Vol. 27, No. 3, pp. 1118-1139 | DONGBIN XIU† AND JAN S. HESTHAVEN‡
This paper introduces a high-order stochastic collocation method for solving differential equations with random inputs. The method combines the advantages of Monte Carlo methods and stochastic Galerkin methods, offering both fast convergence and straightforward numerical implementation. Unlike stochastic Galerkin methods, which require solving a set of coupled deterministic equations, stochastic collocation methods solve a set of decoupled deterministic problems at each interpolation point. The paper discusses the construction of collocation points, including tensor product nodal sets and sparse grids, and compares their performance in terms of accuracy and computational efficiency. Numerical examples demonstrate the effectiveness of the proposed method, particularly in high-dimensional random spaces, where it outperforms Monte Carlo methods and is more efficient than stochastic Galerkin methods. The method is also applied to time-dependent problems, showing its versatility and practicality.This paper introduces a high-order stochastic collocation method for solving differential equations with random inputs. The method combines the advantages of Monte Carlo methods and stochastic Galerkin methods, offering both fast convergence and straightforward numerical implementation. Unlike stochastic Galerkin methods, which require solving a set of coupled deterministic equations, stochastic collocation methods solve a set of decoupled deterministic problems at each interpolation point. The paper discusses the construction of collocation points, including tensor product nodal sets and sparse grids, and compares their performance in terms of accuracy and computational efficiency. Numerical examples demonstrate the effectiveness of the proposed method, particularly in high-dimensional random spaces, where it outperforms Monte Carlo methods and is more efficient than stochastic Galerkin methods. The method is also applied to time-dependent problems, showing its versatility and practicality.