SAV方法求解Allen-Cahn方程的数值比较

The Numerical Comparison of SAV Methods for the Allen-Cahn Equation

该文研究基于标量辅助变量(SAV)格式下求解Allen-Cahn方程的数值比较.首先给出1维Allen-Cahn方程的SAV格式;然后,对方程的时间方向采用2阶向后差分(BDF2)格式和Crank-Nicolson(CN)格式离散,对方程的空间方向采用重心Lagrange插值配点法和2阶中心差分法离散,用离散正弦变换(DST)、快速傅里叶变换(FFT)求解差分导出的线性代数方程组;最后,通过数值算例验证重心Lagrange插值配点法是指数收敛,与差分格式比较,配点格式用较少的点就能达到较高的精度且耗时少,并进一步验证几种SAV离散格式都满足能量递减规律.

The numerical comparison of the Allen-Cahn equation based on the SAV approach is studied in this paper.Firstly,tthe SAV scheme of the one-dimensional Allen-Cahn equation is given.Then,the equations are discretized by BDF2 and CN for the time direction,and in the spatial direction using the barycentric Lagrange interpolation collocation method and the second-order central difference method,and DST/FFT are used to solve linear algebraic equations which is derived by the finite difference method.Finally,numerical examples are given to verify the exponential convergence of the barycentric Lagrange interpolation collocation method.Compared with the difference scheme,the collocation scheme can achieve high accuracy with fewer points and less time consuming.Several SAV schemes satisfy the law of energy decreasing.