Cut two 1/2-inch plywood frame pieces for the arched portion of the door. Hold the plywood frame in the door to verify if looks the way you desire.
Rip down a 2-by-4 board reducing the width by 1 inch. Use this board between the two plywood pieces. By reducing the width of the 2-by-4, the frame will be the same width as the stud wall.
Cut the 2-by-4 to lengths needed to create braces between two pieces of plywood. Place pieces of 2-by-4 on about every 6 inches in the sloped portion of the arch. Fasten the arch framework in place so the plywood sheets are flush with the outside of the stud walls.
Cover the outside of the walls with drywall. The drywall can be left square with the sheet extending past the arch framework. Cut the drywall with a jigsaw to fit the curve of the arch.
Cover the bottom part of the arch with 1/4-inch drywall. This thinner drywall bends to accommodate the curve of the arch. Fasten in place with 1 1/2-inch drywall screws.
Place a flexible outside corner bead on the arch corners. Place straight outside corner bead on the outside of the lower portion of the doorway that is straight.
Apply joint compound to the corners and the screw holes. Sand the compound smooth and apply a second and third coat of joint compound. At this point the arch should be ready to paint or texture.