An exponentially accurate spectral reconstruction technique for the single-phase one-dimensional Stefan problem with constant coefficients

摘要

The Stefan problem represents a large class of physical phenomena ranging from heat diffusion during phase change to the shoreline movement problem. The numerical solution of the Stefan problem requires special attention to the accuracy of the scheme to restrict the propagation of error and consequently avoid non-physical solutions or an increased computational cost. We propose a novel fixed grid scheme using pseudospectral methods, referred to as the spectral reconstruction technique, to obtain an accurate solution of the one-dimensional single-phase Stefan problem with constant coefficients. The technique requires the solution in the spatial or the temporal direction to be decomposed into an infinitely-differentiable smooth function and a step function centered at the interface. The infinitely-differentiable smooth function is treated as a sum of Chebyshev polynomials, while a weighted Heaviside step function is used to impose the Stefan condition at the interface exactly. The weighting function is expressed in a weak form using the interface jump conditions. In this article, we use the spectral reconstruction technique in the spatial direction and two schemes for the temporal direction. At first, the Crank-Nicolson method is used for temporal discretization and then the spectral reconstruction technique. Three instances of the one dimensional Stefan problem, where the solution varies within one of the phases only, are studied: the Stefan melting problem, the Frank sphere solidification problem, and the variable flux shoreline movement problem. We present quantitative comparisons of the computed interface location with existing literature for the melting and the shoreline movement problem. The convergence of the numerical method is clarified via presenting the errors in the maximum norm for the spectral discretization scheme. Between the two temporal schemes, the Crank-Nicolson time integration is easier to implement and requires less memory, but yields algebraic convergence along with non-physical oscillations during interface grid crossing. The spectral reconstruction temporal scheme achieves exponential convergence in the maximum norm while requiring variable grid time integration.