Large error when integrating around point of discontinuity

[flin2]

Describe the bug 🐞

The numerical solution has a large discrepancy from the analytical solution when the bound is close to the point of discontinuity.

Expected behavior

I expect the difference should be really small (much smaller than 0.00059).

Minimal Reproducible Example 👇

using Integrals

function f(x)
    if x < 1
        return exp(x)
    elseif x >=1
        return exp(2x)
    end
end

function F(x0, x1)
    if x0 < 1 && x1 < 1
        return exp(x1) - exp(x0)
    elseif x0 < 1 && x1 >= 1
        return exp(2x1) / 2 - exp(2) / 2 + exp(1) - exp(x0)
    elseif x0 >= 1 && x1 >= 1
        return exp(2x1) / 2 - exp(2x0) / 2
    end
end


x0 = 0.997
x1 = 2.0

prob = IntegralProblem((x,p) -> f(x), (x0,x1))
sol1 = solve(prob, QuadGKJL(), reltol=1e-12, abstol=1e-12).u
sol0 = F(x0, x1)

sol1 - sol0
(sol1 - sol0) ./ sol0

Environment (please complete the following information):

• Output of using Pkg; Pkg.status()

[de52edbc] Integrals v4.6.0

• Output of versioninfo()

Julia Version 1.11.5
Commit 760b2e5b739 (2025-04-14 06:53 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: macOS (arm64-apple-darwin24.0.0)
  CPU: 14 × Apple M4 Pro
  WORD_SIZE: 64
  LLVM: libLLVM-16.0.6 (ORCJIT, apple-m1)
Threads: 1 default, 0 interactive, 1 GC (on 10 virtual cores)

[lxvm]

Hi, the large error you are seeing is due to the fact that the integration algorithm in QuadGKJL assumes that the function you are integrating is smooth and infinitely differentiable and your function does not satisfy these assumptions because it is discontinuous. Specifically, there is a problem with putting the point of discontinuity at the edge of the integration interval because QuadGKJL does not sample points too close to the edge of the interval and thus probably does not 'see' the discontinuity. You might be able to get around this by increasing the order keyword to QuadGKJL, however I would not recommend this because it may be unreliable