A new central compact finite difference formula for improving robustness in weighted compact nonlinear schemes

Abstract

The weighted compact nonlinear scheme (WCNS) is a typical well-known finite difference method that shows high-order accuracy and high resolution when applied to problems involving hyperbolic conservation laws. However, when applied to flow problems with severe discontinuities, the original WCNS often suffers from negative densities or negative pressures. We have found that the robustness of the WCNS can be remarkably improved by changing the original central finite difference scheme employed in the WCNS to an alternative one. In this article, a new cell-edge- and cell-node-type alternate central compact scheme (ACCS) is proposed, and a new WCNS (i.e., alternate central weighted compact nonlinear scheme (ACWCNS)) is developed by combining the ACCS with a nonlinear reconstruction (weighted interpolation) method. To elucidate the basic characteristics of the ACWCNS, we perform truncation error analysis, wavenumber analysis, semi-discrete eigenvalue analysis, and accuracy validation for 1D advection equations. Subsequently, the ACWCNS is applied to several 1D and 2D benchmark problems of compressible flows with severe discontinuities. When solving such problems, it is difficult to preserve the positivity of both density and pressure using the conventional WCNS; nevertheless, the ACWCNS is able to provide stable solutions. In the smooth regions of the solutions, the spatial resolution of the scheme is found to deteriorate slightly as the robustness of the method improved. To compensate for this, the nonlinear reconstruction method in the ACWCNS is modified and the obtained computational results are compared with those obtained by alternative approaches.