Growing interest in the simulation of first order transient systems, typical of those encountered in transient heat conduction, flow transport, and fluid dynamics, has prompted the development of a variety of time integration methods for solving these systems numerically. The primary contribution of this thesis is the design and development of a new time integration/discretization framework, under the class of single step single solve algorithms which are the most popular, for use in such first order transient systems with computationally attractive features. These include second order accuracy, unconditional stability, zero-order overshoot, and controllable numerical dissipation with a new selective control feature which overcomes the restrictions in the existing and current state-of-the-art methods. Throughout the thesis, we demonstrate the capability and advantage of the newly developed framework, termed GS4-1, in comparison to existing methods using various types of numerical examples (both linear and nonlinear). The numerical results consistently demonstrate the roles played by the new feature in improving the numerical solutions of both the primary variable and its time derivative which is important to correctly capture the dynamics of the problems, in contrast to the existing methods without such a feature. Additionally, a breakthrough contribution presented in this thesis is the development of an isochronous integration framework (iIntegrator), stemming from the novel relations between the newly developed GS4-1 framework and the existing GS4-2 framework (for second order dynamic systems). Such a development enables the use of the same computational framework to solve both first and second order dynamic systems without having to resort to the individual GS4-1 and GS4-2 frameworks; hence the practicality in the computational and implementation aspects. Finally, the application of the new GS4-1 framework to silica particle deposition, which is a practical problem of interest, is presented with the focus primarily on the physics of the problem. In this part of the thesis, a numerical model of the problem is presented and employed to investigate the effects of the flow and physicochemical parameters on the rate of deposition. The results of the parametric studies undertaken based on the employed numerical model enable some recommendations for the mitigation of the problem, and therefore serve as additional valuable contribution of the thesis.