Convergence and Stability Analysis of Discontinuous Galerkin Time-Domain Method Based on Improved Energy Technique

This communication proposes an improved energy technique for the convergence and stability analysis of upwind-flux-based discontinuous Galerkin time-domain (DGTD) method. By formulating the discrete energy in the energy-based stability analysis technique into a global quadratic form, the global stability condition of the upwind flux-based DGTD method is analytically derived for the first time. In order to achieve the highly efficient stability condition prediction for large and multiscale complicated problems, an elementwise stability condition has been developed by applying matrix decomposition technique into the obtained global stability condition. Due to the sole approximation in the matrix decomposition, the proposed elementwise stability estimation method is more accurate than the traditional energy-based stability analysis techniques, where multiple approximations are employed to achieve the elementwise operations. The numerical examples show the good time step bound estimation accuracy of the global and elementwise stability conditions compared to the widely used energy-based techniques, which significantly enhances the DGTD method’s solution efficiency on time integration.