Enhancing stability of correction procedure via reconstruction using summation-by-parts operators I: Artificial dissipation

Abstract

The correction procedure via reconstruction (CPR, also known as flux reconstruction) is a framework of high order semidiscretisations used for the numerical solution of hyperbolic conservation laws. Using a reformulation of these schemes relying on summation-by-parts (SBP) operators and simultaneous approximation terms (SATs), artificial dissipation / spectral viscosity operators are investigated in this first part of a series. Semidiscrete stability results for linear advection and Burgers' equation as model problems are extended to fully discrete stability by an explicit Euler method. As second part of this series, Glaubitz, Ranocha, Öffner, and Sonar (Enhancing stability of correction procedure via reconstruction using summation-byparts operators II: Modal filtering, 2016) investigate connections to modal filters and their application instead of artificial dissipation.

Extended skew-symmetric form for summation-by-parts operators and varying Jacobians

“…As expected, momentum is conserved for all bases and the discrete energy (entropy) is constant until t ≈ 0.5 and decays afterwards, as can be seen in Figure 1. These results are obtained using general SBP CPR methods (19) with both correction terms for divergence and restriction (18). Ignoring a non-trivial correction term for a nodal basis leads to physically useless results, as shown for example by [19,Figure 11].…”

“…Therefore, some form of additional shock capturing should be performed, e.g. artificial dissipation or modal filtering [18,6]. However, this is not the target of this investigation.…”

“…Using these correction terms, Theorem 1 is generalised by (10), then a general SBP CPR method with C = M −1 R T B and correction terms (18) for both divergence and restriction to the boundary…”