diff --git a/AMSS_NCKU_source/bssn_class.C b/AMSS_NCKU_source/bssn_class.C
index e14092b..553cc72 100644
--- a/AMSS_NCKU_source/bssn_class.C
+++ b/AMSS_NCKU_source/bssn_class.C
@@ -3302,22 +3302,11 @@ void bssn_class::Step(int lev, int YN)
 #endif
   }
 
-  // check error information (combined Patch + Shell Patch check)
+  // Non-blocking error reduction overlapped with Sync to hide Allreduce latency
+  MPI_Request err_req;
   {
     int erh = ERROR;
-    MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-  }
-
-  if (ERROR)
-  {
-    Parallel::Dump_Data(GH->PatL[lev], StateList, 0, PhysTime, dT_lev);
-    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
-    if (myrank == 0)
-    {
-      if (ErrorMonitor->outfile)
-        ErrorMonitor->outfile << "find NaN in state variables at t = " << PhysTime << ", lev = " << lev << endl;
-      MPI_Abort(MPI_COMM_WORLD, 1);
-    }
+    MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD, &err_req);
   }
 #endif
 
@@ -3334,11 +3323,25 @@ void bssn_class::Step(int lev, int YN)
     {
       prev_clock = curr_clock;
       curr_clock = clock();
-      cout << " Shell stuff synchronization used " 
-           << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC) 
+      cout << " Shell stuff synchronization used "
+           << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC)
            << " seconds! " << endl;
     }
   }
+
+  // Complete non-blocking error reduction and check
+  MPI_Wait(&err_req, MPI_STATUS_IGNORE);
+  if (ERROR)
+  {
+    Parallel::Dump_Data(GH->PatL[lev], StateList, 0, PhysTime, dT_lev);
+    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
+    if (myrank == 0)
+    {
+      if (ErrorMonitor->outfile)
+        ErrorMonitor->outfile << "find NaN in state variables at t = " << PhysTime << ", lev = " << lev << endl;
+      MPI_Abort(MPI_COMM_WORLD, 1);
+    }
+  }
 #endif
 
 #if (MAPBH == 0)
@@ -3655,23 +3658,11 @@ void bssn_class::Step(int lev, int YN)
         sPp = sPp->next;
       }
     }
-    // check error information (combined Patch + Shell Patch check)
+    // Non-blocking error reduction overlapped with Sync to hide Allreduce latency
+    MPI_Request err_req_cor;
     {
       int erh = ERROR;
-      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-    }
-    if (ERROR)
-    {
-      Parallel::Dump_Data(GH->PatL[lev], SynchList_pre, 0, PhysTime, dT_lev);
-      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
-      if (myrank == 0)
-      {
-        if (ErrorMonitor->outfile)
-          ErrorMonitor->outfile << "find NaN in RK4 substep#" << iter_count
-                                << " variables at t = " << PhysTime
-                                << ", lev = " << lev << endl;
-        MPI_Abort(MPI_COMM_WORLD, 1);
-      }
+      MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD, &err_req_cor);
     }
 #endif
 
@@ -3688,11 +3679,27 @@ void bssn_class::Step(int lev, int YN)
       {
         prev_clock = curr_clock;
         curr_clock = clock();
-        cout << " Shell stuff synchronization used " 
-             << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC) 
+        cout << " Shell stuff synchronization used "
+             << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC)
              << " seconds! " << endl;
       }
     }
+
+    // Complete non-blocking error reduction and check
+    MPI_Wait(&err_req_cor, MPI_STATUS_IGNORE);
+    if (ERROR)
+    {
+      Parallel::Dump_Data(GH->PatL[lev], SynchList_pre, 0, PhysTime, dT_lev);
+      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
+      if (myrank == 0)
+      {
+        if (ErrorMonitor->outfile)
+          ErrorMonitor->outfile << "find NaN in RK4 substep#" << iter_count
+                                << " variables at t = " << PhysTime
+                                << ", lev = " << lev << endl;
+        MPI_Abort(MPI_COMM_WORLD, 1);
+      }
+    }
 #endif
 
 #if (MAPBH == 0)
@@ -4146,22 +4153,11 @@ void bssn_class::Step(int lev, int YN)
   }
 #endif
   }
-  // check error information (combined Patch + Shell Patch check)
+  // Non-blocking error reduction overlapped with Sync to hide Allreduce latency
+  MPI_Request err_req;
   {
     int erh = ERROR;
-    MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-  }
-  if (ERROR)
-  {
-    Parallel::Dump_Data(GH->PatL[lev], StateList, 0, PhysTime, dT_lev);
-    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
-    if (myrank == 0)
-    {
-      if (ErrorMonitor->outfile)
-        ErrorMonitor->outfile << "find NaN in state variables at t = " << PhysTime
-                              << ", lev = " << lev << endl;
-      MPI_Abort(MPI_COMM_WORLD, 1);
-    }
+    MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD, &err_req);
   }
 #endif
 
@@ -4178,9 +4174,24 @@ void bssn_class::Step(int lev, int YN)
     {
       prev_clock = curr_clock;
       curr_clock = clock();
-      cout << " Shell stuff synchronization used " 
-      << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC) 
-      << " seconds! " << endl;
+      cout << " Shell stuff synchronization used "
+           << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC)
+           << " seconds! " << endl;
+    }
+  }
+
+  // Complete non-blocking error reduction and check
+  MPI_Wait(&err_req, MPI_STATUS_IGNORE);
+  if (ERROR)
+  {
+    Parallel::Dump_Data(GH->PatL[lev], StateList, 0, PhysTime, dT_lev);
+    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
+    if (myrank == 0)
+    {
+      if (ErrorMonitor->outfile)
+        ErrorMonitor->outfile << "find NaN in state variables at t = " << PhysTime
+                              << ", lev = " << lev << endl;
+      MPI_Abort(MPI_COMM_WORLD, 1);
     }
   }
 #endif
@@ -4483,23 +4494,11 @@ void bssn_class::Step(int lev, int YN)
         sPp = sPp->next;
       }
     }
-    // check error information (combined Patch + Shell Patch check)
+    // Non-blocking error reduction overlapped with Sync to hide Allreduce latency
+    MPI_Request err_req_cor;
     {
       int erh = ERROR;
-      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-    }
-    if (ERROR)
-    {
-      Parallel::Dump_Data(GH->PatL[lev], SynchList_pre, 0, PhysTime, dT_lev);
-      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
-      if (myrank == 0)
-      {
-        if (ErrorMonitor->outfile)
-          ErrorMonitor->outfile << "find NaN in RK4 substep#" << iter_count
-                                << " variables at t = " << PhysTime
-                                << ", lev = " << lev << endl;
-        MPI_Abort(MPI_COMM_WORLD, 1);
-      }
+      MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD, &err_req_cor);
     }
 #endif
 
@@ -4516,11 +4515,27 @@ void bssn_class::Step(int lev, int YN)
       {
         prev_clock = curr_clock;
         curr_clock = clock();
-        cout << " Shell stuff synchronization used " 
-             << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC) 
+        cout << " Shell stuff synchronization used "
+             << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC)
              << " seconds! " << endl;
       }
     }
+
+    // Complete non-blocking error reduction and check
+    MPI_Wait(&err_req_cor, MPI_STATUS_IGNORE);
+    if (ERROR)
+    {
+      Parallel::Dump_Data(GH->PatL[lev], SynchList_pre, 0, PhysTime, dT_lev);
+      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
+      if (myrank == 0)
+      {
+        if (ErrorMonitor->outfile)
+          ErrorMonitor->outfile << "find NaN in RK4 substep#" << iter_count
+                                << " variables at t = " << PhysTime
+                                << ", lev = " << lev << endl;
+        MPI_Abort(MPI_COMM_WORLD, 1);
+      }
+    }
 #endif
     // for black hole position
     if (BH_num > 0 && lev == GH->levels - 1)
@@ -4886,11 +4901,19 @@ void bssn_class::Step(int lev, int YN)
 
   //   misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"after Predictor rhs calculation");
 
-  // check error information
+  // Non-blocking error reduction overlapped with Sync to hide Allreduce latency
+  MPI_Request err_req;
   {
     int erh = ERROR;
-    MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, GH->Commlev[lev]);
+    MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, GH->Commlev[lev], &err_req);
   }
+
+  //   misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"before Predictor sync");
+
+  Parallel::Sync(GH->PatL[lev], SynchList_pre, Symmetry);
+
+  // Complete non-blocking error reduction and check
+  MPI_Wait(&err_req, MPI_STATUS_IGNORE);
   if (ERROR)
   {
     Parallel::Dump_Data(GH->PatL[lev], StateList, 0, PhysTime, dT_lev);
@@ -4902,10 +4925,6 @@ void bssn_class::Step(int lev, int YN)
     }
   }
 
-  //   misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"before Predictor sync");
-
-  Parallel::Sync(GH->PatL[lev], SynchList_pre, Symmetry);
-
 #if (MAPBH == 0)
   // for black hole position
   if (BH_num > 0 && lev == GH->levels - 1)
@@ -5083,22 +5102,11 @@ void bssn_class::Step(int lev, int YN)
 
     //   misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"before Corrector error check");
 
-    // check error information
+    // Non-blocking error reduction overlapped with Sync to hide Allreduce latency
+    MPI_Request err_req_cor;
     {
       int erh = ERROR;
-      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, GH->Commlev[lev]);
-    }
-    if (ERROR)
-    {
-      Parallel::Dump_Data(GH->PatL[lev], SynchList_pre, 0, PhysTime, dT_lev);
-      if (myrank == 0)
-      {
-        if (ErrorMonitor->outfile)
-          ErrorMonitor->outfile << "find NaN in RK4 substep#" << iter_count 
-                                << " variables at t = " << PhysTime 
-                                << ", lev = " << lev << endl;
-        MPI_Abort(MPI_COMM_WORLD, 1);
-      }
+      MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, GH->Commlev[lev], &err_req_cor);
     }
 
     //    misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"before Corrector sync");
@@ -5107,6 +5115,21 @@ void bssn_class::Step(int lev, int YN)
 
     //    misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"after Corrector sync");
 
+    // Complete non-blocking error reduction and check
+    MPI_Wait(&err_req_cor, MPI_STATUS_IGNORE);
+    if (ERROR)
+    {
+      Parallel::Dump_Data(GH->PatL[lev], SynchList_pre, 0, PhysTime, dT_lev);
+      if (myrank == 0)
+      {
+        if (ErrorMonitor->outfile)
+          ErrorMonitor->outfile << "find NaN in RK4 substep#" << iter_count
+                                << " variables at t = " << PhysTime
+                                << ", lev = " << lev << endl;
+        MPI_Abort(MPI_COMM_WORLD, 1);
+      }
+    }
+
 #if (MAPBH == 0)
     // for black hole position
     if (BH_num > 0 && lev == GH->levels - 1)
@@ -5390,21 +5413,11 @@ void bssn_class::SHStep()
 #if (PSTR == 1 || PSTR == 2)
 //   misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"before Predictor's error check");
 #endif
-  // check error information
+  // Non-blocking error reduction overlapped with Synch to hide Allreduce latency
+  MPI_Request err_req;
   {
     int erh = ERROR;
-    MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-  }
-
-  if (ERROR)
-  {
-    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
-    if (myrank == 0)
-    {
-      if (ErrorMonitor->outfile)
-        ErrorMonitor->outfile << "find NaN in state variables on Shell Patches at t = " << PhysTime << endl;
-      MPI_Abort(MPI_COMM_WORLD, 1);
-    }
+    MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD, &err_req);
   }
 
   {
@@ -5416,12 +5429,25 @@ void bssn_class::SHStep()
     {
       prev_clock = curr_clock;
       curr_clock = clock();
-      cout << " Shell stuff synchronization used " 
-           << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC) 
+      cout << " Shell stuff synchronization used "
+           << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC)
            << " seconds! " << endl;
     }
   }
 
+  // Complete non-blocking error reduction and check
+  MPI_Wait(&err_req, MPI_STATUS_IGNORE);
+  if (ERROR)
+  {
+    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
+    if (myrank == 0)
+    {
+      if (ErrorMonitor->outfile)
+        ErrorMonitor->outfile << "find NaN in state variables on Shell Patches at t = " << PhysTime << endl;
+      MPI_Abort(MPI_COMM_WORLD, 1);
+    }
+  }
+
   // corrector
   for (iter_count = 1; iter_count < 4; iter_count++)
   {
@@ -5564,21 +5590,11 @@ void bssn_class::SHStep()
         sPp = sPp->next;
       }
     }
-    // check error information
+    // Non-blocking error reduction overlapped with Synch to hide Allreduce latency
+    MPI_Request err_req_cor;
     {
       int erh = ERROR;
-      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-    }
-    if (ERROR)
-    {
-      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
-      if (myrank == 0)
-      {
-        if (ErrorMonitor->outfile)
-          ErrorMonitor->outfile << "find NaN on Shell Patches in RK4 substep#" << iter_count 
-                                << " variables at t = " << PhysTime << endl;
-        MPI_Abort(MPI_COMM_WORLD, 1);
-      }
+      MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD, &err_req_cor);
     }
 
     {
@@ -5590,12 +5606,26 @@ void bssn_class::SHStep()
       {
         prev_clock = curr_clock;
         curr_clock = clock();
-        cout << " Shell stuff synchronization used " 
-             << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC) 
+        cout << " Shell stuff synchronization used "
+             << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC)
              << " seconds! " << endl;
       }
     }
 
+    // Complete non-blocking error reduction and check
+    MPI_Wait(&err_req_cor, MPI_STATUS_IGNORE);
+    if (ERROR)
+    {
+      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
+      if (myrank == 0)
+      {
+        if (ErrorMonitor->outfile)
+          ErrorMonitor->outfile << "find NaN on Shell Patches in RK4 substep#" << iter_count
+                                << " variables at t = " << PhysTime << endl;
+        MPI_Abort(MPI_COMM_WORLD, 1);
+      }
+    }
+
     sPp = SH->PatL;
     while (sPp)
     {