A comparative analysis of diagonally implicit Runge – Kutta methods for one-dimensional model gas dynamics equations

Several diagonally implicit Runge – Kutta methods (DIRK) of different orders and stability properties are compared. Numerical comparisons are conducted using onedimensional model equations: the linear advection equation and Burgers’ equation. It is shown that, for the linear advection equation with homogeneous boundary conditions, the characteristic wavelength grows indefinitely, limiting the equation’s usefulness for method comparison. Therefore, the methods are compared using Burgers’ equation, whose properties allow this issue to be avoided. Various time-step sizes and initial wavelengths are considered. It is demonstrated that high-order DIRK methods can accurately (i.e., without excessive dissipation or significant spurious oscillations) propagate relatively long-wave perturbations at Courant numbers significantly exceeding unity. This result further confirms the suitability of high-order DIRK methods for hybrid RANS/LES simulations, where such methods can considerably reduce computational time without compromising the accuracy of large-scale flow structures in near-wall regions. The primary parameter for comparing methods was chosen as the minimum wavelength that the method can propagate satisfactorily. Results indicate that increasing the order of accuracy beyond second-order has only a minor influence on this parameter, whereas methods possessing superior stability properties allow for smaller wavelengths to be adequately resolved. Furthermore, it is demonstrated that the backward Euler method is excessively dissipative and thus unsuitable for non-stationary simulations.