diff --git a/hydrogym/firedrake/solvers/bdf_ext.py b/hydrogym/firedrake/solvers/bdf_ext.py
index 756243d..ec2dbc3 100644
--- a/hydrogym/firedrake/solvers/bdf_ext.py
+++ b/hydrogym/firedrake/solvers/bdf_ext.py
@@ -156,7 +156,7 @@ def step(self, iter, control=None):
     self.flow.set_control(bc_scale)
 
     # Solve the linear problem
-    if iter > self.k - 1:
+    if (self.k == 1) or (iter > self.k - 1):
       self.petsc_solver.solve()
     else:
       self.startup_solvers[iter - 1].solve()
diff --git a/test/test_cyl.py b/test/test_cyl.py
index bc6f76c..e41fca4 100644
--- a/test/test_cyl.py
+++ b/test/test_cyl.py
@@ -1,3 +1,4 @@
+import pytest
 import firedrake as fd
 import numpy as np
 
@@ -41,10 +42,11 @@ def test_steady_rotation(tol=1e-3):
   assert abs(CD - 1.49) < tol  # Re = 100
 
 
-def test_integrate():
+@pytest.mark.parametrize("k", [1, 3])
+def test_integrate(k):
   flow = hgym.Cylinder(mesh="medium")
   dt = 1e-2
-  hgym.integrate(flow, t_span=(0, 10 * dt), dt=dt)
+  hgym.integrate(flow, t_span=(0, 10 * dt), dt=dt, order=k)
 
 
 # Simple opposition control on lift