Abstract
A novel space-time discontinuous Galerkin (DG) spectral element method is presented to solve the one dimensional Stefan problem in an Eulerian coordinate system. This method employs the level set procedure to describe the time-evolving interface. To deal with the prior unknown interface, a backward transformation and a forward transformation are introduced in the space-time mesh. By combining an Eulerian description, i.e., a fixed frame of reference, with a Lagrangian description, i.e., a moving frame of reference, the issue of dealing with implicitly defined arbitrary shaped space-time elements is avoided. The backward transformation maps the unknown time-varying interface in the fixed frame of reference to a known stationary interface in the moving frame of reference. In the moving frame of reference, the transformed governing equations, written in the space-time framework, are discretized by a DG spectral element method in each space-time slab. The forward transformation is used to update the level set function and then to project the solution in each phase back from the moving frame of reference to the fixed Eulerian grid. Two options for calculating the interface velocity are presented, and both options exhibit spectral accuracy. Benchmark tests indicate that the method converges with spectral accuracy in both space and time for the temperature distribution and the interface velocity. A Picard iteration algorithm is introduced in order to solve the nonlinear algebraic system of equations and it is found that just a few iterations lead to convergence.
Highlights
The Stefan problem is a moving boundary problem that models phase change; e.g., the freezing and thawing process for a solid-liquid system [49, 38, 43, 7]
Chen et al [11] introduced a simple approach in an Eulerian coordinate system for the Stefan problem, which involved the level set method and a finite difference approach for solving the heat equation in each phase and for evaluating the interface velocity
The auxiliary variables introduced in a discontinuous Galerkin (DG) method for discretizing the diffusion term [15, 14] can be used directly to compute the interface velocity in the weak formulation, which enables the temperature distribution and the interface velocity to be computed simultaneously. Motivated by these properties of space-time DG finite element methods, we present a novel space-time DG spectral element method for solving the one dimensional Stefan problem, which leads to spectral accuracy in both space and time
Summary
The Stefan problem is a moving boundary problem that models phase change; e.g., the freezing and thawing process for a solid-liquid system [49, 38, 43, 7]. Chen et al [11] introduced a simple approach in an Eulerian coordinate system for the Stefan problem, which involved the level set method and a finite difference approach for solving the heat equation in each phase and for evaluating the interface velocity. The auxiliary variables introduced in a DG method for discretizing the diffusion term [15, 14] can be used directly to compute the interface velocity in the weak formulation, which enables the temperature distribution and the interface velocity to be computed simultaneously Motivated by these properties of space-time DG finite element methods, we present a novel space-time DG spectral element method for solving the one dimensional Stefan problem, which leads to spectral accuracy in both space and time.
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