Second Derivative Multistep Methods Research Articles

Explicit strong stability preserving second derivative multistep methods for the Euler and Navier–Stokes equations

In this paper, we develop the explicit second derivative multistep methods (SDMMs) for the Euler and Navier–Stokes equations. The SDMMs use the calculated results from the previous steps, which can improve computational efficiency. The third-order and fourth-order SDMMs temporal discretization methods are constructed based on the order conditions. Additionally, the strong stability preserving (SSP) condition for the SDMMs is proposed, which provides the corresponding parameters for optimal SSP SDMMs. For numerical experiments, the SDMMs can achieve the order of accuracy as designed on the smooth regions, and the optimal SSP coefficient for the SDMMs is shown to be roughly the same in theory as in the numerical results. Moreover, the SDMMs have approximately twice the computational efficiency compared with the third-order and fourth-order two-derivative Runge–Kutta methods, and the SDMMs are also more efficient than the classical three-stage third-order SSP Runge–Kutta method.

Optimal subsets in the stability regions of multistep methods

In this work, we study the stability regions of linear multistep or multiderivative multistep methods for initial value problems by using techniques that are straightforward to implement in modern computer algebra systems. In many applications, one is interested in (i) checking whether a given subset of the complex plane (e.g., a sector, disk, or parabola) is included in the stability region of the numerical method, (ii) finding the largest subset of a certain shape contained in the stability region of a given method, or (iii) finding the numerical method in a parametric family of multistep methods whose stability region contains the largest subset of a given shape. First, we describe a simple procedure to exactly calculate the stability angle α in the definition of A(α)-stability by representing the root locus curve of the multistep method as an implicit algebraic curve. As an illustration, we consider two finite families of implicit multistep methods. We exactly compute the stability angles for the k-step BDF methods (3 ≤ k ≤ 6) and discover that the values of tan(α) are surprisingly simple algebraic numbers of degree 2, 2, 4, and 2, respectively. In contrast, the corresponding values of tan(α) for the k-step second-derivative multistep methods of Enright (3 ≤ k ≤ 7) are much more complicated; the smallest algebraic degree here is 22. Next, we determine the exact value of the stability radius in the BDF family for each 3 ≤ k ≤ 6, that is, the radius of the largest disk in the left half of the complex plane, symmetric with respect to the real axis, touching the imaginary axis and lying in the stability region of the corresponding method. These radii turn out to be algebraic numbers of degree 2, 3, 5, and 5, respectively. Finally, we demonstrate how some Schur–Cohn-type theorems of recursive nature and not relying on the root locus curve method can be used to exactly solve some optimization problems within infinite parametric families of multistep methods. As an example, we choose a two-parameter family of implicit-explicit (IMEX) methods: we identify the unique method having the largest stability angle in the family, then we find the unique method in the same family whose stability region contains the largest parabola.

Singulo oscillatory stiff rational integrators

In this paper, we derive a general singulo oscillatory – stiff rational integrator of order (S+3) for s = 0, 1, 2, 3, 4,… for the solution of initial value problems in ordinary differential systems that are singular, oscillatory or stiff. We compared our integrators with certain maximum order second derivative hybrid multi-step methods, certain Tau and Euler methods, the adaptive implicit and classical Runge-Kutta methods and some existing conventional methods. Our results show good improvement over the existing methods compared with. Key words: Rational integrator, initial value problems, stability, convergence, consistency.

New efficient second derivative multistep methods for stiff systems

The aim of this paper is to select from the large family of possible general linear methods, just a single class which has considerable potential for efficient implementation. This class has possible applications depending on stiff nature of a problem to be performed. A special class of second derivative multistep method (SDMM) is derived. The stability analysis of this class which is depending on free parameters is discussed. The stability regions are plotted for certain choices of parameters. A good comparison between the results of this class and the results due to Gear and Enright is recommended during some numerical tests.