针对一维四阶线性方程,研究了一种隐显式Runge-Kutta全离散局部间断Galerkin方法的稳定性和最优误差估计。空间离散采用局部间断Galerkin方法,时间离散选用强稳定显式Runge-Kutta方法和具有L稳定对角隐式Runge-Kutta方法相结合的三阶隐显式Runge-Kutta方法,数值流通量采用广义交替数值流通量,从而得到全离散LDG格式,分析了该格式的稳定性,同时引入全局Gauss-Radau投影,证明该格式具有k+1阶收敛。最后通过数值实验验证理论结果的正确性。The stability and error estimation of an implicit-explicit Runge-Kutta fully discrete local discontinuous Galerkin method for one-dimensional fourth-order linear equations are studied. The local discontinuity Galerkin method is used for spatial discretization, and the third-order implicit-explicit Runge-Kutta method combining the strong-stability-preserving explicit Runge-Kutta method and the implicit Runge-Kutta method with L-stable diagonal implicit is used for time marching, and the numerical circulation adopts the generalized alternating numerical flux, so as to obtain the fully discrete LDG scheme, and the stability of the scheme is analyzed, and the generalized Gauss-Radau projection is introduced to prove that the scheme has k+1order convergence. Finally, the theoretical results are verified by numerical experiments.
双曲偏微分方程是重要的偏微分方程之一。提出求解电报方程的Chebyshev谱法,采用Chebyshev-Gauss-Lobatto配点,利用Chebyshev多项式构造导数矩阵,将电报方程近似为常微分方程,证明了电报方程的离散Chebyshev谱法的误差估计,采用Runge-Kutta进行求解。将该法得到的数值结果与精确解进行比较,验证了方法的有效性,数据结果的误差与其他方法相比有较高的精确度。Hyperbolic partial differential equation is one of the important partial differential equations. The Chebyshev spectral method is proposed to solve the telegraph equation. Chebyshev-gauss-lobatto is used to assign points, the derivative matrix is constructed by Chebyshev polynomial, and the telegraph equation is approximated as an ordinary differential equation. The error estimation of the discrete Chebyshev spectral method for the telegraph equation was proved. Runge-Kutta was used to solve the problem. The numerical results obtained by the method are compared with the exact solution, and the effectiveness of the method is verified. The error of the data results is more accurate than that of other methods.
