Euler Equations Roe Scheme With High Precision Semi-lagrangian Method

High-order accuracy and high-resolution schemes play a more and more important role in the numerical simulations of multi-scale and multi-material flows. Among the computational fluid dynamical (CFD) methods, some are based on the finite volume method. So how to compute numerical fluxes at the cell boundaries is one of the crucial issues and those could be suitable for the general equation of state are more useful in the simulations of practical problems. In this thesis, we analyzed the numerical properties of several Godunov-type methods, improved the Roe method and constructed a new Semi-Lagrangian method.The main results in this thesis consist of the following parts:1. Under the framework of finite volume method, there are various choices to compute numerical flux, such as the most frequently used approximate Riemann solvers Roe-Pike, HLLC and HLL. More systematic and careful studies should be paid on them. Based on the reconstruction and oscillation-free methods of MUSCL and PPM method, the numerical properties of high-order Roe-Pike, HLLC and HLL approximate Riemann solvers were compared and studied. This work was of potential use for applications and designs of new methods.2. As to the Roe approximate Riemann solver, the process which obtains the Roe-average formulae of complicated hyperbolic equations tends to be rather cumbersome. To simplify it, a modification of the Roe method was presented, which was under the framework of finite volume method to preserve the numerical conservation. The modified Roe method is very simple and efficient. In addition, its characteristic velocities were modified, so the performance of the modified Roe method in computing low-density problem and sonic point glitch problem was improved.3. An algorithm for 1D and 2D Euler equations was proposed by combining the Roe scheme and the semi-Lagrangian method with the characteristic theory. The finite volume method is written in a conservative form and thus guarantees the numerical conservation. We calculated the flux at the cell boundary by using its value at the boundary's center, which could make the algorithm second-order accuracy in time. The semi-Lagrangian method with characteristic theory was applied to trace the fluid particles. The characteristic velocities were modified based on the Roe average to locate the departure points. The application of the ENO reconstruction made the algorithm free of limiters. The high order algorithm is simple and convenient to implement. The numerical results showed that it was competitive comparing with other schemes.