Related papers: Bigeometric Calculus and Runge Kutta Method
The article considers symmetric general linear methods, a class of numerical time integration methods which, like symmetric Runge--Kutta methods, are applicable to general time--reversible differential equations, not just those derived from…
The Butcher theory provides a powerful tool for analyzing order conditions of Runge-Kutta schemes for ordinary differential equations (ODEs); however, such a theory has not yet been well established for backward stochastic differential…
A practical and new Runge--Kutta numerical scheme for stochastic differential equations is explored. Numerical examples demonstrate the strong convergence of the method. The first order strong convergence is then proved using Ito integrals…
Linearly implicit Runge-Kutta methods with approximate matrix factorization can solve efficiently large systems of differential equations that have a stiff linear part, e.g. reaction-diffusion systems. However, the use of approximate…
We present a C++ implementation of a fifth order semi-implicit Runge-Kutta algorithm for solving Ordinary Differential Equations. This algorithm can be used for studying many different problems and in particular it can be applied for…
It is well known that symplectic Runge-Kutta and Partitioned Runge-Kutta methods exactly preserve {\em quadratic} first integrals (invariants of motion) of the system being integrated. While this property is often seen as a mere curiosity…
In the paper explicit functional continuous Runge-Kutta and Runge-Kutta-Nystr\"om methods for retarded functional differential equations are considered. New methods for first order equations as well as for second order equations of the…
Gamma distributed delay differential equations (DDEs) arise naturally in many modelling applications. However, appropriate numerical methods for generic Gamma distributed DDEs are not currently available. Accordingly, modellers often resort…
In this technical note a general procedure is described to construct internally consistent splitting methods for the numerical solution of differential equations, starting from matching pairs of explicit and diagonally implicit Runge-Kutta…
Finite element discretization of time dependent problems also require effective time-stepping schemes. While implicit Runge-Kutta methods provide favorable accuracy and stability problems, they give rise to large and complicated systems of…
We propose an experimental study of adaptive time-stepping methods for efficient modeling of the aggregation-fragmentation kinetics. Precise modeling of this phenomena usually requires utilization of the large systems of nonlinear ordinary…
A rigorous connection between large deviations theory and Gamma-convergence is established. Applications include representations formulas for rate functions, a contraction principle for measurable maps, a large deviations principle for…
The Wright function, which arises in the theory of the space-time fractional diffusion equation, is an interesting mathematical object which has diverse connections with other special and elementary functions. The Wright function provides a…
Rational methods are intended to time integrate linear homogeneous problems. However, their scope can be extended so as to cover linear nonhomogeneous problems. In this paper the integration of semilinear problems is considered. The…
Symplectic partitioned Runge--Kutta methods can be obtained from a variational formulation where all the terms in the discrete Lagrangian are treated with the same quadrature formula. We construct a family of symplectic methods allowing the…
The purpose of the article is to derive equations that determine the trajectory of a non-conservative natural system in configuration space in non-stationary external fields. A theorem on the change in the kinetic energy of the system is…
Finite element methods provide accurate and efficient methods for the numerical solution of partial differential equations by means of restricting variational problems to finite-dimensional approximating spaces. However, they do not…
The differential equation in the external invariant p^2 satisfied by the master integral of the general massive 2-loop 4-denominator self-mass diagram is exploited and the expansion of the master integral at p^2=0 is obtained analytically.…
Runge's method is a tool to figure out integral points on curves effectively in terms of height. This method has been generalised to varieties of any dimension, unfortunately its conditions of application are often too restrictive. In this…
Nonlinear parabolic equations are central to numerous applications in science and engineering, posing significant challenges for analytical solutions and necessitating efficient numerical methods. Exponential integrators have recently…