High-order compact fully discrete schemes for inviscid compressible flows simulations

In this paper, we propose an single-stage compact fully discrete scheme based on the weighted essentially non-oscillatory reconstruction and the solution formula method. The novelty of this scheme relies on three aspects. First, unlike existing single-stage compact schemes that usually require additional computation and evolution of the derivatives, the proposed scheme construct the compact stencil by utilizing the (quasi-)exact solution of the Hamilton–Jacobi equation, which is time efficient as it does not require cumbersome Lax–Wendroff method. Next, to enhance the numerical stabilities for non-smooth flows, we propose a modified entropy condition that specifically designed for the single-stage compact fully discrete scheme. In addition, we also develop a consistent evolution strategy for extending this compact fully discrete scheme to multi-dimensional situations, which can further improve the efficiency of it. A number of one-dimensional and multi-dimensional numerical examples have been tested to demonstrate its accuracy, correctness, and efficiency. For most of the cases, this single-stage compact fully discrete scheme significantly outperforms the original non-compact fully discrete scheme and semi-discrete scheme. The acceleration of this scheme compared to the s-stage semi-discrete scheme can achieve 60%–80% of the theoretical value, i.e., the stage number s, which is considerably larger than that of the existing single-stage compact fully discrete scheme.