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Merged
merged 4 commits into from
Aug 19, 2025
Merged

Improve computation of epsilon in jacobian_fd #2522

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51d10e9
improve computation of epsilon in jacobian_fd
JoshuaLampert Aug 15, 2025
700df52
add link to comment
JoshuaLampert Aug 15, 2025
fb398e4
Merge branch 'main' into jacobian_fd-epsilon
JoshuaLampert Aug 15, 2025
c0fb7bb
add tests
JoshuaLampert Aug 18, 2025
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8 changes: 7 additions & 1 deletion src/semidiscretization/semidiscretization.jl
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Expand Up @@ -240,7 +240,13 @@ function jacobian_fd(semi::AbstractSemidiscretization;
# use second order finite difference to estimate Jacobian matrix
for idx in eachindex(u0_ode)
# determine size of fluctuation
epsilon = sqrt(eps(typeof(u0_ode[idx])))
# This is the approach used by FiniteDiff.jl to compute the
# step size, which assures that the finite difference is accurate
# for very small and very large absolute values `u0_ode[idx]`.
# See https://github.com/trixi-framework/Trixi.jl/pull/2514#issuecomment-3190534904.
absstep = sqrt(eps(typeof(u0_ode[idx])))
relstep = absstep
epsilon = max(relstep * abs(u0_ode[idx]), absstep)

# plus fluctuation
u_ode[idx] = u0_ode[idx] + epsilon
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31 changes: 31 additions & 0 deletions test/test_special_elixirs.jl
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Expand Up @@ -110,6 +110,37 @@ end
@test A * x ≈ Ax
end

@testset "Test Jacobian of DG (1D)" begin
@timed_testset "Linear advection" begin
trixi_include(@__MODULE__,
joinpath(EXAMPLES_DIR, "tree_1d_fdsbp",
"elixir_advection_upwind.jl"),
tspan = (0.0, 0.0))

A, _ = linear_structure(semi)

J = jacobian_ad_forward(semi)
@test Matrix(A) ≈ J
λ = eigvals(J)
@test maximum(real, λ) < 10 * sqrt(eps(real(semi)))

J = jacobian_fd(semi)
@test Matrix(A) ≈ J
λ = eigvals(J)
@test maximum(real, λ) < 10 * sqrt(eps(real(semi)))

# See https://github.com/trixi-framework/Trixi.jl/pull/2514
@test count(real.(λ) .>= -10) > 5
# See https://github.com/trixi-framework/Trixi.jl/pull/2522
t0 = zero(real(semi))
u0_ode = 1e9 * compute_coefficients(t0, semi)
J = jacobian_fd(semi; t0, u0_ode)
λ = eigvals(J)
@test count((-200 .<= real.(λ) .<= -10) .&& (-100 .<= imag.(λ) .<= 100)) == 0
@test count(isapprox.(imag.(λ), 0.0, atol = 10 * sqrt(eps(real(semi))))) == 2
end
end

@testset "Test Jacobian of DG (2D)" begin
@timed_testset "Linear advection" begin
trixi_include(@__MODULE__,
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