Rosenbrock-W methods for stochastic Galerkin systems

Abstract

The stochastic Galerkin approach is a widely used method for quantifying uncertainty in dynamical systems. A polynomial chaos expansion of physical variables is used to represent uncertainty. Thus the evolution equations for the dynamics of physical quantities are replaced by evolution equations for the polynomial chaos coefficients. The dimension of the stochastic Galerkin system is m times larger than the dimension of the original physical system, where m is the number of polynomial chaos coefficients. The cost of implicit time integration methods, required for stiff dynamical systems, increases considerably, for example, by a factor m3 when direct linear solvers are used.This work studies efficient implicit methods for the time integration of stiff stochastic Galerkin systems consisting of ordinary differential equations. Linearly-implicit Rosenbrock–Wanner schemes are considered. A block-diagonal approximation of the Jacobian of the stochastic Galerkin system is employed, where the diagonal blocks are the Jacobian of the physical system, evaluated at the mean solution value. We perform a theoretical study of numerical stability for this approximation, using a stochastic extension of the Dahlquist test equation, and propose a concept of linear stochastic Galerkin stability. A rigorous analysis is done for methods of order one and two. A third-order scheme is designed to investigate the stability properties by numerical experiments. Numerical results confirm the theoretical findings.