Multiderivative Combined Dissipative Compact Scheme Satisfying Geometric Conservation Law I: Basic Formulations and Performance Evaluation (Advances in Applied Mathematics and Mechanics, 2019)
In order to improve the resolution power of the hybrid cell-edge and cell-node dissipative compact schemes (HDCS), a series of multiderivative combined dissipative compact schemes (MDCS) have been proposed in this paper. The design concept of the HDCS has been followed to develop the MDCS satisfying the geometric conservation law (GCL) and possessing inherent dissipation. Evaluations of multiple derivatives are included in the MDCS for the purpose of increasing the scheme resolution. The performance of the MDCS is evaluated by theoretical analysis and numerical tests. The multiple derivatives demonstrate their capability in significantly improving resolution power of the MDCS. A MDCS can achieve much higher resolution power than a HDCS with the same order of accuracy. Based on the solutions of the transition and turbulence decay in three-dimensional Taylor-Green vortex, a fifth-order MDCS with three tri-diagonal operators is recommended. This MDCS has better performance than the fifth-order HDCS in resolving multiple-scales turbulent structures.