指数时程差分与有理谱配点法求解奇异摄动Burgers-Huxley问题

THE EXPONENTIAL TIME DIFFERENCING AND RATIONAL SPECTRAL COLLOCATION METHOD FOR SINGULARLY PERTURBED BURGERS-HUXLEY PROBLEM

带小参数ε的Burgers-Huxley方程是一类非线性、非定常奇异摄动初边值问题,本文用指数时程差分与有理谱配点法求其数值解.对空间方向的边界层,用带sinh变换的有理谱配点法使Chebyshev节点在边界层处加密,只需取较少节点即可达到较高精度;时间方向采用指数时程差分与4阶Runge-Kutta法相结合的格式,并用围线积分计算矩阵函数的方法克服了求解奇异摄动问题时遇到的的数值不稳定难题.数值实验表明,本文提出的方法在求解左、右边界层和内部层的奇异摄动Bugers-Huxley问题都有较高的精度.

The exponential time differencing （ETD） and rational spectral collocation （RSC） method are applied to numerically solve the Burgers-Huxley equation with small parameter ε, which is a nonlinear, unsteady, singularly perturbed, initial and boundary problem. The RSC method with sinh transform is employed to treat the boundary or interior layer in spatial direction while the combination of ETD and forth-order Runge-Kutta （ETDRK4） method is used to discretize the time variable. The matrix function in ETDRK4 is computed by contour integral in the complex plan, which overcomes the problem of instability. Numerical experiments illustrate the high accuracy and efficiency of our method.