feat: integrate callback saving into integrator init and finalize

Author: AayushSabharwal

lib/OrdinaryDiffEqCore/src/integrators/integrator_utils.jl

@@ -240,6 +240,7 @@ function solution_endpoint_match_cur_integrator!(integrator)
             copyat_or_push!(integrator.sol.alg_choice, integrator.saveiter,
                 integrator.cache.current)
         end
+        SciMLBase.save_final_discretes!(integrator, integrator.opts.callback)
     end
 end
 

lib/OrdinaryDiffEqCore/src/solve.jl

@@ -745,4 +745,8 @@ function initialize_callbacks!(integrator, initialize_save = true)
 
     # reset this as it is now handled so the integrators should proceed as normal
     integrator.u_modified = false
+
+    if initialize_save
+        SciMLBase.save_discretes_if_enabled!(integrator, integrator.opts.callback; skip_duplicates = true)
+    end
 end