Solving the sod shock tube problem using localized differential quadrature (LDQ) method

Abstract

The localized differential quadrature (LDQ) method is a numerical technique with high accuracy for solving most kinds of nonlinear problems in engineering and can overcome the difficulties of other methods (such as difference method) to numerically evaluate the derivatives of the functions. Its high efficiency and accuracy attract many engineers to apply the method to solve most of the numerical problems in engineering. However, difficulties can still be found in some particular problems. In the following study, the LDQ was applied to solve the Sod shock tube problem. This problem is a very particular kind of problem, which challenges many common numerical methods. Three different examples were given for testing the robustness and accuracy of the LDQ. In the first example, in which common initial conditions and solving methods were given, the numerical oscillations could be found dramatically; in the second example, the initial conditions were adjusted appropriately and the numerical oscillations were less dramatic than that in the first example; in the third example, the momentum equation of the Sod shock tube problem was corrected by adding artificial viscosity, causing the numerical oscillations to nearly disappear in the process of calculation. The numerical results presented demonstrate the detailed difficulties encountered in the calculations, which need to be improved in future work. However, in summary, the localized differential quadrature is shown to be a trustworthy method for solving most of the nonlinear problems in engineering.