Remove the drywall around the doorway with a utility knife, exposing the wall studs.
Draw the arch of your design on a piece of cardboard, and tape it to the top of the doorway, ensuring there's enough headroom underneath.
Trace the arch onto two pieces of 1/2-inch-thick plywood, and cut them out using a saw.
Saw a piece of 2-by-4-by-72-inch lumber into a 2-by-2 1/2-inch piece. From this piece, cut three blocks to go around the top and sides of the door frame.
Nail the blocks along the top and sides of the door frame, using 1-inch nails.
Position the plywood boards on either side of the blocks and nail them into place, ensuring both the pieces of plywood are flush.
Place the blocks along the inside circumference of the arch, spaced 6 inches apart, and screw them into place using the 1 5/8-inch screws.
Nail the hardboard into the blocks, ensuring there's no space left between the plywood pieces.
Soak the drywall in water for an hour. Bend the drywall to the exact circumference of your arch and nail it into place over the hardboard, every 6 inches.
Apply 1/2-inch drywall over the entire arch frame.
Staple plastic corner beads along the edge of the arch on both sides of the doorway, every 3 inches.
Mix the joint compound and apply it to the arch and frame, covering the beads as well.
Sand the archway after the joint compound has set, and paint it to match your decor.