Numerical study of uncertainty quantification techniques for implicit stiff systems

Galerkin polynomial chaos and collocation methods have been widely adopted for uncertainty quantification purpose. However, when the stiff system is involved, the computational cost can be prohibitive, since stiff numerical integration requires the solution of a nonlinear system of equations at every time step. Applying the Galerkin polynomial chaos to stiff system will cause a computational cost increase from O(n3) to O(S3n3). This paper explores uncertainty quantification techniques for stiff chemical systems using Galerkin polynomial chaos, collocation and collocation least-square approaches. We propose a modification in the implicit time stepping process. The numerical test results show that with the modified approach, the run time of the Galerkin polynomial chaos is reduced. We also explore different methods of choosing collocation points in collocation implementations and propose a collocation least-square approach. We conclude that the collocation least-square for uncertainty quantification is at least as accurate as the Galerkin approach, and is more efficient with a well-chosen set of collocation points.