Energy-preserving schemes for conservative PDEs based on periodic quasi-interpolation methods

Abstract

In this paper, we present a novel energy-preserving scheme for solving time-dependent partial differential equations with periodic solutions on non-uniform grids. The proposed scheme combines the periodic quasi-interpolation approaches with the discrete gradient method. One of the main advantages of our method is its ability to ensure energy conservation without relying on the requirement of an anti-symmetric differentiation matrix. We rigorously verify the energy conservation properties of the full-discrete scheme, as well as establish the convergence results. Furthermore, we extend the capabilities of our method by incorporating an adaptive strategy, allowing for the design of an adaptive energy-preserving scheme. This adaptive feature enhances the accuracy and efficiency of the numerical solution. To demonstrate the effectiveness and energy-preserving ability of the proposed scheme, we present several numerical examples. These examples showcase the superior accuracy and robustness of our method.