A Central Compact Hybrid-Variable Method With Spectral-Like Resolution

Numerical methods for hyperbolic conservation laws have been a driving force for theresearch in scientific computing and simulation science in the past decades, as many physical, biological, and engineering systems are governed by these equations, such as fluid mechanics, tumor growth, and virtual wind tunnel simulations. Despite the existence of many schemes in the literature, people have never stopped searching for more accurate and efficient methods for these problems. Indeed, the increasing complexity of systems in emerging applications demands better resolution of sub-grid scale phenomenon whereas classical methods usually fail to deliver high-fidelity simulation results of such systems within realistic computational time. In this work, we present a central compact hybrid-variable method (CHVM) with spectral-like accuracy for first-order hyperbolic problems with moderate or less discontinuities. It incorporates the compact difference strategy and a recently proposed hybrid-variable discretization technique to achieve even higher accuracy on a given stencil of grid cells. The CHVM is first constructed for the one-dimensional (1D) model linear advection equations, in which case the accuracy and stability analysis are conducted. We also pair our method with various implicit time-integrators to analyze the performance of the method. Then we extend our studies to 1D nonlinear problems such as the Burgers\u27 equation and the Euler equations. A novel Gauss-Seidel type low-pass high-order filter is constructed to suppress spurious oscillations near discontinuities for nonlinear problems. The performance of the proposed method is assessed by extensive benchmark tests