Search results for "Runge–Kutta method"
showing 10 items of 14 documents
Modelling phase transition kinetics of chenodeoxycholic acid with the Runge–Kutta method
2009
Abstract The phase transition kinetics of two chenodeoxycholic acid polymorphic modifications— form I (stable at high temperature), form III (stable at low temperature) and the amorphous phase has been examined under various conditions of temperature and relative humidity. Form III conversion to form I was examined at high temperature conditions and was found to be non-spontaneous, requiring seed crystals for initiation. The formation kinetic model of form I was created incorporating the three-dimensional seed crystal growth, the phase transition rate proportion to the surface area of form I crystals, and the influence of the amorphous phase surface area changes with an empirical stage poin…
NUMERICAL ALGORITHMS
2013
For many systems of differential equations modeling problems in science and engineering, there are natural splittings of the right hand side into two parts, one non-stiff or mildly stiff, and the other one stiff. For such systems implicit-explicit (IMEX) integration combines an explicit scheme for the non-stiff part with an implicit scheme for the stiff part. In a recent series of papers two of the authors (Sandu and Zhang) have developed IMEX GLMs, a family of implicit-explicit schemes based on general linear methods. It has been shown that, due to their high stage order, IMEX GLMs require no additional coupling order conditions, and are not marred by order reduction. This work develops a …
A Magnetohydrodynamic Generator for Marine Energy Harvesting
2018
In this article we present an approach to the description of Magneto-hydrodynamic Marine Energy Harvesting (MHMEH) system. Preliminarly, a general discussion on the principle of operation is presented. Successively, in order to move beyond the analytical model, a 3-D MHD modeling tool and a Runge Kutta method based solver are presented and they are used to investigate an alternative MHD solutions. Some numerical analyses are given.
Equilibrium real gas computations using Marquina's scheme
2003
Marquina's approximate Riemann solver for the compressible Euler equations for gas dynamics is generalized to an arbitrary equilibrium equation of state. Applications of this solver to some test problems in one and two space dimensions show the desired accuracy and robustness
High-order Runge–Kutta–Nyström geometric methods with processing
2001
Abstract We present new families of sixth- and eighth-order Runge–Kutta–Nystrom geometric integrators with processing for ordinary differential equations. Both the processor and the kernel are composed of explicitly computable flows associated with non trivial elements belonging to the Lie algebra involved in the problem. Their efficiency is found to be superior to other previously known algorithms of equivalent order, in some case up to four orders of magnitude.
The interrelation between stochastic differential inclusions and set-valued stochastic differential equations
2013
Abstract In this paper we connect the well established theory of stochastic differential inclusions with a new theory of set-valued stochastic differential equations. Solutions to the latter equations are understood as continuous mappings taking on their values in the hyperspace of nonempty, bounded, convex and closed subsets of the space L 2 consisting of square integrable random vectors. We show that for the solution X to a set-valued stochastic differential equation corresponding to a stochastic differential inclusion, there exists a solution x for this inclusion that is a ‖ ⋅ ‖ L 2 -continuous selection of X . This result enables us to draw inferences about the reachable sets of solutio…
Partially Implicit Runge-Kutta Methods for Wave-Like Equations
2014
Runge-Kutta methods are used to integrate in time systems of differential equations. Implicit methods are designed to overcome numerical instabilities appearing during the evolution of a system of equations. We will present partially implicit Runge-Kutta methods for a particular structure of equations, generalization of a wave equation; the partially implicit term refers to this structure, where the implicit term appears only in a subset of the system of equations. These methods do not require any inversion of operators and the computational costs are similar to those of explicit Runge-Kutta methods. Partially implicit Runge-Kutta methods are derived up to third-order of convergence. We ana…
Minimally implicit Runge-Kutta methods for Resistive Relativistic MHD
2016
The Relativistic Resistive Magnetohydrodynamic (RRMHD) equations are a hyperbolic system of partial differential equations used to describe the dynamics of relativistic magnetized fluids with a finite conductivity. Close to the ideal magnetohydrodynamic regime, the source term proportional to the conductivity becomes potentially stiff and cannot be handled with standard explicit time integration methods. We propose a new class of methods to deal with the stiffness fo the system, which we name Minimally Implicit Runge-Kutta methods. These methods avoid the development of numerical instabilities without increasing the computational costs in comparison with explicit methods, need no iterative …
Generalized singly-implicit Runge-Kutta methods with arbitrary knots
1985
The aim of this paper is to derive Butcher's generalization of singly-implicit methods without restrictions on the knots. Our analysis yields explicit computable expressions for the similarity transformations involved which allow the efficient implementation of the first phase of the method, i.e. the solution of the nonlinear equations. Furthermore, simple formulas for the second phase of the method, i.e. computation of the approximations at the next nodal point, are established. Finally, the matrix which governs the stability of the method is studied.
A computational magnetohydrodynamic model of a marine propulsion system
2016
In this article we present an approach to the description of Magnetohydrodynamic Propulsion. Preliminarly, an analytical model which includes an electromagnetic model and a thermal model is presented. Successively, in order to move beyond the analytical model, a 3-D MHD modeling tool and a Runge Kutta method based solver are presented and they are used to investigate an alternative MHD solutions. Some numerical analysis are given.