end], mass[2:end] .- mass[1])
../_images/2021-10-15-sparse-linalg_17_0.svg

Conservation of energy with RK4

Hamiltonian structure

We can express the total energy for our system as a sum of kinetic and potential energy.

\[H(\rho, \dot\rho) = \underbrace{\frac 1 2 \int_\Omega (\dot\rho)^2}_{\text{kinetic}} + \underbrace{\frac{a^2}{2} \int_\Omega \lVert \nabla \rho \rVert^2}_{\text{potential}}\]

where we identify \(\rho\) as a generalized position and \(\dot\rho\) as generalized momentum. Hamilton’s equations state that the equations of motion are

\[\begin{split} \begin{pmatrix} \rho \\ \dot\rho \end{pmatrix}_t = \begin{bmatrix} \frac{\partial H}{\partial \dot\rho} \\ -\frac{\partial H}{\partial \rho} \end{bmatrix} = \begin{bmatrix} \dot\rho \\ - a^2 L \rho \end{bmatrix} \end{split}\]

where we have used the weak form to associate \(\int \nabla v \cdot \nabla u = v^T L u\).

function energy(sol, n)
    L = laplacian_matrix(n)
    rho = sol[1:end÷2]
    rhodot = sol[end÷2+1:end]
    kinetic = .5 * norm(rhodot)^2
    potential = .5 * rho' * L * rho
    kinetic + potential
end
ehist = [energy(solhist[:,i], n) for i in 1:length(thist)]
plot(thist, ehist)
../_images/2021-10-15-sparse-linalg_20_0.svg

Velocity Verlet integrator

function wave_verlet(n, u0; tfinal=1., h=0.1)
    L = laplacian_matrix(n)
    u = copy(u0)
    t = 0.
    thist = [t]
    uhist = [u0]
    irho = 1:n^2
    irhodot = n^2+1:2*n^2
    accel = -L * u[irho]
    while t < tfinal
        tnext = min(t+h, tfinal)
        h = tnext - t
        u[irho] += h * u[irhodot] + h^2/2 * accel
        accel_next = -L * u[irho]
        u[irhodot] += h/2 * (accel + accel_next)
        accel = accel_next
        t = tnext
        push!(thist, t)
        push!(uhist, copy(u))
    end
    thist, hcat(uhist...)
end

wave_verlet (generic function with 1 method)

thist, solhist = wave_verlet(n, sol0, h=.04)
@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/1RWWg/src/animation.jl:114