Efficient high-order discontinuous Galerkin schemes with first-order hyperbolic advection-diffusion system approach

We propose arbitrary high-order discontinuous Galerkin (DG) schemes that are designed based on a first-order hyperbolic advection-diffusion formulation of the target governing equations. We present, in details, the efficient construction of the proposed high-order schemes (called DG-H), and show that these schemes have the same number of global degrees-of-freedom as comparable conventional high-order DG schemes, produce the same or higher order of accuracy solutions and solution gradients, are exact for exact polynomial functions, and do not need a second-derivative diffusion operator. We demonstrate that the constructed high-order schemes give excellent quality solution and solution gradients on irregular triangular elements. We also construct a Weighted Essentially Non-Oscillatory (WENO) limiter for the proposed DG-H schemes and apply it to discontinuous problems. We also make some accuracy comparisons with conventional DG and interior penalty schemes. A relative qualitative cost analysis is also reported, which indicates that the high-order schemes produce orders of magnitude more accurate results than the low-order schemes for a given CPU time. Furthermore, we show that the proposed DG-H schemes are nearly as efficient as the DG and Interior-Penalty (IP) schemes as these schemes produce results that are relatively at the same error level for approximately a similar CPU time. We propose an efficient high-order discontinuous-Galerkin (DG) schemes with the first-order hyperbolic advection-diffusion formulation.The proposed DG-H schemes have the same DoF as classical DG schemes for the same accuracy without a second-order diffusion operator.DG-H ( P k ) is (k + 2)-order accurate for solution and (k + 1)-order accurate for gradients on irregular elements for advection problems.DG-H ( P k ) gives (k + 1)-order accurate for both solution and gradients on irregular elements for general advection-diffusion problems.The DG-H schemes are nearly as efficient as DG and IP schemes with relatively the same error for nearly similar CPU time.

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