In this thesis we derive time dependent equations that govern the physics of a thermal fluid flowing through a one dimensional pipe. We begin with the conservation laws described by the 3D compressible Navier Stokes equations. This model includes all residual terms resulting from the 1D flow approximations. The final model assumes that all the residual terms are negligible which is a standard assumption in industry. Steady state equations are obtained by assuming the temporal derivatives are zero. We develop a semi-discrete model by applying a linear discontinuous Galerkin method in the spatial dimension. The resulting finite dimensional model is a differential algebraic equation (DAE) which is solved using standard integrators. We investigate two methods for solving the corresponding steady state equations. The first method requires making an initial guess and employs a Newton based solver. The second method is based on a pseudo-transient continuation method. In this method one initializes the dynamic model and integrates forward for a fixed time period to obtain a profile that initializes a Newton solver. We observe that non-uniform meshing can significantly reduce model size while retaining accuracy. For comparison, we employ the same initialization for the pseudo-transient algorithm and the Newton solver. We demonstrate that for the systems considered here, the pseudo-transient initialization algorithm produces initializations that reduce iteration counts and function evaluations when compared to the Newton solver. Several numerical experiments were conducted to illustrate the ideas. Finally, we close with suggestions for future research. / Master of Science / In this thesis we derive time dependent equations that govern the physics of a fluid flowing through a one dimensional pipe. This model includes all error terms that result from 1D modeling approximations. The final model assumes that all of these error terms are negligible which is a standard assumption in industry. Steady state equations result when all time dependence is removed from the 1D equations. We approximate the true solution by a discontinuous piece-wise linear function. Standard techniques are used to solve for this approximate solution. We investigate two methods for solving the steady state equations. In the first method, one makes an educated guess about the solution profile and uses Newton’s method to solve for the true solution. The second method, pseudo-transient initialization, attempts to improve this initial guess through dynamic simulation. In this method, an initial guess is treated as the initial conditions for dynamic simulation. The dynamic simulation is then run for a fixed amount of time. The solution at the end of the simulation is the improved initial guess for Newton’s method and is used to solve for the steady state profile. To test the pseudo-transient initialization, we determine the number of function evaluations required to obtain the steady state solution for an initial guess with and without performing pseudo-transient initialization on it. We demonstrate that for the systems considered here, the pseudo-transient initialization algorithm reduced overall computational costs. Also, we observe that non-uniform meshing can significantly reduce model size while retaining accuracy. Several numerical experiments were conducted to illustrate these ideas. Finally, we close with suggestions for future research.

The main objective of this research work is to apply the discontinuous Galerkin method to a classical partial differential equation to investigate the properties of the numerical solution and compare the numerical solution to the analytical solution by using discontinuous Galerkin method. This scheme is applied to 1-D non-linear conservation equation (Burgers equation) in which the governing differential equation is simplified model of the inviscid Navier-stokes equations. In this work three cases are studied. They are sinusoidal wave profile, initial shock discontinuity and initial linear distribution. A grid and time step refinement is performed. Riemann fluxes at each element interfaces are calculated. This scheme is applied to forward differentiation method (Euler's method) and to second order Runge-kutta method of this work.

In this thesis, we study modelling of mortality in high ages by several approaches. Some of mentioned models take into account the phenomenon of mortality deceleration. Further, we present several ways of estimating of exposed to risk in (almost) extinct cohorts. We focus especially on the survivor ratio method but we also mention the MD method and the DG method. Finally, we perform a numerical study.