We apply the ultraspherical spectral method to solving time-dependent PDEs by proposing two approaches to discretization based on the method of lines and show that these approaches produce approximately same results. We analyze the stability, the error, and the computational cost of the proposed method. In addition, we show how adaptivity can be incorporated to offer adequate spatial resolution efficiently. Both linear and nonlinear problems are considered. We also explore time integration using exponential integrators with the ultraspherical spatial discretization. Comparisons with the Chebyshev pseudospectral method are given along the discussion and they show that the ultraspherical spectral method is a competitive candidate for the spatial discretization of time-dependent PDEs.

翻译：本文提出两种基于直线法的离散化方式，将超球面谱方法应用于求解含时偏微分方程，并证明这两种方式可得到近似一致的结果。我们分析了所提方法的稳定性、误差及计算成本，同时展示了如何融入自适应机制以有效提供充足的空间分辨率。研究涵盖线性和非线性问题，并探讨了结合超球面空间离散化的指数积分器进行时间积分的方法。通过对比切比雪夫伪谱方法，我们论证了超球面谱方法作为含时偏微分方程空间离散化候选方案的竞争力。