Brusselator Work-Precision Diagrams
using OrdinaryDiffEq, DiffEqDevTools, Sundials, ParameterizedFunctions, Plots,
ODEInterfaceDiffEq, LSODA, SparseArrays, LinearSolve,
LinearAlgebra, IncompleteLU, AlgebraicMultigrid, Symbolics, ModelingToolkit,
RecursiveFactorization
gr()
const N = 8
xyd_brusselator = range(0,stop=1,length=N)
brusselator_f(x, y, t) = (((x-0.3)^2 + (y-0.6)^2) <= 0.1^2) * (t >= 1.1) * 5.
limit(a, N) = a == N+1 ? 1 : a == 0 ? N : a
function brusselator_2d_loop(du, u, p, t)
A, B, alpha, dx = p
alpha = alpha/dx^2
@inbounds for I in CartesianIndices((N, N))
i, j = Tuple(I)
x, y = xyd_brusselator[I[1]], xyd_brusselator[I[2]]
ip1, im1, jp1, jm1 = limit(i+1, N), limit(i-1, N), limit(j+1, N), limit(j-1, N)
du[i,j,1] = alpha*(u[im1,j,1] + u[ip1,j,1] + u[i,jp1,1] + u[i,jm1,1] - 4u[i,j,1]) +
B + u[i,j,1]^2*u[i,j,2] - (A + 1)*u[i,j,1] + brusselator_f(x, y, t)
du[i,j,2] = alpha*(u[im1,j,2] + u[ip1,j,2] + u[i,jp1,2] + u[i,jm1,2] - 4u[i,j,2]) +
A*u[i,j,1] - u[i,j,1]^2*u[i,j,2]
end
end
p = (3.4, 1., 10., step(xyd_brusselator))
input = rand(N,N,2)
output = similar(input)
sparsity_pattern = Symbolics.jacobian_sparsity(brusselator_2d_loop,output,input,p,0.0)
jac_sparsity = Float64.(sparse(sparsity_pattern))
f = ODEFunction{true, SciMLBase.FullSpecialize}(brusselator_2d_loop;jac_prototype=jac_sparsity)
function init_brusselator_2d(xyd)
N = length(xyd)
u = zeros(N, N, 2)
for I in CartesianIndices((N, N))
x = xyd[I[1]]
y = xyd[I[2]]
u[I,1] = 22*(y*(1-y))^(3/2)
u[I,2] = 27*(x*(1-x))^(3/2)
end
u
end
u0 = init_brusselator_2d(xyd_brusselator)
prob = ODEProblem(f,u0,(0.,11.5),p);prob_mtk = ODEProblem(complete(modelingtoolkitize(prob)),[],(0.0,11.5),jac=true,sparse=true);Also comparing with MethodOfLines.jl:
using MethodOfLines, DomainSets
@parameters x y t
@variables u(..) v(..)
Dt = Differential(t)
Dx = Differential(x)
Dy = Differential(y)
Dxx = Differential(x)^2
Dyy = Differential(y)^2
∇²(u) = Dxx(u) + Dyy(u)
brusselator_f(x, y, t) = (((x-0.3)^2 + (y-0.6)^2) <= 0.1^2) * (t >= 1.1) * 5.
x_min = y_min = t_min = 0.0
x_max = y_max = 1.0
t_max = 11.5
α = 10.
u0_mol(x,y,t) = 22(y*(1-y))^(3/2)
v0_mol(x,y,t) = 27(x*(1-x))^(3/2)
eq = [Dt(u(x,y,t)) ~ 1. + v(x,y,t)*u(x,y,t)^2 - 4.4*u(x,y,t) + α*∇²(u(x,y,t)) + brusselator_f(x, y, t),
Dt(v(x,y,t)) ~ 3.4*u(x,y,t) - v(x,y,t)*u(x,y,t)^2 + α*∇²(v(x,y,t))]
domains = [x ∈ Interval(x_min, x_max),
y ∈ Interval(y_min, y_max),
t ∈ Interval(t_min, t_max)]
bcs = [u(x,y,0) ~ u0_mol(x,y,0),
u(0,y,t) ~ u(1,y,t),
u(x,0,t) ~ u(x,1,t),
v(x,y,0) ~ v0_mol(x,y,0),
v(0,y,t) ~ v(1,y,t),
v(x,0,t) ~ v(x,1,t)]
@named pdesys = PDESystem(eq,bcs,domains,[x,y,t],[u(x,y,t),v(x,y,t)])
# Method of lines discretization
dx = 1/N
dy = 1/N
order = 2
discretization = MOLFiniteDifference([x=>dx, y=>dy], t; approx_order = order, jac = true, sparse = true, wrap = Val(false))
# Convert the PDE system into an ODE problem
prob_mol = discretize(pdesys,discretization)ODEProblem with uType Vector{Float64} and tType Float64. In-place: true
timespan: (0.0, 11.5)
u0: 128-element Vector{Float64}:
0.7957923865311465
0.7957923865311465
0.7957923865311465
0.7957923865311465
0.7957923865311465
0.7957923865311465
0.7957923865311465
0.7957923865311465
1.7861773953054048
1.7861773953054048
⋮
0.0
0.9766542925609525
2.1921268033293604
3.063590342214851
3.3750000000000004
3.063590342214851
2.1921268033293604
0.9766542925609525
0.0using Base.Experimental: Const, @aliasscope
macro vp(expr)
nodes = (Symbol("llvm.loop.vectorize.predicate.enable"), 1)
if expr.head != :for
error("Syntax error: loopinfo needs a for loop")
end
push!(expr.args[2].args, Expr(:loopinfo, nodes))
return esc(expr)
end
struct Brusselator2DLoop <: Function
N::Int
s::Float64
end
function (b::Brusselator2DLoop)(du, unc, p, t)
N = b.N
s = b.s
A, B, alpha, dx = p
alpha = alpha/abs2(dx)
u = Base.Experimental.Const(unc)
Base.Experimental.@aliasscope begin
@inbounds @fastmath begin
b = ((abs2(-0.3) + abs2(-0.6)) <= abs2(0.1)) * (t >= 1.1) * 5.0
du1 = alpha*(u[N,1,1] + u[2,1,1] + u[1,2,1] + u[1,N,1] - 4u[1,1,1]) +
B + abs2(u[1,1,1])*u[1,1,2] - (A + 1)*u[1,1,1] + b
du2 = alpha*(u[N,1,2] + u[2,1,2] + u[1,2,2] + u[1,N,2] - 4u[1,1,2]) +
A*u[1,1,1] - abs2(u[1,1,1])*u[1,1,2]
du[1,1,1] = du1
du[1,1,2] = du2
@vp for i = 2:N-1
x = (i-1)*s
ip1 = i+1
im1 = i-1
b = ((abs2(x-0.3) + abs2(-0.6)) <= abs2(0.1)) * (t >= 1.1) * 5.0
du1 = alpha*(u[im1,1,1] + u[ip1,1,1] + u[i,2,1] + u[i,N,1] - 4u[i,1,1]) +
B + abs2(u[i,1,1])*u[i,1,2] - (A + 1)*u[i,1,1] + b
du2 = alpha*(u[im1,1,2] + u[ip1,1,2] + u[i,2,2] + u[i,N,2] - 4u[i,1,2]) +
A*u[i,1,1] - abs2(u[i,1,1])*u[i,1,2]
du[i,1,1] = du1
du[i,1,2] = du2
end
b = ((abs2(0.7) + abs2(-0.6)) <= abs2(0.1)) * (t >= 1.1) * 5.0
du1 = alpha*(u[N-1,1,1] + u[1,1,1] + u[N,2,1] + u[N,N,1] - 4u[N,1,1]) +
B + abs2(u[N,1,1])*u[N,1,2] - (A + 1)*u[N,1,1] + b
du2 = alpha*(u[N-1,1,2] + u[1,1,2] + u[N,2,2] + u[N,N,2] - 4u[N,1,2]) +
A*u[N,1,1] - abs2(u[N,1,1])*u[N,1,2]
du[N,1,1] = du1
du[N,1,2] = du2
for j = 2:N-1
y = (j-1)*s
jp1 = j+1
jm1 = j-1
b0 = ((abs2(-0.3) + abs2(y-0.6)) <= abs2(0.1)) * (t >= 1.1) * 5.0
du[1,j,1] = alpha*(u[N,j,1] + u[2,j,1] + u[1,jp1,1] + u[1,jm1,1] - 4u[1,j,1]) +
B + abs2(u[1,j,1])*u[1,j,2] - (A + 1)*u[1,j,1] + b0
du[1,j,2] = alpha*(u[N,j,2] + u[2,j,2] + u[1,jp1,2] + u[1,jm1,2] - 4u[1,j,2]) +
A*u[1,j,1] - abs2(u[1,j,1])*u[1,j,2]
@vp for i = 2:N-1
x = (i-1)*s
b = ((abs2(x-0.3) + abs2(y-0.6)) <= abs2(0.1)) * (t >= 1.1) * 5.0
du1 = alpha*(u[i-1,j,1] + u[i+1,j,1] + u[i,jp1,1] + u[i,jm1,1] - 4u[i,j,1]) +
B + abs2(u[i,j,1])*u[i,j,2] - (A + 1)*u[i,j,1] + b
du2 = alpha*(u[i-1,j,2] + u[i+1,j,2] + u[i,jp1,2] + u[i,jm1,2] - 4u[i,j,2]) +
A*u[i,j,1] - abs2(u[i,j,1])*u[i,j,2]
du[i,j,1] = du1
du[i,j,2] = du2
end
bN = ((abs2(0.7) + abs2(y-0.6)) <= abs2(0.1)) * (t >= 1.1) * 5.0
du[N,j,1] = alpha*(u[N-1,j,1] + u[1,j,1] + u[N,jp1,1] + u[N,jm1,1] - 4u[N,j,1]) +
B + abs2(u[N,j,1])*u[N,j,2] - (A + 1)*u[N,j,1] + bN
du[N,j,2] = alpha*(u[N-1,j,2] + u[1,j,2] + u[N,jp1,2] + u[N,jm1,2] - 4u[N,j,2]) +
A*u[N,j,1] - abs2(u[N,j,1])*u[N,j,2]
end
b = ((abs2(-0.3) + abs2(0.4)) <= abs2(0.1)) * (t >= 1.1) * 5.0
du1 = alpha*(u[N,N,1] + u[2,N,1] + u[1,1,1] + u[1,N-1,1] - 4u[1,N,1]) +
B + abs2(u[1,N,1])*u[1,N,2] - (A + 1)*u[1,N,1] + b
du2 = alpha*(u[N,N,2] + u[2,N,2] + u[1,1,2] + u[1,N-1,2] - 4u[1,N,2]) +
A*u[1,N,1] - abs2(u[1,N,1])*u[1,N,2]
du[1,N,1] = du1
du[1,N,2] = du2
@vp for i = 2:N-1
x = (i-1)*s
ip1 = i+1
im1 = i-1
b = ((abs2(x-0.3) + abs2(0.4)) <= abs2(0.1)) * (t >= 1.1) * 5.0
du1 = alpha*(u[im1,N,1] + u[ip1,N,1] + u[i,1,1] + u[i,N-1,1] - 4u[i,N,1]) +
B + abs2(u[i,N,1])*u[i,N,2] - (A + 1)*u[i,N,1] + b
du2 = alpha*(u[im1,N,2] + u[ip1,N,2] + u[i,1,2] + u[i,N-1,2] - 4u[i,N,2]) +
A*u[i,N,1] - abs2(u[i,N,1])*u[i,N,2]
du[i,N,1] = du1
du[i,N,2] = du2
end
b = ((abs2(0.7) + abs2(0.4)) <= abs2(0.1)) * (t >= 1.1) * 5.0
du1 = alpha*(u[N-1,N,1] + u[1,N,1] + u[N,1,1] + u[N,N-1,1] - 4u[N,N,1]) +
B + abs2(u[N,N,1])*u[N,N,2] - (A + 1)*u[N,N,1] + b
du2 = alpha*(u[N-1,N,2] + u[1,N,2] + u[N,1,2] + u[N,N-1,2] - 4u[N,N,2]) +
A*u[N,N,1] - abs2(u[N,N,1])*u[N,N,2]
du[N,N,1] = du1
du[N,N,2] = du2
end
end
end
function fast_bruss(N)
xyd_brusselator = range(0,stop=1,length=N)
brusselator_2d_loop = Brusselator2DLoop(N,Float64(step(xyd_brusselator)))
p = (3.4, 1., 10., step(xyd_brusselator))
input = rand(N,N,2)
output = similar(input)
sparsity_pattern = Symbolics.jacobian_sparsity(brusselator_2d_loop,output,input,p,0.0)
jac_sparsity = Float64.(sparse(sparsity_pattern))
f = ODEFunction(brusselator_2d_loop;jac_prototype=jac_sparsity)
u0 = zeros(N, N, 2)
@inbounds for I in CartesianIndices((N, N))
x = xyd_brusselator[I[1]]
y = xyd_brusselator[I[2]]
u0[I,1] = 22*(y*(1-y))^(3/2)
u0[I,2] = 27*(x*(1-x))^(3/2)
end
return ODEProblem(f,u0,(0.,11.5),p)
end
fastprob = fast_bruss(N)ODEProblem with uType Array{Float64, 3} and tType Float64. In-place: true
timespan: (0.0, 11.5)
u0: 8×8×2 Array{Float64, 3}:
[:, :, 1] =
0.0 0.942661 2.02828 2.66625 2.66625 2.02828 0.942661 0.0
0.0 0.942661 2.02828 2.66625 2.66625 2.02828 0.942661 0.0
0.0 0.942661 2.02828 2.66625 2.66625 2.02828 0.942661 0.0
0.0 0.942661 2.02828 2.66625 2.66625 2.02828 0.942661 0.0
0.0 0.942661 2.02828 2.66625 2.66625 2.02828 0.942661 0.0
0.0 0.942661 2.02828 2.66625 2.66625 2.02828 0.942661 0.0
0.0 0.942661 2.02828 2.66625 2.66625 2.02828 0.942661 0.0
0.0 0.942661 2.02828 2.66625 2.66625 2.02828 0.942661 0.0
[:, :, 2] =
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
1.1569 1.1569 1.1569 1.1569 1.1569 1.1569 1.1569 1.1569
2.48926 2.48926 2.48926 2.48926 2.48926 2.48926 2.48926 2.48926
3.27221 3.27221 3.27221 3.27221 3.27221 3.27221 3.27221 3.27221
3.27221 3.27221 3.27221 3.27221 3.27221 3.27221 3.27221 3.27221
2.48926 2.48926 2.48926 2.48926 2.48926 2.48926 2.48926 2.48926
1.1569 1.1569 1.1569 1.1569 1.1569 1.1569 1.1569 1.1569
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0sol = solve(prob,CVODE_BDF(),abstol=1/10^14,reltol=1/10^14)
sol2 = solve(prob_mtk,CVODE_BDF(linear_solver = :KLU),abstol=1/10^14,reltol=1/10^14)
sol3 = solve(prob_mol,CVODE_BDF(linear_solver = :KLU),abstol=1/10^14,reltol=1/10^14,wrap=Val(false))retcode: Success
Interpolation: 3rd order Hermite
t: 7088-element Vector{Float64}:
0.0
1.4238870862469186e-15
1.4240294749555432e-11
1.5662900337424728e-10
2.9901771199893913e-10
5.160801983264086e-10
8.725711849204022e-10
1.4781897844618872e-9
2.519675048784877e-9
4.3190349346343956e-9
⋮
11.495686108181351
11.496274749829004
11.496863391476657
11.49745203312431
11.498040674771962
11.498629316419615
11.499217958067268
11.49980659971492
11.5
u: 7088-element Vector{Vector{Float64}}:
[0.7957923865311465, 0.7957923865311465, 0.7957923865311465, 0.79579238653
11465, 0.7957923865311465, 0.7957923865311465, 0.7957923865311465, 0.795792
3865311465, 1.7861773953054048, 1.7861773953054048 … 0.9766542925609525,
0.0, 0.9766542925609525, 2.1921268033293604, 3.063590342214851, 3.375000000
0000004, 3.063590342214851, 2.1921268033293604, 0.9766542925609525, 0.0]
[0.7957923865313211, 0.7957923865313222, 0.795792386531323, 0.795792386531
3233, 0.795792386531323, 0.7957923865313222, 0.7957923865313211, 0.79579238
65313202, 1.786177395305144, 1.7861773953051494 … 0.9766542925611731, 1.7
838787595843077e-12, 0.9766542925611701, 2.192126803329047, 3.0635903422143
41, 3.374999999999433, 3.063590342214341, 2.192126803329047, 0.976654292561
1701, 1.7800261566757715e-12]
[0.7957923882778083, 0.7957923882887697, 0.7957923882966287, 0.79579238829
94371, 0.7957923882966287, 0.7957923882887697, 0.7957923882778083, 0.795792
3882690007, 1.7861773926974722, 1.7861773927526943 … 0.9766542947672135,
1.7840571189693174e-8, 0.9766542947374913, 2.1921268001941354, 3.0635903371
106386, 3.374999994323757, 3.0635903371106386, 2.1921268001941354, 0.976654
2947374913, 1.7802041309561312e-8]
[0.7957924057426853, 0.7957924058632492, 0.7957924059496904, 0.79579240598
05794, 0.7957924059496904, 0.7957924058632492, 0.7957924057426853, 0.795792
4056458101, 1.786177366620756, 1.7861773672281442 … 0.9766543168276247, 1
.9622842530506597e-7, 0.9766543165007099, 2.1921267688450223, 3.06359028607
36164, 3.3749999375669977, 3.0635902860736164, 2.1921267688450223, 0.976654
3165007099, 1.958046352534055e-7]
[0.795792423207569, 0.7957924234377355, 0.7957924236027589, 0.795792423661
7286, 0.7957924236027589, 0.7957924234377355, 0.795792423207569, 0.79579242
30226263, 1.7861773405440422, 1.7861773417035964 … 0.9766543388880443, 3.
746162540951588e-7, 0.976654338263937, 2.192126737495912, 3.063590235036595
6, 3.3749998808102393, 3.0635902350365956, 2.192126737495912, 0.97665433826
3937, 3.738072040103176e-7]
[0.7957924498316827, 0.7957924502289312, 0.7957924505137483, 0.79579245061
55252, 0.7957924505137483, 0.7957924502289312, 0.7957924498316827, 0.795792
4495124866, 1.7861773007917632, 1.7861773027930594 … 0.9766543725177618,
6.465570447995342e-7, 0.9766543714406037, 2.1921266897062024, 3.06359015723
39215, 3.37499979428819, 3.0635901572339215, 2.1921266897062024, 0.97665437
14406037, 6.451606907088272e-7]
[0.7957924935576395, 0.795792494229294, 0.7957924947108534, 0.795792494882
9345, 0.7957924947108534, 0.795792494229294, 0.7957924935576395, 0.79579249
30179533, 1.7861772355049113, 1.7861772388886359 … 0.9766544277493292, 1.
0931768703959916e-6, 0.9766544259281074, 2.1921266112191384, 3.063590029455
275, 3.3749996521893415, 3.063590029455275, 2.1921266112191384, 0.976654425
9281074, 1.0908159626472516e-6]
[0.7957925678408412, 0.7957925689786661, 0.7957925697944583, 0.79579257008
59744, 0.7957925697944583, 0.7957925689786661, 0.7957925678408412, 0.795792
5669265793, 1.7861771245934985, 1.786177130325737 … 0.9766545215786651, 1
.8519088939775023e-6, 0.9766545184934051, 2.192126477882781, 3.063589812380
733, 3.3749994107871655, 3.063589812380733, 2.192126477882781, 0.9766545184
934051, 1.8479093730188316e-6]
[0.7957926955863502, 0.7957926975258519, 0.7957926989164267, 0.79579269941
33361, 0.7957926989164267, 0.7957926975258519, 0.7957926955863502, 0.795792
6940279286, 1.7861769338587068, 1.786176943629694 … 0.976654682937829, 3.
1567028571118565e-6, 0.9766546776788025, 2.1921262485837283, 3.063589439076
6394, 3.374998995646712, 3.0635894390766394, 2.1921262485837283, 0.97665467
76788025, 3.149885411590933e-6]
[0.7957929162914903, 0.795792919616041, 0.7957929219996626, 0.795792922851
4283, 0.7957929219996626, 0.795792919616041, 0.7957929162914903, 0.79579291
36201628, 1.7861766043291272, 1.786176621077798 … 0.9766549617170381, 5.4
10973999659674e-6, 0.9766549527024425, 2.19212585242728, 3.0635887941244206
, 3.374998278414355, 3.0635887941244206, 2.19212585242728, 0.97665495270244
25, 5.3992880839237515e-6]
⋮
[4.073542147517943, 4.073678256826177, 4.073678256826177, 4.07354214751794
25, 4.073369276100977, 4.073256699954195, 4.073256699954196, 4.073369276100
977, 4.073684991585881, 4.073870247312897 … 1.1851911298275786, 1.1851854
979836303, 1.185186253669876, 1.1851814266888845, 1.1851814266888845, 1.185
186253669876, 1.1851926200032414, 1.185196887133102, 1.185196887133102, 1.1
851926200032414]
[4.075236586856852, 4.075372679859942, 4.075372679859942, 4.07523658685685
2, 4.075063737468067, 4.07495117632641, 4.074951176326411, 4.07506373746806
7, 4.075379416824593, 4.075564653507394 … 1.1817785913918286, 1.181772981
8020342, 1.1817737345428994, 1.1817689266686777, 1.1817689266686775, 1.1817
737345428994, 1.1817800756316128, 1.1817843258181333, 1.1817843258181333, 1
.1817800756316128]
[4.076902965031779, 4.077039041822214, 4.077039041822214, 4.07690296503177
85, 4.076730137546133, 4.076617591324378, 4.076617591324378, 4.076730137546
133, 4.077045780979023, 4.077230998725429 … 1.1783931250209605, 1.1783875
375594715, 1.1783882873714766, 1.1783834984958645, 1.1783834984958643, 1.17
83882873714766, 1.1783946033585904, 1.178398836697857, 1.178398836697857, 1
.1783946033585904]
[4.0785414099893, 4.078677470659699, 4.078677470659699, 4.078541409989299,
4.078368604281576, 4.078256072894378, 4.078256072894378, 4.078368604281576
, 4.078684211995861, 4.078869410913847 … 1.1750346192486743, 1.1750290537
894597, 1.1750298006891513, 1.1750250307038326, 1.1750250307038328, 1.17502
98006891513, 1.1750360917179217, 1.1750403083061565, 1.1750403083061562, 1.
1750360917179214]
[4.0801520507468645, 4.080288095389964, 4.080288095389964, 4.0801520507468
64, 4.079979266691691, 4.079866750053601, 4.079866750053601, 4.079979266691
691, 4.080294838892661, 4.080480019090341 … 1.1717029614621586, 1.1716974
178790238, 1.1716981618829718, 1.1716934106794916, 1.1716934106794918, 1.17
16981618829718, 1.1717044280968363, 1.1717086280303821, 1.171708628030382,
1.1717044280968363]
[4.081735017358575, 4.0818710460672145, 4.0818710460672145, 4.081735017358
575, 4.081562254830442, 4.081449752855921, 4.081449752855921, 4.08156225483
0442, 4.0818777917236195, 4.082062953309229 … 1.168398037935694, 1.168392
5161023, 1.168393257227095, 1.1683885246968762, 1.1683885246968762, 1.16839
32572270952, 1.1683994987696518, 1.168403682144958, 1.168403682144958, 1.16
83994987696518]
[4.08329044088123, 4.083426453748336, 4.083426453748336, 4.08329044088123,
4.083117699754509, 4.083005212357935, 4.083005212357935, 4.083117699754509
, 4.0834332015456125, 4.083618344627493 … 1.1651197338640091, 1.165114233
653892, 1.1651149719161427, 1.165110257950501, 1.165110257950501, 1.1651149
719161427, 1.165121188931128, 1.1651253558447368, 1.165125355844737, 1.1651
211889311281]
[4.084818453340643, 4.0849544504592155, 4.084954450459216, 4.0848184533406
42, 4.084645733489605, 4.08453326058529, 4.08453326058529, 4.08464573348960
6, 4.084961200384522, 4.085146325071104 … 1.1618679333953803, 1.161862454
6819691, 1.1618631900983, 1.1618584945884618, 1.1618584945884618, 1.1618631
900982999, 1.161869382729568, 1.1618735332780994, 1.1618735332780996, 1.161
869382729568]
[4.085314527232731, 4.085450519197376, 4.0854505191973765, 4.0853145272327
3, 4.085141814344439, 4.085029346182895, 4.085029346182895, 4.0851418143444
4, 4.0854572698190506, 4.0856423884854705 … 1.1608053075612759, 1.1607998
358830436, 1.160800570367968, 1.160795880898096, 1.160795880898096, 1.16080
0570367968, 1.1608067550192933, 1.1608109002121196, 1.1608109002121196, 1.1
608067550192935]test_sol = [sol,sol2,sol,sol3]
probs = [prob,prob_mtk,fastprob,prob_mol];plot(sol, idxs = 1)
plot(sol, idxs = 10)
Setup Preconditioners
OrdinaryDiffEq
function incompletelu(W,du,u,p,t,newW,Plprev,Prprev,solverdata)
if newW === nothing || newW
Pl = ilu(convert(AbstractMatrix,W), τ = 50.0)
else
Pl = Plprev
end
Pl,nothing
end
function algebraicmultigrid(W,du,u,p,t,newW,Plprev,Prprev,solverdata)
if newW === nothing || newW
Pl = aspreconditioner(ruge_stuben(convert(AbstractMatrix,W)))
else
Pl = Plprev
end
Pl,nothing
endalgebraicmultigrid (generic function with 1 method)Sundials
const jaccache = prob_mtk.f.jac(prob.u0,prob.p,0.0)
const W = I - 1.0*jaccache
prectmp = ilu(W, τ = 50.0)
const preccache = Ref(prectmp)
function psetupilu(p, t, u, du, jok, jcurPtr, gamma)
if !jok
prob_mtk.f.jac(jaccache,u,p,t)
jcurPtr[] = true
# W = I - gamma*J
@. W = -gamma*jaccache
idxs = diagind(W)
@. @view(W[idxs]) = @view(W[idxs]) + 1
# Build preconditioner on W
preccache[] = ilu(W, τ = 5.0)
end
end
function precilu(z,r,p,t,y,fy,gamma,delta,lr)
ldiv!(z,preccache[],r)
end
prectmp2 = aspreconditioner(ruge_stuben(W, presmoother = AlgebraicMultigrid.Jacobi(rand(size(W,1))), postsmoother = AlgebraicMultigrid.Jacobi(rand(size(W,1)))))
const preccache2 = Ref(prectmp2)
function psetupamg(p, t, u, du, jok, jcurPtr, gamma)
if !jok
prob_mtk.f.jac(jaccache,u,p,t)
jcurPtr[] = true
# W = I - gamma*J
@. W = -gamma*jaccache
idxs = diagind(W)
@. @view(W[idxs]) = @view(W[idxs]) + 1
# Build preconditioner on W
preccache2[] = aspreconditioner(ruge_stuben(W, presmoother = AlgebraicMultigrid.Jacobi(rand(size(W,1))), postsmoother = AlgebraicMultigrid.Jacobi(rand(size(W,1)))))
end
end
function precamg(z,r,p,t,y,fy,gamma,delta,lr)
ldiv!(z,preccache2[],r)
endprecamg (generic function with 1 method)Compare Problem Implementations
abstols = 1.0 ./ 10.0 .^ (5:8)
reltols = 1.0 ./ 10.0 .^ (1:4);
setups = [
Dict(:alg => KenCarp47(linsolve=KLUFactorization())),
Dict(:alg => KenCarp47(linsolve=KLUFactorization()), :prob_choice => 2),
Dict(:alg => KenCarp47(linsolve=KLUFactorization()), :prob_choice => 3),
Dict(:alg => KenCarp47(linsolve=KLUFactorization()), :prob_choice => 4),
Dict(:alg => KenCarp47(linsolve=KrylovJL_GMRES())),
Dict(:alg => KenCarp47(linsolve=KrylovJL_GMRES()), :prob_choice => 2),
Dict(:alg => KenCarp47(linsolve=KrylovJL_GMRES()), :prob_choice => 3),
Dict(:alg => KenCarp47(linsolve=KrylovJL_GMRES()), :prob_choice => 4),]
names = ["KenCarp47 KLU","KenCarp47 KLU MTK","KenCarp47 KLU FastBruss", "KenCarp47 KLU MOL",
"KenCarp47 GMRES", "KenCarp47 GMRES MTK", "KenCarp47 GMRES FastBruss", "KenCarp47 GMRES MOL"];
wp = WorkPrecisionSet(probs,abstols,reltols,setups;names = names,
save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10,wrap=Val(false))
plot(wp)
High Tolerances
This is the speed when you just want the answer.
abstols = 1.0 ./ 10.0 .^ (5:8)
reltols = 1.0 ./ 10.0 .^ (1:4);
setups = [
Dict(:alg=>CVODE_BDF(linear_solver = :KLU), :prob_choice => 2),
Dict(:alg=>CVODE_BDF(linear_solver = :GMRES)),
Dict(:alg=>CVODE_BDF(linear_solver = :GMRES), :prob_choice => 2),
Dict(:alg=>CVODE_BDF(linear_solver=:GMRES,prec=precilu,psetup=psetupilu,prec_side=1)),
Dict(:alg=>CVODE_BDF(linear_solver=:GMRES,prec=precamg,psetup=psetupamg,prec_side=1)),
Dict(:alg=>CVODE_BDF(linear_solver=:GMRES,prec=precilu,psetup=psetupilu,prec_side=1), :prob_choice => 2),
Dict(:alg=>CVODE_BDF(linear_solver=:GMRES,prec=precamg,psetup=psetupamg,prec_side=1), :prob_choice => 2),
]
names = ["CVODE MTK KLU","CVODE GMRES","CVODE MTK GMRES", "CVODE iLU GMRES", "CVODE AMG GMRES", "CVODE iLU MTK GMRES", "CVODE AMG MTK GMRES"];
wp = WorkPrecisionSet(probs,abstols,reltols,setups;names=names,
save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)
plot(wp)
setups = [
Dict(:alg=>KenCarp47(linsolve=KLUFactorization())),
Dict(:alg=>KenCarp47(linsolve=KLUFactorization()), :prob_choice => 2),
Dict(:alg=>KenCarp47(linsolve=UMFPACKFactorization())),
Dict(:alg=>KenCarp47(linsolve=UMFPACKFactorization()), :prob_choice => 2),
Dict(:alg=>KenCarp47(linsolve=KrylovJL_GMRES())),
Dict(:alg=>KenCarp47(linsolve=KrylovJL_GMRES()), :prob_choice => 2),
Dict(:alg=>KenCarp47(linsolve=KrylovJL_GMRES(),precs=incompletelu,concrete_jac=true)),
Dict(:alg=>KenCarp47(linsolve=KrylovJL_GMRES(),precs=incompletelu,concrete_jac=true), :prob_choice => 2),
Dict(:alg=>KenCarp47(linsolve=KrylovJL_GMRES(),precs=algebraicmultigrid,concrete_jac=true)),
Dict(:alg=>KenCarp47(linsolve=KrylovJL_GMRES(),precs=algebraicmultigrid,concrete_jac=true), :prob_choice => 2),
]
names = ["KenCarp47 KLU","KenCarp47 KLU MTK","KenCarp47 UMFPACK", "KenCarp47 UMFPACK MTK", "KenCarp47 GMRES",
"KenCarp47 GMRES MTK", "KenCarp47 iLU GMRES", "KenCarp47 iLU GMRES MTK", "KenCarp47 AMG GMRES",
"KenCarp47 AMG GMRES MTK"];
wp = WorkPrecisionSet(probs,abstols,reltols,setups;names = names,
save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)
plot(wp)
setups = [
Dict(:alg=>TRBDF2()),
Dict(:alg=>KenCarp4()),
Dict(:alg=>KenCarp47()),
# Dict(:alg=>QNDF()), # bad
Dict(:alg=>FBDF()),
]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;
save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)
plot(wp)
setups = [
Dict(:alg=>KenCarp47(linsolve=KLUFactorization()), :prob_choice => 2),
Dict(:alg=>KenCarp47(linsolve=KrylovJL_GMRES()), :prob_choice => 2),
Dict(:alg=>FBDF(linsolve=KLUFactorization()), :prob_choice => 2),
Dict(:alg=>FBDF(linsolve=KrylovJL_GMRES()), :prob_choice => 2),
Dict(:alg=>CVODE_BDF(linear_solver = :KLU), :prob_choice => 2),
Dict(:alg=>CVODE_BDF(linear_solver=:GMRES,prec=precilu,psetup=psetupilu,prec_side=1), :prob_choice => 2),
]
names = ["KenCarp47 KLU MTK", "KenCarp47 GMRES MTK",
"FBDF KLU MTK", "FBDF GMRES MTK",
"CVODE MTK KLU", "CVODE iLU MTK GMRES"
];
wp = WorkPrecisionSet(probs,abstols,reltols,setups;names = names,
save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)
plot(wp)
Low Tolerances
This is the speed at lower tolerances, measuring what's good when accuracy is needed.
abstols = 1.0 ./ 10.0 .^ (7:12)
reltols = 1.0 ./ 10.0 .^ (4:9)
setups = [
Dict(:alg=>CVODE_BDF(linear_solver = :KLU), :prob_choice => 2),
Dict(:alg=>CVODE_BDF(linear_solver = :GMRES)),
Dict(:alg=>CVODE_BDF(linear_solver = :GMRES), :prob_choice => 2),
Dict(:alg=>CVODE_BDF(linear_solver=:GMRES,prec=precilu,psetup=psetupilu,prec_side=1)),
Dict(:alg=>CVODE_BDF(linear_solver=:GMRES,prec=precamg,psetup=psetupamg,prec_side=1)),
Dict(:alg=>CVODE_BDF(linear_solver=:GMRES,prec=precilu,psetup=psetupilu,prec_side=1), :prob_choice => 2),
Dict(:alg=>CVODE_BDF(linear_solver=:GMRES,prec=precamg,psetup=psetupamg,prec_side=1), :prob_choice => 2),
]
names = ["CVODE MTK KLU","CVODE GMRES","CVODE MTK GMRES", "CVODE iLU GMRES", "CVODE AMG GMRES", "CVODE iLU MTK GMRES", "CVODE AMG MTK GMRES"];
wp = WorkPrecisionSet(probs,abstols,reltols,setups;names = names,
save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)
plot(wp)
setups = [
Dict(:alg=>KenCarp47(linsolve=KLUFactorization()), :prob_choice => 2),
Dict(:alg=>KenCarp47(linsolve=KrylovJL_GMRES()), :prob_choice => 2),
Dict(:alg=>FBDF(linsolve=KLUFactorization()), :prob_choice => 2),
Dict(:alg=>FBDF(linsolve=KrylovJL_GMRES()), :prob_choice => 2),
Dict(:alg=>Rodas5P(linsolve=KrylovJL_GMRES()), :prob_choice => 2),
Dict(:alg=>CVODE_BDF(linear_solver = :KLU), :prob_choice => 2),
Dict(:alg=>CVODE_BDF(linear_solver=:GMRES,prec=precilu,psetup=psetupilu,prec_side=1), :prob_choice => 2),
]
names = ["KenCarp47 KLU MTK", "KenCarp47 GMRES MTK",
"FBDF KLU MTK", "FBDF GMRES MTK",
"Rodas5P GMRES MTK",
"CVODE MTK KLU", "CVODE iLU MTK GMRES"
];
wp = WorkPrecisionSet(probs,abstols,reltols,setups;names = names,
save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)
plot(wp)
Appendix
These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: https://github.com/SciML/SciMLBenchmarks.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.
To locally run this benchmark, do the following commands:
using SciMLBenchmarks
SciMLBenchmarks.weave_file("benchmarks/StiffODE","Bruss.jmd")Computer Information:
Julia Version 1.10.9
Commit 5595d20a287 (2025-03-10 12:51 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 128 × AMD EPYC 7502 32-Core Processor
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-15.0.7 (ORCJIT, znver2)
Threads: 1 default, 0 interactive, 1 GC (on 128 virtual cores)
Environment:
JULIA_CPU_THREADS = 128
JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953
Package Information:
Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/StiffODE/Project.toml`
⌃ [2169fc97] AlgebraicMultigrid v0.6.0
⌃ [6e4b80f9] BenchmarkTools v1.5.0
⌃ [f3b72e0c] DiffEqDevTools v2.45.0
⌃ [5b8099bc] DomainSets v0.7.14
[5a33fad7] GeometricIntegratorsDiffEq v0.2.5
[40713840] IncompleteLU v0.2.1
[7f56f5a3] LSODA v0.7.5
⌅ [7ed4a6bd] LinearSolve v2.34.0
⌃ [94925ecb] MethodOfLines v0.11.3
⌃ [961ee093] ModelingToolkit v9.32.0
⌃ [09606e27] ODEInterfaceDiffEq v3.13.3
⌃ [1dea7af3] OrdinaryDiffEq v6.89.0
⌃ [65888b18] ParameterizedFunctions v5.17.0
⌃ [91a5bcdd] Plots v1.40.8
[132c30aa] ProfileSVG v0.2.2
[f2c3362d] RecursiveFactorization v0.2.23
[31c91b34] SciMLBenchmarks v0.1.3
⌃ [90137ffa] StaticArrays v1.9.7
⌃ [c3572dad] Sundials v4.25.0
⌅ [0c5d862f] Symbolics v5.36.0
⌃ [a759f4b9] TimerOutputs v0.5.24
[37e2e46d] LinearAlgebra
[2f01184e] SparseArrays v1.10.0
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.And the full manifest:
Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/StiffODE/Manifest.toml`
⌃ [47edcb42] ADTypes v1.7.1
[a4c015fc] ANSIColoredPrinters v0.0.1
[621f4979] AbstractFFTs v1.5.0
[1520ce14] AbstractTrees v0.4.5
⌃ [7d9f7c33] Accessors v0.1.37
⌃ [79e6a3ab] Adapt v4.0.4
⌃ [2169fc97] AlgebraicMultigrid v0.6.0
[66dad0bd] AliasTables v1.1.3
⌅ [4c88cf16] Aqua v0.5.6
[ec485272] ArnoldiMethod v0.4.0
⌃ [4fba245c] ArrayInterface v7.16.0
⌃ [4c555306] ArrayLayouts v1.10.3
[13072b0f] AxisAlgorithms v1.1.0
⌃ [aae01518] BandedMatrices v1.7.4
⌃ [6e4b80f9] BenchmarkTools v1.5.0
[e2ed5e7c] Bijections v0.1.9
[d1d4a3ce] BitFlags v0.1.9
[62783981] BitTwiddlingConvenienceFunctions v0.1.6
⌃ [8e7c35d0] BlockArrays v1.1.0
[fa961155] CEnum v0.5.0
[2a0fbf3d] CPUSummary v0.2.6
[00ebfdb7] CSTParser v3.4.3
⌃ [d360d2e6] ChainRulesCore v1.24.0
[fb6a15b2] CloseOpenIntervals v0.1.13
⌃ [944b1d66] CodecZlib v0.7.6
⌃ [35d6a980] ColorSchemes v3.26.0
⌅ [3da002f7] ColorTypes v0.11.5
⌅ [c3611d14] ColorVectorSpace v0.10.0
⌅ [5ae59095] Colors v0.12.11
[861a8166] Combinatorics v1.0.2
⌅ [a80b9123] CommonMark v0.8.12
[38540f10] CommonSolve v0.2.4
[bbf7d656] CommonSubexpressions v0.3.1
[f70d9fcc] CommonWorldInvalidations v1.0.0
⌃ [a09551c4] CompactBasisFunctions v0.2.12
[34da2185] Compat v4.16.0
[b152e2b5] CompositeTypes v0.1.4
[a33af91c] CompositionsBase v0.1.2
[2569d6c7] ConcreteStructs v0.2.3
⌃ [f0e56b4a] ConcurrentUtilities v2.4.2
[8f4d0f93] Conda v1.10.2
⌅ [187b0558] ConstructionBase v1.5.6
⌅ [7ae1f121] ContinuumArrays v0.18.4
[d38c429a] Contour v0.6.3
[adafc99b] CpuId v0.3.1
[a8cc5b0e] Crayons v4.1.1
⌅ [717857b8] DSP v0.7.9
[9a962f9c] DataAPI v1.16.0
⌃ [864edb3b] DataStructures v0.18.20
[e2d170a0] DataValueInterfaces v1.0.0
⌃ [55939f99] DecFP v1.3.2
[8bb1440f] DelimitedFiles v1.9.1
⌃ [2b5f629d] DiffEqBase v6.155.0
⌅ [459566f4] DiffEqCallbacks v3.9.1
⌃ [f3b72e0c] DiffEqDevTools v2.45.0
⌃ [77a26b50] DiffEqNoiseProcess v5.23.0
[163ba53b] DiffResults v1.1.0
[b552c78f] DiffRules v1.15.1
⌅ [a0c0ee7d] DifferentiationInterface v0.5.17
⌃ [b4f34e82] Distances v0.10.11
⌃ [31c24e10] Distributions v0.25.111
[ffbed154] DocStringExtensions v0.9.3
⌅ [e30172f5] Documenter v0.27.25
⌃ [5b8099bc] DomainSets v0.7.14
⌅ [7c1d4256] DynamicPolynomials v0.5.7
⌅ [06fc5a27] DynamicQuantities v0.13.2
[4e289a0a] EnumX v1.0.4
⌅ [f151be2c] EnzymeCore v0.7.8
⌃ [460bff9d] ExceptionUnwrapping v0.1.10
⌃ [d4d017d3] ExponentialUtilities v1.26.1
[e2ba6199] ExprTools v0.1.10
⌅ [6b7a57c9] Expronicon v0.8.5
⌃ [c87230d0] FFMPEG v0.4.1
⌃ [7a1cc6ca] FFTW v1.8.0
[7034ab61] FastBroadcast v0.3.5
[9aa1b823] FastClosures v0.3.2
[442a2c76] FastGaussQuadrature v1.0.2
[29a986be] FastLapackInterface v2.0.4
⌅ [057dd010] FastTransforms v0.16.4
⌃ [5789e2e9] FileIO v1.16.3
[1a297f60] FillArrays v1.13.0
[64ca27bc] FindFirstFunctions v1.4.1
⌃ [6a86dc24] FiniteDiff v2.24.0
[53c48c17] FixedPointNumbers v0.8.5
⌃ [08572546] FlameGraphs v1.0.0
[1fa38f19] Format v1.3.7
⌃ [f6369f11] ForwardDiff v0.10.36
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Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`
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