| This research effort explores the technique of higher-order implicit time integration methods for a simulation of unsteady compressible flows by Navier-Stokes Equations. Based on the concept of Runge-Kutta Butcher Tableau, the higher-order implicit methods are studied and tested to experience its accuracy in time.;First of all, a simple case of one-dimensional advection equation is analyzed. A few fundamental implicit time integration techniques, the backward differentiation formulas (BDF1, or the first-order Euler Implicit), the second-order Euler Implicit (BDF2), and the second-order Crank Nicolson (CN2) schemes are first analyzed. Then, the Explicit Singly Diagonal Implicit Runge-Kutta with third-order (ESDIRK3) and fourth-order (ESDIRK4) are analyzed to check accuracies in time. These five schemes are tested with different values of time step to observe the accuracy of the numerical solutions.;Since the orders of accuracy in temporal discretization are analyzed the spatial discretization also needs to be higher-order. In this work, the discontinuous Galerkin method (DGM) with Taylor basis is used. The DGM is a technique based on the finite element method (FEM) and the finite volume method (FVM) that can achieve higher-order spatial discretization depending on the order of polynomials. To update the solution, two techniques called pseudo-time approach and p-Multigrid method are used. From the obtained results it was found that the higher-order implicit time integration methods achieved the desired order of accuracy in time which was proven in theory.;After the technique was verified for the one-dimensional case, the research further explores the two-dimensional unsteady flows. All the results are obtained based on the techniques used for one-dimensional case. Based on the results this research is concluded that the temporal accuracies were achieved based on the theory for both one-dimensional as well as two-dimensional unsteady compressible Navier-Stokes equations. |
