On the Theory of a Space-Time Solution Algorithm for Computational Science and Engineering
A Thesis Presented for the Master of Science in Computational Engineering Degree, University of Tennessee at Chattanooga
Arash Ghasemi, May 2013
A new mechanism for accelerating the numerical solution of Partial Differential Equations based on the simultaneous space and time discretization is presented. Under certain conditions, the space-time discretization can be made more efficient than a classical timemarching scheme. The minimum requirements for the space-time discretization in this case are: (1) unconditional stability in time and iteration spaces, (2) arbitrary formal order of accuracy in time, (3) arbitrary number of levels in time, and (4) the coefficients of the temporal discretization must be known a priori. To satisfy these requirements, a discretized version of Picard iteration is derived along with various integration operators. Several test cases are investigated with the space-time scheme, including simple Ordinary Differential Equations, the linear convection equation and the two-dimensional compressible Navier-Stokes equations. A thorough comparison is made between classical time-marching methods and the newly developed method. The results reveal that for a given absolute error, the newly developed method reaches the prescribed tolerance using less operations than traditional and implicit Runge-Kutta schemes.