一种求解Benjamin-Bona-Mahony-Burgers方程的改进三角样条方法

An Improved Trigonometric Spline Method for Solving the Benjamin-Bona-Mahony-Burgers Equation

摘要: 针对Benjamin-Bona-Mahony-Burgers equation的高精度数值求解问题，本文构造了一种融合Hermite插值与紧致差分思想的三次三角样条配置方法。时间离散采用Crank-Nicolson格式以保证二阶时间精度；空间离散方面，通过在三次三角样条基函数中引入Hermite插值修正二阶导数，并结合一阶导数的四阶紧致差分格式，实现空间四阶高精度逼近。在理论分析方面，基于傅里叶方法证明了所构造格式的无条件稳定性，并通过数值收敛实验验证了该方法在时间方向具有二阶收敛精度、在空间方向具有四阶收敛精度。数值实验结果表明，该方法在保持稳定性的同时，相较已有方法在空间精度与整体误差控制方面具有更优表现，尤其在细网格条件下展现出良好的收敛性与稳定性。

Abstract: For the high-accuracy numerical solution of the Benjamin-Bona-Mahony-Burgers equation, a cubic trigonometric spline collocation method incorporating Hermite interpolation and compact finite difference techniques is proposed. In the temporal direction, the Crank-Nicolson scheme is employed to achieve second-order accuracy. In the spatial discretization, Hermite interpolation is introduced into the cubic trigonometric spline basis to improve the approximation of the second derivative, while a fourth-order compact finite difference scheme is applied to the first derivative, resulting in an overall fourth-order spatial accuracy. From a theoretical perspective, the unconditional stability of the proposed scheme is established via Fourier analysis, and numerical convergence tests verify that the method achieves second-order convergence in time and fourth-order convergence in space. Numerical experiments confirm that the proposed method not only maintains stability but also outperforms existing methods in terms of spatial accuracy and overall error control. In particular, it exhibits excellent convergence behavior and stability under fine mesh conditions.