feat: integrate callback saving into integrator init and finalize

Author: AayushSabharwal

src/integrators/integrator_utils.jl

@@ -234,6 +234,7 @@ end
             copyat_or_push!(integrator.sol.u, integrator.saveiter,
                 integrator.u[integrator.opts.save_idxs], Val{false})
         end
+        SciMLBase.save_final_discretes!(integrator, integrator.opts.callback)
     end
     if (!isnothing(integrator.W) && integrator.W.curt != integrator.t) ||
        (!isnothing(integrator.P) && integrator.P.curt != integrator.t)

src/solve.jl

@@ -791,4 +791,8 @@ 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