Superconvergence analysis of symmetric Gauss-type exponential collocation integrators for solving the multidimensional nonlinear first-order partial differential equations - Wu, Xiumei; Fang, Yonglei; Liu, Changying; Song, Yuanling (Journal of mathematical chemistry, 2025)
The main objective of this research is to develop and analyze high-order symmetric Gauss-type exponential collocation time-stepping methods for solving systems of nonlinear first-order partial differential equations (PDEs). Initially, the nonlinear PDEs are reformulated as an abstract Hamiltonian ordinary differential equation (ODE) system in an appropriate infinite-dimensional function space. Subsequently, the Gauss-type exponential collocation time integrators are derived. The symmetry, local error bounds and nonlinear stability of the proposed time integrators are rigorously analysed in details. Furthermore, the rigourous convergence analysis demonstrates that Gauss-type exponential collocation time integrators can achieve superconvergence. Numerical experiments verify our theoretical analysis results, and demonstrate the remarkable superiority in comparison with the traditional temporal integration methods.