Projection-Based Reconstruction for Achieving High-Order Accuracy from Low-Order DGSEM Simulations
For computational scientists using high-order methods for conservation laws, this work reduces computational cost and memory footprint while maintaining high-order accuracy, though it is an incremental improvement over existing reconstruction techniques.
This paper develops a corrected PnPm (cPnPm) approach for DGSEM-LGL discretizations that recovers the accuracy of a high-order m-th-order approximation while evolving only low-order n-th-order degrees of freedom, achieving m+1-th convergence order for smooth solutions. Numerical experiments show competitive accuracy relative to computational cost, with efficiency gains for viscous flows.
High-order discontinuous Galerkin spectral element methods (DGSEM) based on Legendre-Gauss-Lobatto (LGL) nodes provide accurate and efficient discretizations for conservation laws. However, their cost, memory footprint, and time-step restrictions increase rapidly when the degree of the polynomial increases. This paper develops a corrected $\mathbb{P}_n\mathbb{P}_m$ ($c\mathbb{P}_n\mathbb{P}_m$) approach for DGSEM-LGL discretizations that aims to recover the accuracy of an $m^{th}$-order approximation while evolving only the degrees of freedom associated with an $n^{th}$-order representation, with $n<m$. The projected evolution of the high-order components is derived first at the continuous level and then in the fully discrete DGSEM-LGL setting. The discrete analysis shows that because LGL quadrature is not exact for the highest Legendre mode, a correction term for that mode is required to preserve the order of convergence. A compact projection-based reconstruction operator is then introduced to recover high-order components without solving the enlarged constrained least-squares systems used in standard reconstruction procedures. For sufficiently smooth solutions, the resulting $c\mathbb{P}_n\mathbb{P}_m$ scheme is shown to achieve the expected $m+1^{th}$ convergence order. Numerical experiments for one- and two-dimensional conservation laws, including Euler, viscous Burgers, and 2D decaying homogeneous isotropic turbulence, confirm theoretical convergence behavior and demonstrate competitive accuracy relative to computational cost, with particularly clear efficiency gains for viscous flows.