Annular extrudate swell simulations at high Weissenberg numbers were made using a differential type constitutive equation. The streamline-upwinding method with a sub-element for extra stress components, which is called SU4x4, is one of the best mixed finite element methods for computation of viscoelastic flows. Planar and capillary extrudate swell calculations at high Weissenberg numbers (We>1000) were accomplished by SU4X4. However, annular extrudate swell simulations at high Fire by SU4X4 were not successful. The calculated We was less than about 4. A new calculation technique using a Newton-Raphson discretization of the equation of motion was developed. This technique is called a "new under-relaxation method." The calculated We of annular extrudate swell simulation by the new under-relaxation method with SU4X4 was about 6 similar to 250 times larger than those by SU4X4. Reasonable calculation results were obtained in an annular flow and a capillary extrudate swell by this method, and the reliability and the utility of the new under-relaxation method are shown. It is now possible to consider the swell shapes of annular extrudate under industrially useful conditions. The calculated swelling ratios were also compared with experimental ones.