diff --git a/examples/structured_2d_dgsem/elixir_euler_vortex_perk4.jl b/examples/structured_2d_dgsem/elixir_euler_vortex_perk4.jl
index 896f6c4f281..7e0098bf318 100644
--- a/examples/structured_2d_dgsem/elixir_euler_vortex_perk4.jl
+++ b/examples/structured_2d_dgsem/elixir_euler_vortex_perk4.jl
@@ -1,5 +1,6 @@
 
-using OrdinaryDiffEq # Required for `CallbackSet`
+# We use time integration methods implemented in Trixi.jl, but we need the `CallbackSet`
+using OrdinaryDiffEq: CallbackSet
 using Trixi
 
 # Ratio of specific heats
@@ -21,7 +22,7 @@ The classical isentropic vortex test case as presented in Section 5.1 of
 function initial_condition_isentropic_vortex(x, t, equations::CompressibleEulerEquations2D)
     # Evaluate error after full domain traversion
     if t == t_end()
-        t = 0
+        t = zero(t)
     end
 
     # Initial center of the vortex
diff --git a/examples/tree_1d_dgsem/elixir_hypdiff_nonperiodic_perk4.jl b/examples/tree_1d_dgsem/elixir_hypdiff_nonperiodic_perk4.jl
index b43ab3c416d..ac249d23c78 100644
--- a/examples/tree_1d_dgsem/elixir_hypdiff_nonperiodic_perk4.jl
+++ b/examples/tree_1d_dgsem/elixir_hypdiff_nonperiodic_perk4.jl
@@ -1,5 +1,6 @@
 
-using OrdinaryDiffEq # Required for `CallbackSet`
+# We use time integration methods implemented in Trixi.jl, but we need the `CallbackSet`
+using OrdinaryDiffEq: CallbackSet
 using Trixi
 
 # Convex and ECOS are imported because they are used for finding the optimal time step and optimal 
diff --git a/src/time_integration/methods_2N.jl b/src/time_integration/methods_2N.jl
index a7536de644b..65a8aa3e26d 100644
--- a/src/time_integration/methods_2N.jl
+++ b/src/time_integration/methods_2N.jl
@@ -103,7 +103,7 @@ mutable struct SimpleIntegrator2N{RealT <: Real, uType, Params, Sol, F, Alg,
     iter::Int # current number of time steps (iteration)
     p::Params # will be the semidiscretization from Trixi.jl
     sol::Sol # faked
-    f::F # `rhs` of the semidiscretization
+    f::F # `rhs!` of the semidiscretization
     alg::Alg
     opts::SimpleIntegrator2NOptions
     finalstep::Bool # added for convenience
diff --git a/src/time_integration/methods_3Sstar.jl b/src/time_integration/methods_3Sstar.jl
index 460b0a105e2..c2ef37d57b6 100644
--- a/src/time_integration/methods_3Sstar.jl
+++ b/src/time_integration/methods_3Sstar.jl
@@ -156,7 +156,7 @@ mutable struct SimpleIntegrator3Sstar{RealT <: Real, uType, Params, Sol, F, Alg,
     iter::Int # current number of time step (iteration)
     p::Params # will be the semidiscretization from Trixi.jl
     sol::Sol # faked
-    f::F # `rhs` of the semidiscretization
+    f::F # `rhs!` of the semidiscretization
     alg::Alg
     opts::SimpleIntegrator3SstarOptions
     finalstep::Bool # added for convenience
diff --git a/src/time_integration/methods_SSP.jl b/src/time_integration/methods_SSP.jl
index e8703aceefe..071bf3e6f50 100644
--- a/src/time_integration/methods_SSP.jl
+++ b/src/time_integration/methods_SSP.jl
@@ -89,7 +89,7 @@ mutable struct SimpleIntegratorSSP{RealT <: Real, uType, Params, Sol, F, Alg,
     iter::Int # current number of time steps (iteration)
     p::Params # will be the semidiscretization from Trixi
     sol::Sol # faked
-    f::F # `rhs` of the semidiscretization
+    f::F # `rhs!` of the semidiscretization
     alg::Alg # SimpleSSPRK33
     opts::SimpleIntegratorSSPOptions
     finalstep::Bool # added for convenience
diff --git a/src/time_integration/paired_explicit_runge_kutta/methods_PERK2.jl b/src/time_integration/paired_explicit_runge_kutta/methods_PERK2.jl
index c08f8e64259..6464271f1af 100644
--- a/src/time_integration/paired_explicit_runge_kutta/methods_PERK2.jl
+++ b/src/time_integration/paired_explicit_runge_kutta/methods_PERK2.jl
@@ -200,7 +200,7 @@ mutable struct PairedExplicitRK2Integrator{RealT <: Real, uType, Params, Sol, F,
     iter::Int # current number of time steps (iteration)
     p::Params # will be the semidiscretization from Trixi
     sol::Sol # faked
-    f::F # `rhs` of the semidiscretization
+    f::F # `rhs!` of the semidiscretization
     alg::Alg # This is our own class written above; Abbreviation for ALGorithm
     opts::PairedExplicitRKOptions
     finalstep::Bool # added for convenience
diff --git a/src/time_integration/paired_explicit_runge_kutta/methods_PERK3.jl b/src/time_integration/paired_explicit_runge_kutta/methods_PERK3.jl
index 849dae87307..d8fac2d8e10 100644
--- a/src/time_integration/paired_explicit_runge_kutta/methods_PERK3.jl
+++ b/src/time_integration/paired_explicit_runge_kutta/methods_PERK3.jl
@@ -191,7 +191,7 @@ mutable struct PairedExplicitRK3Integrator{RealT <: Real, uType, Params, Sol, F,
     iter::Int # current number of time steps (iteration)
     p::Params # will be the semidiscretization from Trixi
     sol::Sol # faked
-    f::F # `rhs` of the semidiscretization
+    f::F # `rhs!` of the semidiscretization
     alg::Alg # This is our own class written above; Abbreviation for ALGorithm
     opts::PairedExplicitRKOptions
     finalstep::Bool # added for convenience
diff --git a/src/time_integration/paired_explicit_runge_kutta/methods_PERK4.jl b/src/time_integration/paired_explicit_runge_kutta/methods_PERK4.jl
index 6c006afc07e..2c613361756 100644
--- a/src/time_integration/paired_explicit_runge_kutta/methods_PERK4.jl
+++ b/src/time_integration/paired_explicit_runge_kutta/methods_PERK4.jl
@@ -189,7 +189,7 @@ mutable struct PairedExplicitRK4Integrator{RealT <: Real, uType, Params, Sol, F,
     iter::Int # current number of time steps (iteration)
     p::Params # will be the semidiscretization from Trixi
     sol::Sol # faked
-    f::F # `rhs` of the semidiscretization
+    f::F # `rhs!` of the semidiscretization
     alg::Alg # This is our own class written above; Abbreviation for ALGorithm
     opts::PairedExplicitRKOptions
     finalstep::Bool # added for convenience