Semi-analytical implicit direct time integration method for 1-D gas dynamic problem

Abstract

Sharp wave treatment for 1-D gas dynamic problem is still a chellenge for modern numerical methods. They often require too many space and time steps, produce spurious oscillation of solution, exhibit a strong numerical dissipation or divergence of results. This paper is further extension of authors’ idea of employment the analytical solution for space coordinate, where time step is a parameter which used in the space solution. Its peculiarity consists in development of additional procedure of linearization of dependence between the pressure and density. It is performed in premise that actual pressure for each space element is close to the basic pressure, attained at previous moment of time. The efficiency of method is tested on the very popular task of Sod, where two different ideal gases in a tube are separated by diaphragm, which is suddenly broken. The problem considered in Lagrangian coordinates formulation. The results obtained show the very good efficiency of method, which requires the essentially lesser time and space steps, leads to no spurious oscillation and give consistent and predictable results with respect to meshing. The accuracy of method is mostly controlled by time step, which should be larger than clearly stated theoretical lower limit. Other advantage of method is that it can calculate the process to any desired moment of time, and space meshing can be variable in time and space and can be easily adapted during the process of calculation.

Implicit direct time integration of the heat conduction problem in the Method of Matched Sections

“…With some small amendment we repeat here the procedure given in [29], [30]. First of all, note that availability of solution at the time i means that ,0 ,0 ,…”

“…Interesting to note, that enriched element interpolation functions as given in [26] were used in analysis of wave propagation in a rod [28]. Alternative idea was proposed in our works [29], [30] where time step in implicit central difference scheme is used directly in element interpolation schemes. The similar idea (without realization) was expressed by Reddy in the textbook [31].…”