Space-Time Spectral Element Methods in Fluid Dynamics and Materials Science

In this manuscript, we propose space-time spectral element methods to solve problems arising from fluid dynamics and materials science. Many engineering applications require one to solve complex problems, such as flows containing multi-scale structure in either space or time or both. It is straightforward that high-order methods are always more accurate and efficient than low-order ones for solving smooth problems. For example, spectral element methods can achieve a given level of accuracy with significantly fewer degrees of freedom compared to methods with algebraic convergence rates, e.g., finite difference methods. However, when it comes to complex problems, a high order method should be augmented with, e.g., a level set method or an artificial viscosity method, in order to address the issues caused by either sharp interfaces or shocks in the solution. Complex problems considered in this work are problems with solutions exhibiting multiple scales, i.e., the Stefan problem, nonlinear hyperbolic problems, and problems with smooth solutions but forces exhibiting disparate temporal scales, such as advection, diffusion and reaction processes. Correspondingly, two families of space-time spectral element methods are introduced in order to achieve spectral accuracy in both space and time. The first category of space-time methods are the fully implicit space-time discontinuous Galerkin spectral element methods. In the fully implicit space-time methods, time is treated as an additional dimension, and the model equation is rewritten into a space-time formulation. The other category of space-time methods are specialized for problems exhibiting multiple time scales: multi-implicit space-time spectral element methods are developed. The method of lines approach is employed in the multi-implicit space-time methods. The model is first discretized by a discontinuous spectral element method in space, and the resulting ordinary differential equations are then solved by a new multi-implicit spectral deferred correction method. A novel fully implicit space-time discontinuous Galerkin (DG) spectral element method is presented to solve the Stefan problem in an Eulerian coordinate system. This method employs a 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 with a Lagrangian description, the issue of dealing with the 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 onto the new corresponding time-dependent domain. Two options for calculating the interface velocity are presented, and both options exhibit spectral accuracy. Benchmark tests in one spatial dimension indicate that the method converges with spectral accuracy in both space and time for the temperature distribution and the interface velocity. The interrelation between the interface position and the temperature makes the Stefan problem a nonlinear problem; 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. We also apply the fully implicit space-time DG spectral element method to solve nonlinear hyperbolic problems. The space-time method is combined with two different approaches for treating problems with discontinuous solutions: (i) space-time dependent artificial viscosity is introduced in order to capture discontinuities/shocks, and (ii) the sharp discontinuity is tracked with space-time spectral accuracy, as it moves through the grid. To capture the discontinuity whose location is initially unknown, an artificial viscosity term is strategically introduced, and the amount of artificial viscosity varies in time within a given space-time slab. It is found that spectral accuracy is recovered everywhere except in the "troublesome element(s)'' where the unresolved steep/sharp gradient exists. When the location of a discontinuity is initially known, a space-time spectrally accurate tracking method has been developed so that the spectral accuracy of the position of the discontinuity and the solution on either side of the discontinuity is preserved. A Picard iteration method is employed to handle nonlinear terms. Within each Picard iteration, a linear system of equations is solved, which is derived from the space-time DG spectral element discretization. Spectral accuracy in both space and time is first demonstrated for the Burgers' equation with a smooth solution. For tests with discontinuities, the present space-time method enables better accuracy at capturing the shock strength in the element containing shock when higher order polynomials in both space and time are used. Moreover, the spectral accuracy of the shock speed and location is demonstrated for the solution of the inviscid Burgers' equation obtained by the shock tracking method, and the sensitivity of the number of Picard iterations to the temporal order is discussed. The dynamics of many physical and biological systems involve two or more processes with a wide difference of characteristic time scales, e.g., problems with advection, diffusion and reaction processes. The computational cost of solving a coupled nonlinear system of equations is expensive for a fully implicit (i.e., "monolithic") space-time method. Thus, we develop another type of a space-time spectral element method, which is referred to as the multi-implicit space-time spectral element method. Rather than coupling space and time together, the method of lines is used to separate the discretization of space and time. The model is first discretized by a discontinuous spectral element method in space and the resulting ordinary differential equations are then solved by a new multi-implicit spectral deferred correction method. The present multi-implicit spectral deferred correction method treats processes with disparate temporal scales independently, but couples them iteratively by a series of deferred correction steps. Compared to lower order operator splitting methods, the splitting error in the multi-implicit spectral deferred correction method is eliminated by exploiting an iterative coupling strategy in the deferred correction procedure. For the spectral element discretization in space, two advective flux reconstructions are proposed: extended element-wise flux reconstruction and non-extended element-wise flux reconstruction. A low-order I-stable building block time integration scheme is introduced as an explicit treatment for the hyperbolic terms in order to obtain a stable and efficient building block for the spectrally accurate space-time scheme along with these two advective flux reconstructions. In other words, we compare the extended element-wise reconstruction with I-stable building block scheme with the non-extended element-wise reconstruction with I-stable building block scheme. Both options exhibit spectral accuracy in space and time. However, the solutions obtained by extended element-wise flux reconstruction are more accurate than those yielded by non-extended element-wise flux reconstruction with the same number of degrees of freedom. The spectral convergence in both space and time is demonstrated for advection-diffusion-reaction problems. Two different coupling strategies in the multi-implicit spectral deferred correction method are also investigated and both options exhibit spectral accuracy in space and time.