Design of order-preserving algorithms for transient first-order systems with controllable numerical dissipation

Using a new design procedure termed as Algorithms by Design, which we have successfully introduced in our previous efforts for second-order systems, alternatively, we advance in this exposition, the design and development of a computational framework that permits order-preserving second-order time accurate, unconditionally stable, zero-order overshooting behavior, and features with controllable numerical dissipation and dispersion via a family of algorithms for effectively solving transient first-order systems. The key feature is the incorporation of a spurious root to introduce controllable numerical dissipation while preserving second-order accuracy (order-preserving feature) resulting in a two-root system, namely, the principal root (ρ_1∞) and a spurious root (ρ_2∞). In contrast to the classical Trapezoidal family of algorithms which are the most popular, the present framework has the same order of computational complexity, but a higher payoff that is a significant advance to the field for tackling a wide class of applications dealing with first-order transient systems. We also present the special case with selection of ρ_1∞ = 1 and any ρ_2∞ leading to the design of a family of generalized single-step single-solve [GS4-1] algorithms recovering the Crank-Nicolson method at one end (ρ_2∞ = 1) and the Midpoint Rule at the other end (ρ_2∞ = 0) and anything in between, all of which have spectral radius features resembling that of the Crank-Nicolson method. More interestingly, with the particular choice of ρ_1∞ = ρ_2∞ = 0, the developed framework additionally inherits L-stable features. We illustrate the successful design of the developed GS4-1 framework using two simple illustrative numerical examples.