Related papers: Numerical evaluation of master integrals from diff…
A new class of third order Runge-Kutta methods for stochastic differential equations with additive noise is introduced. In contrast to Platen's method, which to the knowledge of the author has been up to now the only known third order…
A novel class of high-order linearly implicit energy-preserving integrating factor Runge-Kutta methods are proposed for the nonlinear Schr\"odinger equation. Based on the idea of the scalar auxiliary variable approach, the original equation…
This is a sequel of our previous paper where we described an algorithm to find a solution of differential equations for master integrals in the form of an $\epsilon$-expansion series with numerical coefficients. The algorithm is based on…
We provide a note on continuous-stage Runge-Kutta methods (csRK) for solving initial value problems of first-order ordinary differential equations. Such methods, as an interesting and creative extension of traditional Runge-Kutta (RK)…
We calculate two-loop massive master integrals for $e^{+}e^{-}\rightarrow2\gamma$ in terms of generalized power series with respect to electron mass. The coefficients of this series are expressed via Goncharov's polylogarithms. Our approach…
Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic…
The nonlinear gyrokinetic equations describe plasma turbulence in laboratory and astrophysical plasmas. To solve these equations, massively parallel codes have been developed and run on present-day supercomputers. This paper describes…
We present analytical results for one-loop five-point master integrals with up to three off-shell legs. The method of canonical differential equations along with the Simplified Differential Equations approach is employed. All necessary…
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…
Singly-TASE operators for the numerical solution of stiff differential equations were proposed by Calvo et al. in J.Sci. Comput. 2023 to reduce the computational cost of Runge-Kutta-TASE (RKTASE) methods when the involved linear systems are…
The aim of this paper is to design the explicit radial basis function (RBF) Runge-Kutta methods for the initial value problem. We construct the two-, three- and four-stage RBF Runge-Kutta methods based on the Gaussian RBF Euler method with…
We review the Laporta algorithm for the reduction of scalar integrals to the master integrals and the differential equations technique for their evaluation. We discuss the use of the basis of harmonic polylogarithms for the analytical…
In this paper we derive and analyze the properties of explicit singly diagonal implicit Runge-Kutta (ESDIRK) integration methods. We discuss the principles for construction of Runge-Kutta methods with embedded methods of different order for…
This paper discusses stochastic numerical methods of Runge-Kutta type with weak and strong convergences for systems of stochastic differential equations in It\^o form. At the beginning we give a brief overview of the stochastic numerical…
Numerical evaluations of Feynman integrals often proceed via a deformation of the integration contour into the complex plane. While valid contours are easy to construct, the numerical precision for a multi-loop integral can depend…
A recently proposed scheme for numerical evaluation of Feynman diagrams is extended to cover all two-loop two-point functions with arbitrary internal and external masses. The adopted algorithm is a modification of the one proposed by F. V.…
A purely numerical method, Direct ComputationMethod is applied to evaluate Feynman integrals. This method is based on the combination of an efficient numerical integration and an efficient extrapolation. In addition, high-precision…
Feynman loop integrals are a key ingredient for the calculation of higher order radiation effects, and are responsible for reliable and accurate theoretical prediction. We improve the efficiency of numerical integration in sector…
In this paper a technique is given to recover the classical order of the method when explicit exponential Runge-Kutta methods integrate reaction-diffusion problems. Although methods of high stiff order for problems with vanishing boundary…
In this paper we discuss a framework for the polynomial approximation to the solution of initial value problems for differential equations. The framework, initially devised for the approximation of ordinary differential equations, is…