2022-10-03 2D Advection-diffusion and waves#

Last time#

  • Implicit Runge-Kutta methods

  • Exploring/discussing tradeoffs

  • SciML benchmarks suite (DifferentialEquations.jl)

Today#

  • FD methods in 2D

  • Cost profile

  • The need for fast algebraic solvers

  • Wave equation and Hamiltonians

  • Symplectic integrators

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)

Extending advection-diffusion to 2D#

1 dimension#

(20)#\[\begin{align} u_t + (- \kappa u_x + w u)_x &= f(x) & \text{ on } \Omega &= (a,b) \\ u(a) &= g_D(a) & u'(b) &= g_N(b) \end{align}\]
  • Cell Peclet number \(\mathrm{Pe}_h = \frac{\lvert w \rvert h}{\kappa}\)

    • \(\mathrm{Pe}_h \lesssim 1\) avoids oscillations

    • \(\mathrm{Pe}_h \gtrsim 1\) is non-stiff for time-dependent model

  • Centered versus upwind for advection

  • Need uniformly bounded \(\kappa \ge \epsilon > 0\)

  • “Strong form” not defined at discontinuities in \(\kappa\)

    • Works okay using divergence form and fluxes at staggered points

2 dimensions#

(21)#\[\begin{align} u_t + \nabla\cdot\big(- \kappa \nabla u + \mathbf{w} u \big) &= f(x,y) & \text{ on } \Omega & \subset \mathbb R^2 \\ u|_{\Gamma_D} &= g_D(x,y) \\ (-\kappa \nabla u + \mathbf w u) \cdot \hat n|_{\Gamma_N} &= g_N(x,y) \end{align}\]
  • \(\Omega\) is some well-connected open set (we will assume simply connected) and the Dirichlet boundary \(\Gamma_D \subset \partial \Omega\) is nonempty.

  • Finite difference methods don’t have an elegant/flexible way to specify boundaries

  • We’ll choose \(\Omega = (-1, 1) \times (-1, 1)\)

On finite difference grids#

  • Non-uniform grids can mesh “special” domains

    • Rare in 3D; overset grids, immersed boundary methods

  • Concept of staggering is complicated/ambiguous

Wesseling 11.4: A boundary-fitted grid around an airfoil.

Time-independent advection-diffusion#

Advection#

\[ \nabla\cdot(\mathbf w u) = \mathbf w \cdot \nabla u + (\nabla\cdot \mathbf w) u\]

If we choose divergence-free \(\mathbf w\), we can use the stencil

\[\begin{split} \mathbf w \cdot \nabla u \approx \frac 1 h \begin{bmatrix} & w_2 & \\ -w_1 & & w_1 \\ & -w_2 & \end{bmatrix} \!:\! \begin{bmatrix} u_{i-1, j+1} & u_{i, j+1} & u_{i+1,j+1} \\ u_{i-1, j} & u_{i, j} & u_{i+1,j} \\ u_{i-1, j-1} & u_{i, j-1} & u_{i+1,j-1} \\ \end{bmatrix} \end{split}\]

Diffusion#

\[ -\nabla\cdot(\kappa \nabla u) = -\kappa \nabla\cdot \nabla u - \nabla\kappa\cdot \nabla u\]
  • When would you trust this decomposition?

  • If we have constant \(\kappa\), we can write

    \[\begin{split} -\kappa \nabla\cdot \nabla u \approx \frac{\kappa}{h^2} \begin{bmatrix} & -1 & \\ -1 & 4 & -1 \\ & -1 & \end{bmatrix} \!:\! \begin{bmatrix} u_{i-1, j+1} & u_{i, j+1} & u_{i+1,j+1} \\ u_{i-1, j} & u_{i, j} & u_{i+1,j} \\ u_{i-1, j-1} & u_{i, j-1} & u_{i+1,j-1} \\ \end{bmatrix} \end{split}\]

Advection-diffusion in code#

function advdiff_matrix(n; kappa=1, wind=[1, 1]/sqrt(2))
    h = 2 / n
    rows = Vector{Int64}()
    cols = Vector{Int64}()
    vals = Vector{Float64}()
    idx(i, j) = (i-1)*n + j
    stencil_advect = [-wind[1], -wind[2], 0, wind[1], wind[2]] / h
    stencil_diffuse = [-1, -1, 4, -1, -1] * kappa / h^2
    for i in 1:n
        for j in 1:n
            if i in [1, n] || j in [1, n]
                push!(rows, idx(i, j))
                push!(cols, idx(i, j))
                push!(vals, 1.)
            else
                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_advect + stencil_diffuse)
            end
        end
    end
    sparse(rows, cols, vals)
end
advdiff_matrix (generic function with 1 method)

Spy the matrix#

A = advdiff_matrix(6, wind=[1, 0], kappa=.001)
my_spy(A)
../_images/2022-10-03-2d_15_0.svg
A = advdiff_matrix(20, wind=[2, 1], kappa=.1)
ev = eigvals(Matrix(-A))
scatter(real(ev), imag(ev))
scatter!([0], [0], color=:black, label=:none, marker=:plus)
../_images/2022-10-03-2d_16_0.svg

Plot a solution#

n = 50
x = LinRange(-1, 1, n)
y = x
f = cos.(pi*x/2) * cos.(pi*y/2)'
heatmap(x, y, f, aspect_ratio=:equal)
../_images/2022-10-03-2d_18_0.svg
A = advdiff_matrix(n, wind=[2,1], kappa=.001)
u = A \ vec(f)
heatmap(x, y, reshape(u, n, n), aspect_ratio=:equal)
../_images/2022-10-03-2d_19_0.svg
  • What happens when advection dominates?

  • As you refine the grid?

Cost breadown and optimization#

using ProfileSVG
function assemble_and_solve(n)
    A = advdiff_matrix(n)
    x = LinRange(-1, 1, n)
    f = cos.(pi*x/2) * cos.(pi*x/2)'
    u = A \ vec(f)
end

@profview assemble_and_solve(400)
Profile results in :-1 #15 in task.jl:484 eventloop in eventloop.jl:8 invokelatest in essentials.jl:726 #invokelatest#2 in essentials.jl:729 execute_request in execute_request.jl:67 softscope_include_string in SoftGlobalScope.jl:65 include_string in loading.jl:1428 eval in boot.jl:368 typeinf_ext_toplevel in typeinfer.jl:996 typeinf_ext_toplevel in typeinfer.jl:1000 typeinf_ext in typeinfer.jl:967 typeinf in typeinfer.jl:213 _typeinf in typeinfer.jl:230 typeinf_nocycle in abstractinterpretation.jl:2462 typeinf_local in abstractinterpretation.jl:2340 abstract_eval_statement in abstractinterpretation.jl:1890 abstract_call in abstractinterpretation.jl:1733 abstract_call in abstractinterpretation.jl:1766 abstract_call_known in abstractinterpretation.jl:1696 abstract_call_gf_by_type in abstractinterpretation.jl:162 abstract_call_method_with_const_args in abstractinterpretation.jl:850 typeinf in typeinfer.jl:213 _typeinf in typeinfer.jl:230 typeinf_nocycle in abstractinterpretation.jl:2462 typeinf_local in abstractinterpretation.jl:2366 abstract_eval_statement in abstractinterpretation.jl:1890 abstract_call in abstractinterpretation.jl:1733 abstract_call in abstractinterpretation.jl:1766 abstract_call_known in abstractinterpretation.jl:1696 abstract_call_gf_by_type in abstractinterpretation.jl:162 abstract_call_method_with_const_args in abstractinterpretation.jl:850 typeinf in typeinfer.jl:213 _typeinf in typeinfer.jl:230 typeinf_nocycle in abstractinterpretation.jl:2462 typeinf_local in abstractinterpretation.jl:2366 abstract_eval_statement in abstractinterpretation.jl:1890 abstract_call in abstractinterpretation.jl:1733 abstract_call in abstractinterpretation.jl:1766 abstract_call_known in abstractinterpretation.jl:1696 abstract_call_gf_by_type in abstractinterpretation.jl:162 abstract_call_method_with_const_args in abstractinterpretation.jl:850 typeinf in typeinfer.jl:213 _typeinf in typeinfer.jl:230 typeinf_nocycle in abstractinterpretation.jl:2462 typeinf_local in abstractinterpretation.jl:2366 abstract_eval_statement in abstractinterpretation.jl:1890 abstract_call in abstractinterpretation.jl:1733 abstract_call in abstractinterpretation.jl:1766 abstract_call_known in abstractinterpretation.jl:1696 _typeinf in typeinfer.jl:257 optimize in optimize.jl:504 run_passes in optimize.jl:539 ssa_inlining_pass! in inlining.jl:82 assemble_inline_todo! in inlining.jl:1359 process_simple! in inlining.jl:1120 call_sig in inlining.jl:961 typeinf_local in abstractinterpretation.jl:2366 abstract_eval_statement in abstractinterpretation.jl:1872 Box in boot.jl:377 abstract_eval_statement in abstractinterpretation.jl:1890 abstract_call in abstractinterpretation.jl:1733 abstract_call in abstractinterpretation.jl:1766 abstract_call_known in abstractinterpretation.jl:1696 abstract_call_gf_by_type in abstractinterpretation.jl:162 abstract_call_method_with_const_args in abstractinterpretation.jl:850 typeinf in typeinfer.jl:213 _typeinf in typeinfer.jl:230 typeinf_nocycle in abstractinterpretation.jl:2462 typeinf_local in abstractinterpretation.jl:2366 abstract_eval_statement in abstractinterpretation.jl:1890 abstract_call in abstractinterpretation.jl:1733 abstract_call in abstractinterpretation.jl:1766 abstract_call_known in abstractinterpretation.jl:1696 abstract_call_gf_by_type in abstractinterpretation.jl:162 abstract_call_method_with_const_args in abstractinterpretation.jl:850 typeinf in typeinfer.jl:213 _typeinf in typeinfer.jl:230 typeinf_nocycle in abstractinterpretation.jl:2462 typeinf_local in abstractinterpretation.jl:2340 abstract_eval_statement in abstractinterpretation.jl:1890 abstract_call in abstractinterpretation.jl:1733 abstract_call in abstractinterpretation.jl:1766 abstract_call_known in abstractinterpretation.jl:1600 abstract_apply in abstractinterpretation.jl:1339 abstract_call in abstractinterpretation.jl:1766 abstract_call_known in abstractinterpretation.jl:1696 abstract_call_gf_by_type in abstractinterpretation.jl:162 abstract_call_method_with_const_args in abstractinterpretation.jl:850 typeinf in typeinfer.jl:213 _typeinf in typeinfer.jl:230 typeinf_nocycle in abstractinterpretation.jl:2462 typeinf_local in abstractinterpretation.jl:2366 abstract_eval_statement in abstractinterpretation.jl:1890 typeinf_local in abstractinterpretation.jl:2366 abstract_eval_statement in abstractinterpretation.jl:1890 abstract_call in abstractinterpretation.jl:1733 abstract_call in abstractinterpretation.jl:1766 abstract_call_known in abstractinterpretation.jl:1696 abstract_call_gf_by_type in abstractinterpretation.jl:162 abstract_call_method_with_const_args in abstractinterpretation.jl:850 typeinf in typeinfer.jl:213 _typeinf in typeinfer.jl:230 typeinf_nocycle in abstractinterpretation.jl:2462 typeinf_local in abstractinterpretation.jl:2366 abstract_eval_statement in abstractinterpretation.jl:1890 abstract_call in abstractinterpretation.jl:1733 abstract_call in abstractinterpretation.jl:1766 abstract_call_known in abstractinterpretation.jl:1696 _typeinf in typeinfer.jl:277 cache_result! in typeinfer.jl:395 transform_result_for_cache in typeinfer.jl:369 maybe_compress_codeinfo in typeinfer.jl:349 assemble_and_solve in In[71]:3 advdiff_matrix in In[2]:1 #advdiff_matrix#1 in In[2]:11 vect in array.jl:126 _array_for in array.jl:679 _array_for in array.jl:676 similar in abstractarray.jl:840 similar in abstractarray.jl:841 Array in boot.jl:468 Array in boot.jl:459 umfpack_free_symbolic in umfpack.jl:742 umfpack_free_numeric in umfpack.jl:754 umfpack_dl_free_numeric in x86_64-linux-gnu.jl:1800 #advdiff_matrix#1 in In[2]:16 repeat in abstractarraymath.jl:356 repeat##kw in abstractarraymath.jl:398 #repeat#1 in abstractarraymath.jl:401 repeat_inner_outer in abstractarraymath.jl:459 repeat_outer in abstractarraymath.jl:479 similar in array.jl:377 Array in boot.jl:459 repeat_outer in abstractarraymath.jl:482 setindex! in array.jl:993 unsafe_copyto! in array.jl:289 append! in array.jl:1108 _growend! in array.jl:1011 vect in array.jl:126 _array_for in array.jl:679 _array_for in array.jl:676 similar in abstractarray.jl:840 similar in abstractarray.jl:841 Array in boot.jl:468 Array in boot.jl:459 #advdiff_matrix#1 in In[2]:17 append! in array.jl:1108 _growend! in array.jl:1011 append! in array.jl:1109 copyto! in array.jl:322 _copyto_impl! in array.jl:331 unsafe_copyto! in array.jl:289 vect in array.jl:126 _array_for in array.jl:679 _array_for in array.jl:676 similar in abstractarray.jl:840 similar in abstractarray.jl:841 Array in boot.jl:468 Array in boot.jl:459 setindex! in array.jl:966 #advdiff_matrix#1 in In[2]:18 append! in array.jl:1108 _growend! in array.jl:1011 append! in array.jl:1109 copyto! in array.jl:322 _copyto_impl! in array.jl:331 unsafe_copyto! in array.jl:289 + in arraymath.jl:16 broadcast_preserving_zero_d in broadcast.jl:849 materialize in broadcast.jl:860 copy in broadcast.jl:885 copyto! in broadcast.jl:913 copyto! in broadcast.jl:960 macro expansion in simdloop.jl:77 macro expansion in broadcast.jl:961 setindex! in array.jl:966 similar in broadcast.jl:211 similar in broadcast.jl:212 similar in abstractarray.jl:840 similar in abstractarray.jl:841 Array in boot.jl:476 Array in boot.jl:468 Array in boot.jl:459 #advdiff_matrix#1 in In[2]:22 sparse in sparsematrix.jl:1041 dimlub in sparsematrix.jl:1037 maximum in reducedim.jl:994 #maximum#780 in reducedim.jl:994 _maximum in reducedim.jl:998 #_maximum#782 in reducedim.jl:998 _maximum in reducedim.jl:999 #_maximum#783 in reducedim.jl:999 mapreduce in reducedim.jl:357 #mapreduce#765 in reducedim.jl:357 _mapreduce_dim in reducedim.jl:365 _mapreduce in reduce.jl:442 mapreduce_impl in reduce.jl:646 _fast in reduce.jl:617 max in promotion.jl:488 ifelse in essentials.jl:489 sparse in sparsematrix.jl:1045 sparse in sparsematrix.jl:849 Array in boot.jl:459 sparse in sparsematrix.jl:856 sparse! in sparsematrix.jl:931 iterate in range.jl:883 == in promotion.jl:477 sparse! in sparsematrix.jl:953 getindex in array.jl:924 sparse! in sparsematrix.jl:954 iterate in range.jl:883 == in promotion.jl:477 sparse! in sparsematrix.jl:972 &lt; in int.jl:83 sparse! in sparsematrix.jl:1003 resize! in array.jl:1236 _growend! in array.jl:1011 sparse! in sparsematrix.jl:1014 setindex! in array.jl:966 sparse! in sparsematrix.jl:1015 setindex! in array.jl:966 assemble_and_solve in In[71]:6 \ in linalg.jl:1561 ishermitian in sparsematrix.jl:3520 is_hermsym in sparsematrix.jl:3529 copy in array.jl:369 \ in linalg.jl:1564 \ in factorization.jl:105 ldiv! in umfpack.jl:676 ldiv! in umfpack.jl:689 _Aq_ldiv_B! in umfpack.jl:706 _AqldivB_kernel! in umfpack.jl:711 solve! in umfpack.jl:417 umfpack_dl_solve in x86_64-linux-gnu.jl:1768 lu in umfpack.jl:195 #lu#1 in umfpack.jl:0 #lu#1 in umfpack.jl:197 copy in array.jl:369 decrement in SuiteSparse.jl:18 decrement! in SuiteSparse.jl:15 #lu#1 in umfpack.jl:202 umfpack_numeric! in umfpack.jl:381 #umfpack_numeric!#14 in umfpack.jl:383 umfpack_symbolic! in umfpack.jl:369 umfpack_dl_symbolic in x86_64-linux-gnu.jl:1736 #umfpack_numeric!#14 in umfpack.jl:385 umfpack_dl_numeric in x86_64-linux-gnu.jl:1752
┌ Warning: The depth of this graph is 96, exceeding the `maxdepth` (=50).
│ The deeper frames will be truncated.
└ @ ProfileSVG /home/jed/.julia/packages/ProfileSVG/ecSyU/src/ProfileSVG.jl:275
┌ Warning: The depth of this graph is 96, exceeding the `maxdepth` (=50).
│ The deeper frames will be truncated.
└ @ ProfileSVG /home/jed/.julia/packages/ProfileSVG/ecSyU/src/ProfileSVG.jl:275

What’s left?#

  • Symmetric Dirichlet boundary conditions

  • Symmetric Neumann boundary conditions

  • Verification with method of manufactured solutions

  • Non-uniform grids

  • Upwinding for advection-dominated problems

  • Variable coefficients

  • Time-dependent problems

  • Fast algebraic solvers

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#

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

contour(LinRange(0, 2, 30), LinRange(0, 2, 30), pressure, xlabel="\$\\rho\$", ylabel="\$T\$")
../_images/2022-10-03-2d_27_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

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