diff --git a/opm/simulators/linalg/HyprePreconditioner.hpp b/opm/simulators/linalg/HyprePreconditioner.hpp
index 9539967c7..17bbfec09 100644
--- a/opm/simulators/linalg/HyprePreconditioner.hpp
+++ b/opm/simulators/linalg/HyprePreconditioner.hpp
@@ -192,8 +192,10 @@ public:
 
     bool hasPerfectUpdate() const override
     {
-        // The Hypre preconditioner can depend on the values of the matrix, so it must be recreated
-        return false;
+        // The Hypre preconditioner can depend on the values of the matrix so it does not have perfect update.
+        // However, copying the matrix to Hypre requires to setup the solver again, so this is handled internally.
+        // So for ISTLSolver, we can return true.
+        return true;
     }
 
 private:
diff --git a/opm/simulators/linalg/OwningTwoLevelPreconditioner.hpp b/opm/simulators/linalg/OwningTwoLevelPreconditioner.hpp
index 5f32cccc6..98ab1cac1 100644
--- a/opm/simulators/linalg/OwningTwoLevelPreconditioner.hpp
+++ b/opm/simulators/linalg/OwningTwoLevelPreconditioner.hpp
@@ -174,7 +174,7 @@ public:
     }
 
     virtual bool hasPerfectUpdate() const override {
-        return false;
+        return twolevel_method_.hasPerfectUpdate();
     }
 
 private:
diff --git a/opm/simulators/linalg/PressureSolverPolicy.hpp b/opm/simulators/linalg/PressureSolverPolicy.hpp
index 684ed1c77..0669a725a 100644
--- a/opm/simulators/linalg/PressureSolverPolicy.hpp
+++ b/opm/simulators/linalg/PressureSolverPolicy.hpp
@@ -88,6 +88,11 @@ namespace Amg
                 return linsolver_->category();
             }
 
+            bool hasPerfectUpdate() const
+            {
+                return linsolver_->preconditioner().hasPerfectUpdate();
+            }
+
             void apply(X& x, X& b, double reduction, Dune::InverseOperatorResult& res) override
             {
                 linsolver_->apply(x, b, reduction, res);
diff --git a/opm/simulators/linalg/twolevelmethodcpr.hh b/opm/simulators/linalg/twolevelmethodcpr.hh
index 97435c82c..fbf8638ed 100644
--- a/opm/simulators/linalg/twolevelmethodcpr.hh
+++ b/opm/simulators/linalg/twolevelmethodcpr.hh
@@ -495,6 +495,12 @@ public:
   {
   }
 
+  bool hasPerfectUpdate() const
+  {
+    // The two-level method has perfect update if both the finesmoother and coarse solver do.
+    return smoother_->hasPerfectUpdate() && coarseSolver_->hasPerfectUpdate();
+  }
+
   void apply(FineDomainType& v, const FineRangeType& d)
   {
     FineDomainType u(v);