Compute third stage time-step for RK3 in a way that reduces the accum…

…ulation of error (#3617)

* Change TimeInterval to avoid roundoff error issue plus some cleanup

* fix typo

* Import fix

* Potentially address overflow problem

* Recompute new third stage time-step to reduce error accumulation

* Make schedule work even with fake models

* Fix tests for TimeInterval

* Fix JLD2 output writer tests

* Fix NetCDF writer test?

src/TimeSteppers/runge_kutta_3.jl

@@ -95,6 +95,9 @@ function time_step!(model::AbstractModel{<:RungeKutta3TimeStepper}, Δt; callbac
     second_stage_Δt = (γ² + ζ²) * Δt
     third_stage_Δt  = (γ³ + ζ³) * Δt
 
+    # Compute the next time step a priori to reduce floating point error accumulation
+    tⁿ⁺¹ = next_time(model.clock, Δt)
+
     #
     # First stage
     #
@@ -132,7 +135,10 @@ function time_step!(model::AbstractModel{<:RungeKutta3TimeStepper}, Δt; callbac
     calculate_pressure_correction!(model, third_stage_Δt)
     pressure_correct_velocities!(model, third_stage_Δt)
 
-    tick!(model.clock, third_stage_Δt)
+    # This formulation of the final time-step reduces the accumulation of round-off error when Δt
+    # is added to model.clock.time.
+    corrected_third_stage_Δt = tⁿ⁺¹ - model.clock.time
+    tick!(model.clock, corrected_third_stage_Δt)
     update_state!(model, callbacks; compute_tendencies)
     step_lagrangian_particles!(model, third_stage_Δt)