非局部粘性水波模型的直接间断Galerkin算法

Direct Discontinuous Galerkin Algorithm for a Nonlocal Viscous Water Wave Model

摘要: 本文基于直接间断有限元(DDG)方法数值求解非局部粘性水波模型。该算法结合L1近似公式与BDF2方法，系统构建了非线性时间分数阶偏微分方程的高效数值算法。首先，运用分部积分法对模型的弱形式进行降阶处理。其次，通过引入边界项和选用合适的数值通量，确保离散格式的稳定性。最后，针对时间导数项应用L1近似公式与BDF2时间差分的离散方法，建立全离散DDG格式。文中详细给出数值格式的构造过程并严格证明该算法的稳定性。数值实验部分选取无已知解析解的水波模型，验证该算法在时空离散上的高精度特性。

Abstract: This paper numerically solves the nonlocal viscous water wave model based on the Direct Discontinuous Galerkin (DDG) method. This algorithm combines the L1 approximation formula with the BDF2 method to systematically construct an efficient numerical algorithm for nonlinear time-fractional partial differential equations. Firstly, the integration by parts method is used to reduce the order of the weak formula. Secondly, the stability of the discrete scheme is ensured by introducing the boundary term and constructing a stable numerical flux. Finally, for the time derivative term, the discrete methods of the L1 approximation formula and the BDF2 time difference are applied to construct the fully discrete DDG scheme. This paper gives a detailed description of the construction process of the numerical scheme and strictly proves the stability of this algorithm. In the numerical experiment part, a water wave model without a known analytical solution is selected to verify the high-precision characteristics of this algorithm in space-time discretization.