On Numerical Integration and Conservation of Cell-Centered Finite Difference Method

Conservation and numerical integration have been important issues for finite difference method related to robustness, reliability and accuracy requirements. In this paper, we discuss the relationship between the discretized Newton-Leibniz formula and four conservation and integration properties, including geometric conservation, flow conservation, surface integration and volume integration, for the multi-block based high-order cell-centered finite difference method. In order to achieve these conservation and integration properties, as well as multi-block compatibility, high-order accuracy, and stability within a unified methodology, we propose a new series of boundary schemes that incorporate all these constraints. To ensure geometric conservation, conservative metrics and Jacobian are adopted for coodinate transformation. To realize flow conservation, the width of the boundary stencil is enlarged to provide more degrees of freedom in order to meet the conservation constraints. To achieve uniformly high-order accuracy with arbitrary multi-block topology, cross-interface interpolation or differencing is avoided by utilizing one-sided scheme. To maintain stability, boundary interpolation scheme is designed as upwindly and compactly as possible. The proposed method is finally tested through a series of numerical cases, including a wave propagation and an isentropic vortex for accuracy verification, several acoustic tests to demonstrate the capability of handling arbitrary multi-block grid topology, a wavy channel and a closed flying wing problem for conservation verification. These numerical tests indicate that the new scheme possesses satisfactory conservation and integration properties while satisfying the requirements for high-order accuracy and stability.