The use of high-order accurate spatial discretizations for simulating steady and unsteady fluid flows has become more widespread over the last decade, For unsteady simulations using high-order spatial discretizations with a wide disparity between temporal and spatial scales, higher-order implicit time-integration approaches are desirable. However, such approaches require the efficient solution of a non-linear problem at each time-step in order to remain competitive. In this work, we investigate the use of spectral multigrid methods for driving implicit time-integration schemes of high-order discontinuous Galerkin discretizations, using the linear two-dimensional wave equation as a model problem.

The method of lines is employed, in which the wave equation is first discretized in space, resulting in a large set of coupled ordinary differential equations, which are then discretized and integrated in time. The Discontinuous Galerkin (DG) method represents a spatial discretization approach based on a finite-element method, which makes use of element based basis functions which are discontinuous across element interfaces [1]. In this approach, the computational domain is partitioned into an ensemble of non-overlapping elements and within each element the solution is approximately by a truncated polynomial expansion. The solution is thus determined by the modal coefficients of the expansion in terms of the basis functions within each element. Because the resulting solution representation is discontinuous across element interfaces, an upwind numerical flux function is used to resolve the discontinuity at element interfaces. The current implementation uses a set of hierarchical basis functions on triangles [2], enabling solutions from 1st order (p=0), up to 4th order (p=3) spatial accuracy.

In order to solve the time-dependent problem, the resulting spatially discretized equations must be integrated in time. Although explicit time-integration schemes have been widely used for DG discretizations, in this work we concentrate on the use of implicit time-integration schemes, which are not restricted by the stability limit of explicit methods, and are more suitable for stiff problems. The implicit time-integration schemes employed in this work range from first to fourth-order accurate in time, including both first and second order accurate multistep backward difference formulas(BDF1, BDF2) and a fourth-order accurate implicit multistage Runge-Kutta scheme. The use of implicit Runge-Kutta schemes represents an attempt to balance the spatial and temporal orders of accuracy. Moreover, even for second-order accurate finite-volume schemes, fourth-order implicit Runge-Kutta schemes have been found to outperform BDF2 schemes for engineering accuracy levels [3].

At each time-step, implicit time-integration methods require the solution of one or more non-linear problems. Efficient non-linear solvers are required for this task in order to result in an overall competitive approach. Our approach consists of using a spectral multigrid strategy [4,5] for solving the implicit system at each time step. The spectral or p-multigrid approach consists of a multigrid method where the coarser levels are constructed by reducing the order of accuracy (p-coarsening) while keeping the spatial grid resolution fixed (as opposed to h-coarsening). Thus, for a p=3 (fourth-order accurate) spatial discretization, three coarser multigrid levels are employed, consisting of p=2, p=1, and p=0 at the coarsest level. At each p-multigrid level, an element-Jacobi scheme is used as a smoother. The element Jacobi smoother can be viewed as an approximate Newton method, where only the Jacobian entries corresponding to the modal coupling within an element are retained, and all other entries are discarded, resulting in a block diagonal matrix, which is easily inverted using Gaussian elimination at the block level. When used as a single grid solver, this smoother is shown to produce p-independent convergence rates, with strong h-dependence. When used as a smoother within the p-multigrid scheme, both h and p independent convergence rates are obtained.

In the paper, we show both p and h independent convergence rates for the implicit systems arising from the various time discretizations using the element Jacobi driven spectral multigrid solver. The overall efficiency and accuracy of the various time-integration schemes are compared by examining the error as a function of time for various time step sizes, where the design accuracy of the respective time-integration schemes is demonstrated. Further work will focus on extending this approach to the unsteady Euler and Navier-Stokes equations.

[1]T. C. Warburton, I. Lomtev, Y. Du, S.J. Sherwin and G.E. Karniadakis, "Galerkin and Discontinuous Galerkin Spectral/hp Methods", Comput. Methods Appl. Mech. Engrg. 175(1999), 343-359.

[2]F. Graham, J. Carey and T. Oden, "Finite Elements A Second Course", Vol.2, 1983, 89~95

[3]G. Jothiprasad , D. J. Mavriplis and D. Caughey, "Higher-Order Time-Integration Schemes for the Unsteady Navier-Stokes Equations on Unstructured Meshes", Journal of Computational Physics, Vol 191, Issue 2, pp. 542-566, November 2003

[4]B. Helenbrook, D. J. Mavriplis and H. Atkins, "Analysis of "p"-Multigrid for Continuous and Discontinuous Finite Element Discretizations", Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, 2003, AIAA Paper 2003-3989,

[5]C. R. Nastase and D. J. Mavriplis, "High-Order Discontinuous Galerkin Methods using a Spectral Multigrid Approach", AIAA Paper 2005-1268, January 2005.