Solving time-dependent PDEs with the ultraspherical spectral method

TL;DR

This paper demonstrates the effectiveness of the ultraspherical spectral method for solving time-dependent PDEs, highlighting its stability, accuracy, and computational efficiency, especially when combined with adaptive spatial resolution and exponential time integrators.

Contribution

It introduces two discretization approaches using the ultraspherical spectral method for time-dependent PDEs and compares their performance with existing methods, including adaptivity and nonlinear problems.

Findings

Ultraspherical spectral method is stable and accurate for time-dependent PDEs.

The method is computationally efficient and competitive with Chebyshev pseudospectral methods.

Adaptive spatial resolution enhances efficiency without sacrificing accuracy.

Abstract

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.