Dispersion Relation Preserving Optimized High-Order Straight and Mixed Second Derivative Schemes: Application to Fluid Flow Equations

In this study, we propose a family of conservative, fourth-order, second derivative schemes, that are parameterized by two variables. The key features of the proposed schemes are high-order accuracy and spectral-like resolution, with the retention of h-elliptic property for both straight and mixed second derivative terms. We employ a seven-point central stencil and propose a three-step procedure to optimize spectral properties of the second derivative scheme. The first step involves establishing a stencil for computing first derivatives at midpoint locations j − 3 2, j − 1 2, j + 1 2, j + 3 2, using all the available information within the prescribed stencil ranging in [j −3, j +3] to compute the final second derivative at location j. This ensures f lux conservation and also makes the scheme free from odd-even decoupling. In the second step, a standard fourth-order derivative scheme is used to compute second derivative using the first derivatives formulated at half-locations in the first step. In the third step, the coefficients used to compute the midpoint derivatives, which are initially unknown, are determined by imposing constraints on the order of accuracy and truncation error of the first derivative terms using two tunable parameters incorporated to adjust the truncation error. Optimizing for the resolving efficiency, the values of these two parameters are determined. For the resulting optimized scheme, we achieve a resolving efficiency of ≈ 82%, and high-wavenumber resolution for both straight and mixed second derivative terms. Numerical experiments conducted on the double periodic shear layer test case demonstrate improved accuracy with the use of our optimized second derivative scheme compared to others in the literature. However, our results suggest that while respecting the h-elliptic property is crucial, the influence of high wavenumber spectral properties on solution accuracy is only marginally significant.