数值求解波动方程是大尺度正演模拟、基于波动方程的地震偏移和反演成像的关键。为了提高长时间波场模拟的计算精度和效率，本项目拟将时间上保持辛结构的龙格-库塔方法与基于内部罚函数的间断有限元方法相结合，提出一种兼具二者优点的保辛间断有限元新方法。该方法不仅具有弱数值频散、易处理复杂边界、容易拓展到高精度等特性，还能在长时间模拟中保持波动方程的辛结构。本项目拟研究该保辛间断有限元方法的时间及空间离散格式；研究相应的理论问题（如稳定性条件、数值频散等）；研究各向异性等复杂介质下的非一致网格算法；将此保辛间断有限元方法与有限差分方法相结合，研究压制数值频散能力强、计算效率高、易处理复杂几何边界的混合方法。该项目的完成，将获得高精度的保辛间断有限元方法及混合方法，有助于解决复杂情形下地震波模拟中存在的数值频散严重、计算效率低等的问题，并能显著提高波场模拟的质量，为复杂油气藏的精准勘探奠定理论和方法基础。

Numerical solving wave equations is the key point of the forward wave field simulation, seismic migration and the inversion imaging. In order to improve the accuracy and efficiency of long-time wave field simulation, we combine the symplectic Runge-Kutta method with the interior penalty discontinuous Galerkin method, and propose a new symplectic discontinuous Galerkin method which has the advantages of both. This new method can effectively suppress numerical dispersion and handle complex boundary. It is also easy to expand to high accuracy and keep the symplectic structure of wave equation in long-term simulation. We intend to study the time and space discretization schemes of the symplectic discontinuous Galerkin method, and investigate the corresponding theoretical problems (such as stability conditions, numerical dispersion, etc.). Afterwards, this new method is applied to simulate several complicated geologic media such as anisotropic medium and viscoelastic medium. We also apply irregular meshes to investigate the wave propagation in complicated geologic models. Moreover, we will combine the new symplectic discontinuous Galerkin method with high-order finite difference method. A hybrid scheme which can suppress numerical dispersion efficiently and can significantly improve the computational efficiency is developed. This hybrid method is applied for simulating the wave propagation in complicated structure media. The accomplishment of this project will achieve a new symplectic discontinuous Galerkin method and a new hybrid method, which will give a solution for the problem of large numerical dispersion and low computational efficiency in numerical simulations for complicated models, and will also improve the wave-field character. The accomplishment of this project will lay a good theoretical and methodological foundation for accurate exploration of complex reservoirs.