A class of single-stage fully-discrete weighted compact nonlinear scheme for hyperbolic conservation laws

In this paper, we propose a single-stage weighted compact nonlinear scheme based on the framework of solution formula method, which can achieve arbitrary consistent spatial and temporal accuracy for hyperbolic conservation laws. The main idea in the construction of the new scheme consists of three parts. Firstly, we construct and discretize the (quasi-) exact solution of the Hamilton–Jacobi equation by a flux linearization technique, due to the flux of it can be written in a same form with conservation law. Once we obtain the numerical flux, it is directly applied to construct conservative schemes. Secondly, we apply a shock detector in the flux linearization procedure to enhance its robustness property in high-order situation. Finally, with a combination of the weighted compact nonlinear scheme, we achieve a single-stage high-order scheme with no need of the time-consuming Runge–Kutta or complicate Lax–Wendroff method. The algorithm was numerically validated by various one-dimensional and multi-dimensional test cases, are thoroughly analyzed through both theoretical considerations and numerical experiments. Numerous numerical results demonstrated that the proposed method had an essentially similar even better performance as that based on Runge–Kutta method, while its computational speed is approximately 2.4 times faster than three-stage TVD-RK3 for Euler equations.