diff --git a/NEWS.md b/NEWS.md
index a54572e4..504f4dd3 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -5,6 +5,7 @@ Main
 -------
 v0.9.0
 - ![][badge-💥breaking] If saveat is a number, then it does not automatically expand to `tspan[1]:saveat:tspan[2]`.
+- ARS343 and SSPKnoth are compatible with ITime. See ClimaUtilities for more information about ITime.
 
 v0.7.18
 -------
diff --git a/src/ClimaTimeSteppers.jl b/src/ClimaTimeSteppers.jl
index 959f9016..e0cafb91 100644
--- a/src/ClimaTimeSteppers.jl
+++ b/src/ClimaTimeSteppers.jl
@@ -128,6 +128,7 @@ include("solvers/rosenbrock.jl")
 
 include("Callbacks.jl")
 
+include("arbitrary_number_types.jl")
 
 benchmark_step(integrator, device) =
     @warn "Must load CUDA, BenchmarkTools, OrderedCollections, StatsBase, PrettyTables to trigger the ClimaTimeSteppersBenchmarkToolsExt extension"
diff --git a/src/arbitrary_number_types.jl b/src/arbitrary_number_types.jl
new file mode 100644
index 00000000..e659dabf
--- /dev/null
+++ b/src/arbitrary_number_types.jl
@@ -0,0 +1,19 @@
+"""
+    SciMLBase.allows_arbitrary_number_types(alg::T)
+        where {T <: ClimaTimeSteppers.RosenbrockAlgorithm}
+
+Return `true`. Enable SSPKnoth to run with `ClimaUtilities.ITime`.
+"""
+function SciMLBase.allows_arbitrary_number_types(alg::T) where {T <: ClimaTimeSteppers.RosenbrockAlgorithm}
+    true
+end
+
+"""
+    SciMLBase.allows_arbitrary_number_types(alg::T)
+        where {T <: ClimaTimeSteppers.IMEXAlgorithm}
+
+Return `true`. Enable ARS343 to run with `ClimaUtilities.ITime`.
+"""
+function SciMLBase.allows_arbitrary_number_types(alg::T) where {T <: ClimaTimeSteppers.IMEXAlgorithm}
+    true
+end
diff --git a/src/integrators.jl b/src/integrators.jl
index 48f52ff6..be8ca418 100644
--- a/src/integrators.jl
+++ b/src/integrators.jl
@@ -81,7 +81,7 @@ function tstops_and_saveat_heaps(t0, tf, tstops, saveat = [])
     return tstops, saveat
 end
 
-compute_tdir(ts) = ts[1] > ts[end] ? sign(ts[end] - ts[1]) : eltype(ts)(1)
+compute_tdir(ts) = ts[1] > ts[end] ? sign(ts[end] - ts[1]) : oneunit(ts[1])
 
 # called by DiffEqBase.init and DiffEqBase.solve
 function DiffEqBase.__init(
@@ -102,8 +102,9 @@ function DiffEqBase.__init(
 )
     (; u0, p) = prob
     t0, tf = prob.tspan
+    t0, tf, dt = promote(t0, tf, dt)
 
-    dt > zero(dt) || error("dt must be positive")
+    dt > zero(oneunit(dt)) || error("dt must be positive")
     _dt = dt
     dt = tf > t0 ? dt : -dt
 
@@ -243,8 +244,8 @@ function __step!(integrator)
     # is taken from OrdinaryDiffEq.jl
     t_plus_dt = integrator.t + integrator.dt
     t_unit = oneunit(integrator.t)
-    max_t_error = 100 * eps(float(integrator.t / t_unit)) * t_unit
-    integrator.t = !isempty(tstops) && abs(first(tstops) - t_plus_dt) < max_t_error ? first(tstops) : t_plus_dt
+    max_t_error = 100 * eps(float(integrator.t / t_unit)) * float(t_unit)
+    integrator.t = !isempty(tstops) && abs(float(first(tstops)) - float(t_plus_dt)) < max_t_error ? first(tstops) : t_plus_dt
 
     # apply callbacks
     discrete_callbacks = integrator.callback.discrete_callbacks
diff --git a/src/solvers/imex_ark.jl b/src/solvers/imex_ark.jl
index 6b5a2d24..87070785 100644
--- a/src/solvers/imex_ark.jl
+++ b/src/solvers/imex_ark.jl
@@ -107,7 +107,7 @@ end
 
     t_exp = t + dt * c_exp[i]
     t_imp = t + dt * c_imp[i]
-    dtγ = dt * a_imp[i, i]
+    dtγ = float(dt) * a_imp[i, i]
 
     if has_T_lim(f) # Update based on limited tendencies from previous stages
         assign_fused_increment!(U, u, dt, a_exp, T_lim, Val(i))
@@ -135,6 +135,10 @@ end
             T_imp!(residual, U′, p, t_imp)
             @. residual = temp + dtγ * residual - U′
         end
+        implicit_equation_jacobian! = (jacobian, Ui) -> begin
+            T_imp!.Wfact(jacobian, Ui, p, dtγ, t_imp)
+        end
+        implicit_equation_cache! = Ui -> cache_imp!(Ui, p, t_imp)
         solve_newton!(
             newtons_method,
             newtons_method_cache,
diff --git a/src/solvers/rosenbrock.jl b/src/solvers/rosenbrock.jl
index 37b9384d..4a5ad657 100644
--- a/src/solvers/rosenbrock.jl
+++ b/src/solvers/rosenbrock.jl
@@ -127,7 +127,7 @@ function step_u!(int, cache::RosenbrockCache{Nstages}) where {Nstages}
 
     # TODO: This is only valid when Γ[i, i] is constant, otherwise we have to
     # move this in the for loop
-    @inbounds dtγ = dt * Γ[1, 1]
+    @inbounds dtγ = float(dt) * Γ[1, 1]
 
     if !isnothing(T_imp!)
         Wfact! = int.sol.prob.f.T_imp!.Wfact
@@ -175,14 +175,14 @@ function step_u!(int, cache::RosenbrockCache{Nstages}) where {Nstages}
         end
 
         if !isnothing(tgrad!)
-            fU .+= γi .* dt .* ∂Y∂t
+            fU .+= γi .* float(dt) .* ∂Y∂t
         end
 
         for j in 1:(i - 1)
-            fU .+= (C[i, j] / dt) .* k[j]
+            fU .+= (C[i, j] / float(dt)) .* k[j]
         end
 
-        fU .*= -dtγ
+        fU .*= -float(dtγ)
 
         if !isnothing(T_imp!)
             if W isa Matrix
diff --git a/src/utilities/fused_increment.jl b/src/utilities/fused_increment.jl
index 07f8b087..bc0c6f5d 100644
--- a/src/utilities/fused_increment.jl
+++ b/src/utilities/fused_increment.jl
@@ -121,7 +121,7 @@ In the edge case (coeffs are zero, `j` range is empty),
 this lowers to `nothing` (no-op)
 """
 @inline function fused_increment!(u, dt, sc, tend, v)
-    bc = fused_increment(u, dt, sc, tend, v)
+    bc = fused_increment(u, float(dt), sc, tend, v)
     if bc isa Base.Broadcast.Broadcasted # Only material if not trivial assignment
         Base.Broadcast.materialize!(u, bc)
     end
@@ -142,7 +142,7 @@ this lowers to
     `@. U = u`
 """
 @inline function assign_fused_increment!(U, u, dt, sc, tend, v)
-    bc = fused_increment(u, dt, sc, tend, v)
+    bc = fused_increment(u, float(dt), sc, tend, v)
     Base.Broadcast.materialize!(U, bc)
     return nothing
 end