一类非线性反应-扩散方程的间断Galerkin谱元方法

Discontinuous Galerkin Spectral Element Methods for Nonlinear Reaction-Diffusion Equations

提出了一类非线性反应-扩散方程的间断Galerkin谱元方法,在每个子区间上,基本格式采用Legendre-Galerkin方法,非线性项采用Chebyshev-Gauss-Lobatto插值,跳跃项利用中心数值流量处理,时间方向应用4阶低存储Runge-Kutta格式离散.该方法处理某些初值间断问题有效,并可并行实现;给出了该方法半离散格式下的稳定性和收敛性分析,利用Chebyshev-Gauss-Lobatto插值算子在不带权意义下的逼近结果,获得了按L2-模的最优误差估计;最后,给出了连续问题和间断问题的数值算例.

Discontinuous Galerkin spectral element methods for nonlinear reaction-diffusion equations are considered.The schemes are basically in the Legendre-Galerkin form.The nonlinear term is interpolated through the Chebyshev-Gauss-Lobatto points.The jump term tackled by the central numerical flux in space variation.The fourth-order low-storage Runge-Kutta scheme is applied for time discrete inside each subinterval.The methods can be used to solve discontinuous initial value problems and implemented in a parallel way.Stability and the optimal rate of convergence in L2-norm for the semi-discrete scheme are shown using the approximate results of the Chebyshev-Gauss-Lobatto interpolation operator without a weight function.Numerical results for the continuous and discontinuous problems are given.