In this paper‎, ‎we propose a modification of the second order method‎ ‎introduced in [‎‎Q. Li and ‎X‎. ‎Y. ‎Wu‎, A two-step explicit $P$-stable method for solving second order initial value problems‎, ‎textit{‎Appl‎. ‎Math‎. ‎Comput‎.}‎ {‎138}‎ (2003)‎, no. 2-3, ‎435--442‎] for the numerical solution of‎ ‎IVPs for second order ODEs‎. ‎The numerical results obtained by the‎ ‎new method for some...

The application of a Method of Lines to a hyperbolic PDE with source terms gives rise to a system of ODEs containing terms that may have very different stiffness properties. In this case, Implicit-Explicit Runge-Kutta (IMEX-RK) schemes are particularly useful as high order time integrators because they allow an explicit handling of the convective terms, which can be discretized using the highly...

A new family of implicit, single-step time integration methods is presented for solving structural dynamics problems. The proposed method is unconditionally stable, second-order accurate and asymptotically annihilating. It is spectrally equivalent to Houbolt's method but is cast in single-step form rather than multi-step form; thus the new algorithm computationally is more convenient. An explic...

This paper investigates parallel predictor-corrector iteration schemes (PC iteration schemes) based on collocation Runge-Kutta corrector methods (RK corrector methods) with continuous output formulas for solving nonstiff initial-value problems (IVPs) for systems of first-order differential equations. The resulting parallel-iterated RK-type PC methods are also provided with continuous output for...

Let χ(n, d) be the number of injective maps σ : S → Zn \ {0} such that (a) S ⊂ Zn of cardinality |S | = n − d, (b) σ(i) , i for all i ∈ S and (c) σ(i) − i . σ( j) − j (mod n) whenever i , j. Let Rk,n be the number of k × n reduced Latin rectangles. We show that Rk,n ≡ χ(p, n − p) (n − p)!(n − p − 1)!2 (n − k)! Rk−p,n−p (mod p) when p is a prime and n ≥ k ≥ p+1. This allows us to calculate expli...

We examine the potential for parallelism in Runge-Kutta (RK) methods based on formulas in standard one-step form. Both negative and positive results are presented. Many of the negative results are based on a theorem that bounds the order of a RK formula in terms of the minimum polynomial for its coeecient matrix. The positive results are largely examples of prototypical formulas which ooer a po...