diff --git a/docs/fig/upwind_benchmark.png b/docs/fig/upwind_benchmark.png
new file mode 100644
index 0000000..7183488
Binary files /dev/null and b/docs/fig/upwind_benchmark.png differ
diff --git a/docs/src/index.md b/docs/src/index.md
index 51eba78..529cdd3 100644
--- a/docs/src/index.md
+++ b/docs/src/index.md
@@ -11,6 +11,7 @@ TrixiEnzyme is not a registered Julia package, and it can be installed by runnin
 
 ## Configuring Batch Size
 
+To utilize `Enzyme.BatchDuplicated`, one can create a tuple containing duals (or shadows).
 TrixiEnzyme.jl performs partial derivative evaluation on one "batch" of the input vector at a time.
 Each differentiation of a batch requires a call to the target function as well as additional memory proportional to the square of the batch's size.
 Thus, a smaller batch size makes better use of memory bandwidth at the cost of more calls to the target function,
@@ -44,5 +45,5 @@ julia> @time jacobian_enzyme_forward(TrixiEnzyme.upwind!, x);
 ```
 
 Benchmark for a 401x401 Jacobian of `TrixiEnzyme.upwind!` (Lower is better):
-![upwind](https://private-user-images.githubusercontent.com/40481594/358694436-21588007-8469-46c5-8b77-e349b27c1c19.png?jwt=eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.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.lr66vXaVZMmUnWw24uh30-u754ckRPYzBctskuEntJc)
+![upwind benchmark](../fig/upwind_benchmark.png)
 `Enyme(@batch)` means applying `Polyester.@batch` to `middlebatches`.
diff --git a/src/TrixiEnzyme.jl b/src/TrixiEnzyme.jl
index 422f38f..2df2696 100644
--- a/src/TrixiEnzyme.jl
+++ b/src/TrixiEnzyme.jl
@@ -10,12 +10,12 @@ export plusTwo, jacobian_enzyme_forward, jacobian_enzyme_forward_closure
 export autodiff, Forward, Reverse, Duplicated, DuplicatedNoNeed, BatchDuplicated, BatchDuplicatedNoNeed, Const
 
 import Trixi
-using Trixi: AbstractEquations, TreeMesh, DGSEM,
+using Trixi: AbstractEquations, TreeMesh, DGSEM, jacobian_ad_forward
              BoundaryConditionPeriodic, SemidiscretizationHyperbolic,
-             VolumeIntegralWeakForm, VolumeIntegralFluxDifferencing,
+             VolumeIntegralWeakForm, VolumeIntegralFluxDifferencing, 
              wrap_array, compute_coefficients, have_nonconservative_terms,
-             boundary_condition_periodic,
-             set_log_type, set_sqrt_type
+             boundary_condition_periodic, LinearScalarAdvectionEquation1D
+             set_log_type, set_sqrt_type, initial_condition_sine_wave, SVector
 import Enzyme
 using Enzyme: autodiff, Forward, Reverse, Duplicated, DuplicatedNoNeed, make_zero, BatchDuplicated, BatchDuplicatedNoNeed, Const
 using Polyester: @batch
diff --git a/test/SemiTest.jl b/test/SemiTest.jl
index ac5d37a..958039b 100644
--- a/test/SemiTest.jl
+++ b/test/SemiTest.jl
@@ -1,6 +1,5 @@
 module SemiTest
 using Test
-using Trixi
 using TrixiEnzyme
 
 # %%
diff --git a/test/UpwindTest.jl b/test/UpwindTest.jl
new file mode 100644
index 0000000..0b7b210
--- /dev/null
+++ b/test/UpwindTest.jl
@@ -0,0 +1,33 @@
+module UpwindTest
+using Test
+using TrixiEnzyme
+using TrixiEnzyme: upwind!
+using ForwardDiff
+
+x = -1:0.005:1
+batch_size = 2
+jacobian_enzyme_forward(TrixiEnzyme.upwind!, x, N=batch_size)
+
+Δt₁ = @elapsed  J1 = jacobian_enzyme_forward(TrixiEnzyme.upwind!, x)
+
+Δt₂ = @elapsed begin
+u = zeros(length(x))
+du = zeros(length(x))
+cfg = ForwardDiff.JacobianConfig(nothing, du, u)
+uEltype = eltype(cfg)
+nan_uEltype=convert(uEltype, NaN)
+numerical_flux=fill(nan_uEltype, length(u))
+
+J2 = ForwardDiff.jacobian(du, u, cfg) do du_ode, u_ode
+    upwind!(du_ode, u_ode, (;v=1.0, numerical_flux));
+end;
+
+end; # 0.326764 seconds (1.09 M allocations: 75.013 MiB, 7.28% gc time, 99.87% compilation time)
+
+@test J1 = J2
+
+@info "Compare the time consumed by Enzyme.jl Δt and ForwardDiff.jl..."
+println("Enzyme: ", Δt₁)
+println("ForwardDiff: ", Δt₂)
+
+end