Increasing the real stability boundary of explicit methods

AbstractBased on the simplest well-known integration rules (such as the forward Euler scheme and the “classical” Runge-Kutta method), an extension is proposed to enlarge the real stability boundary. The main characteristic of the resulting schemes is that the computational complexity is hardly increased

Blooming in a non-local, coupled phytoplankton-nutrient model

Recently, it has been discovered that the dynamics of phytoplankton concentrations in an ocean exhibit a rich variety of patterns, ranging from trivial states to oscillating and even chaotic behavior [J. Huisman, N. N. Pham Thi, D. M. Karl, and B. P. Sommeijer, Nature, 439 (2006), pp. 322–325]. This paper is a first step towards understanding the bifurcational structure associated with nonlocal coupled phytoplankton-nutrient models as studied in that paper. Its main subject is the linear stability analysis that governs the occurrence of the first nontrivial stationary patterns, the deep chlorophyll maxima (DCMs) and the benthic layers (BLs). Since the model can be scaled into a system with a natural singularly perturbed nature, and since the associated eigenvalue problem decouples into a problem of Sturm–Liouville type, it is possible to obtain explicit (and rigorous) bounds on, and accurate approximations of, the eigenvalues. The analysis yields bifurcation-manifolds in parameter space, of which the existence, position, and nature are confirmed by numerical simulations. Moreover, it follows from the simulations and the results on the eigenvalue problem that the asymptotic linear analysis may also serve as a foundation for the secondary bifurcations, such as the oscillating DCMs, exhibited by the model

A note on a diagonally implicit Runge-Kutta-Nyström method

AbstractIt is shown that it is possible to obtain fourth-order accurate diagonally implicit Runge-Kutta-Nyström methods with only 2 stages. The scheme with the largest interval of periodicity, i.e. (0, 12), is given. Furthermore, the requirement of P-stability decreases the order to 2

Parallel-iterated Runge-Kutta methods for stiff ordinary differential equations

AbstractFor the numerical integration of a stiff ordinary differential equation, fully implicit Runge-Kutta methods offer nice properties, like a high classical order and high stage order as well as an excellent stability behaviour. However, such methods need the solution of a set of highly coupled equations for the stage values and this is a considerable computational task. This paper discusses an iteration scheme to tackle this problem. By means of a suitable choice of the iteration parameters, the implicit relations for the stage values, as they occur in each iteration, can be uncoupled so that they can be solved in parallel. The resulting scheme can be cast into the class of Diagonally Implicit Runge-Kutta (DIRK) methods and, similar to these methods, requires only one LU factorization per step (per processor). The stability as well as the computational efficiency of the process strongly depends on the particular choice of the iteration parameters and on the number of iterations performed. We discuss several choices to obtain good stability and fast convergence. Based on these approaches, we wrote two codes possessing local error control and stepsize variation. We have implemented both codes on an ALLIANT FX/4 machine (four parallel vector processors and shared memory) and measured their speedup factors for a number of test problems. Furthermore, the performance of these codes is compared with the performance of the best stiff ODE codes for sequential computers, like SIMPLE, LSODE and RADAU5

Observations on gynes and drones around nuptial flights in the stingless bees Tetragonisca angustula and Melipona beecheii (Hymenoptera, Apidae, Meliponinae)

The nuptial flight of gynes of Tetragonisca angustula and Melipona beecheii was studied. The moment of nuptial flight was found to be related to the ambient temperature, and the duration of the nuptial flight for M. beecheii was longer in November (rainy season) than in March (dry season). A repeated mating flight was recorded for two gynes of T. angustula. Three of five T. angustula queens and all six M. beecheii queens were mated successfully. Behavioural data of drones and gynes shortly before and after the nuptial flight are presented. Drones of T. angustula participated in a congregation for up to three days. The importance of pheromones for the attraction of drones and gynes is discussed. An hypothesis explaining the observed seasonal occurrence of male congregations near nests of T. angustula is presented

The deposition of anal excretions by Melipona favosa foragers (Apidae: Meliponinae): behavioural observations concerning the location of food sources

Melipona favosa consistently deposited anal excretions while foraging. Anal depositions were released more frequently and by more bees on artificial food sources at a greater distance from the nest. Our hypothesis that these deposits serve as scent marks is supported by experimental evidence regarding the choices made by foraging bees arriving at food sources either with or without anal excretions. The clearly reduced visitation rate in the experimental situation without depositions indicates the importance of these cues during visitation of the food source

An implicit-explicit Runge-Kutta-Chebyshev scheme for diffusion-reaction equations

An implicit-explicit (IMEX) extension of the explicit Runge-Kutta-Chebyshev (RKC) scheme designed for parabolic PDEs is proposed for diffusion-reaction problems with severely stiff reaction terms. The IMEX scheme treats these reaction terms implicitly and diffusion terms explicitly. Within the setting of linear stability theory, the new IMEX scheme is unconditionally stable for reaction terms having a Jacobian matrix with a real spectrum. For diffusion terms the stability characteristics remain unchanged. A numerical comparison for a stiff, nonlinear radiation-diffusion problem between an RKC solver, an IMEX-RKC solver and the popular implicit BDF solver VODPK using the Krylov solver GMRES illustrates the excellent performance of the new scheme