Flux reconstruction using Jacobi correction functions in discontinuous spectral element method

A flux reconstruction approach is implemented in the spectral difference (SD) framework, which we refer to it as discontinuous spectral element method (DSEM), to improve the accuracy and stability of numerical simulations. A new class of correction functions is developed based on a weighted orthogonality condition, which leads to the introduction of Jacobi correction functions. Jacobi correction functions can construct the well-known DSEM and staggered-grid version of nodal discontinuous Galerkin spectral element method (DGSEM) as well as a broad range of other high-order numerical schemes with a variety of numerical characteristics, such as high numerical dissipation to suppress aliasing-driven errors, super accuracy, and solution boundedness in shock prediction. Three single-parameter families of Jacobi correction functions are recognized, and a von Neumann analysis is performed to acquire their wave propagation properties. The order of accuracy and stability criterion of a broad range of schemes are calculated, and the most super-accurate and stable numerical schemes are identified and later validated through a set of numerical experiments on a one-dimensional advection equation. The solution unboundedness issue of the high-order DSEM scheme is discussed through simulating the Burgers equation. An accuracy test performed on the Burgers equation revealed that the new staggered-grid flux reconstruction approach could acquire a higher accuracy than that of the energy stable flux reconstruction framework on a collocated grid. The two-dimensional extension of the new framework is also examined by simulation of a non-linear Euler system of equations for the isentropic vortex in free-stream flow. The staggered-grid flux reconstruction using the Jacobi correction function can precisely reconstruct the DSEM approach for non-linear two-dimensional hyperbolic equations for all polynomial orders.