Reconstructed discontinuous Galerkin methods for linear advection–diffusion equations based on first-order hyperbolic system

High-Order Hyperbolic Navier-Stokes Reconstructed Discontinuous Galerkin Method

“…However, HNS(P0P1+P0) has one-order lower order of accuracy in viscous dominated regions, e.g., first-order accurate in a boundary-layer, as already pointed out for advection-diffusion problems in Ref. [8]. The efficient hyperbolic DG method can be extended systematically to higher-order, such as HNS(P 0 P 2 +P 1 ), HNS(P 0 P 3 +P 2 ), and so on.…”

“…The rDG method is a general framework for constructing efficient high-order schemes with reconstruction techniques, having the finite-volume (FV) and discontinuous Galerkin (DG) methods as special cases [3,4]. As we have shown in our previous developments [5,6,7,8,9], the two approaches can be combined systematically to simplify the discretization of diffusion/viscous terms, improve gradient accuracy, accelerate iterative convergence, and achieve higher-order accuracy with fewer numbers of degrees of freedom than DG methods. These advantages have been demonstrated for diffusion with scalar and tensor diffusion coefficients [5,6], nonlinear diffusion [7] advection-diffusion equations [8], and the NS equations [9].…”

“…The polynomials used in the flux and source evaluations are upgraded by reconstructing higher-order derivatives from the underlying DG polynomials. The resulting hyperbolic rDG schemes are known to outperform conventional DG schemes of the same number of discrete unknowns: higher-order inviscid and gradient accuracy and stiffness due to second-derivative viscous terms completely eliminated [5,6,7,8,9]. However, since the HNS system has the conservative variables as the primary variables and the gradients of the primitive variables as the gradient variables, the derivative replacement in the hyperbolic DG can be quite complicated, especially for higher-order discretizations [10].…”

“…In the previous papers, the Petrov-Galerkin was employed for diffusion in Refs. [5,6], and later, the Galerkin formulation was found to be more robust for advection-diffusion problems [8]. Further later, it was discovered that the Galerkin formulation enables accurate unsteady advection-diffusion computations by explicit time-stepping schemes [11].…”

Explicit Hyperbolic Reconstructed Discontinuous Galerkin Methods for Time-Dependent Problems

“…The rDG method is a general framework for constructing efficient high-order schemes with reconstruction techniques, having the finite-volume (FV) and DG schemes as special cases. As we have shown in our previous developments [8][9][10][11][12], the two approaches can be combined in a systematic manner to simplify the discretization of diffusion terms, improve gradient accuracy, accelerate iterative convergence, and achieve higher-order accuracy than DG methods with fewer numbers of degrees of freedom. Specifically, we have developed hyperbolic rDG schemes for diffusion with scalar and tensor diffusion coefficients [8,9], nonlinear diffusion [10] advection-diffusion equations [11], and the Navier-Stokes equations [12].…”