Uncertainty quantification of coupled odes with stochastic galerkin methods using adaptive higher order Runge–Kutta methods

Many real world problems are so complex that simplifications of these problems are needed. Otherwise the computing costs would be so high that specific problems, for example uncertainty quantification, could not be solved. In this paper we consider a system of coupled ODEs and discretise the subsystems in time with adaptive high order Runge–Kutta methods. This approach is called ”partitioned method”, and we use a Block Gauss-Seidel method for solving the final linear or non-linear systems. The motivation for using high order methods is the computation of very accurate numerical results. Moreover, these time integration methods are more effective than lower order methods, and in the case of the partitioned approach they need less iterations than lower order methods. For the stochastic discretisation we use a stochastic Galerkin method which only needs a few solutions of the deterministic ODE system. We show that using higher order methods in time leads to a better and more accurate quantification of uncertainties because we can expect a higher accuracy and a faster convergence for the deterministic problem. Numerical results show the advantages of the novel approach