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 (r1) and a spurious root (r2) 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 r1 1 and any r2 leading to the design of a family of generalized single step single solve [GS4 1] algorithms recovering the Crank Nicolson method at one end (r2 1) and the Midpoint Rule at the other end (r2 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 r1 r2 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. Copyright (c) 2011 John Wiley Sons, Ltd.