Related papers: Numerical analysis of the master equation
Simulation of complex dynamical systems arising in many applications is computationally challenging due to their size and complexity. Model order reduction, machine learning, and other types of surrogate modeling techniques offer cheaper…
A novel optimization procedure for the generation of stability polynomials of stabilized explicit Runge-Kutta methods is devised. Intended for semidiscretizations of hyperbolic partial differential equations, the herein developed approach…
This paper deals with stability of classical Runge-Kutta collocation methods. When such methods are embedded in linearly implicit methods as developed in [12] and used in [13] for the time integration of nonlinear evolution PDEs, the…
To achieve efficient and accurate long-time integration, we propose a fast, accurate, and stable high-order numerical method for solving fractional-in-space reaction-diffusion equations. The proposed method is explicit in nature and…
When applied to stiff, linear differential equations with time-dependent forcing, Runge-Kutta methods can exhibit convergence rates lower than predicted by the classical order condition theory. Commonly, this order reduction phenomenon is…
The properties of the Bigeometric or proportional derivative are presented and discussed explicitly. Based on this derivative, the Bigeometric Taylor theorem is worked out. As an application of this calculus, the Bigeometric Runge-Kutta…
A space-time fully adaptive multiresolution method for evolutionary non-linear partial differential equations is presented introducing an improved local time-stepping method. The space discretisation is based on classical finite volumes,…
We investigate the computational performance of various numerical methods for the integration of the equations of motion and the variational equations for some typical classical many-body models of condensed matter physics: the…
An explicit stabilized additive Runge-Kutta scheme is proposed. The method is based on a splitting of the problem in severely stiff and mildly stiff subproblems, which are then independently solved using a Runge-Kutta-Chebyshev scheme. The…
This work presents a new evolutionary optimization algorithm in theoretical mathematics with important applications in scientific computing. The use of the evolutionary algorithm is justified by the difficulty of the study of the…
We propose an implementation of symplectic implicit Runge-Kutta schemes for highly accurate numerical integration of non-stiff Hamiltonian systems based on fixed point iteration. Provided that the computations are done in a given floating…
A standard approach to solve ordinary differential equations, when they describe dynamical systems, is to adopt a Runge-Kutta or related scheme. Such schemes, however, are not applicable to the large class of equations which do not…
We propose a novel method to compute multi-loop master integrals by constructing and numerically solving a system of ordinary differential equations, with almost trivial boundary conditions. Thus it can be systematically applied to problems…
Exponential Runge--Kutta methods have shown to be competitive for the time integration of stiff semilinear parabolic PDEs. The current construction of stiffly accurate exponential Runge--Kutta methods, however, relies on a convergence…
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…
In current research, we analyse dissipation and dispersion characteristics of most accurate two and three stage Gauss-Legendre implicit Runge-Kutta (R-K) methods. These methods, known for their $A$-stability and immense accuracy, are…
Multiphysics systems are driven by multiple processes acting simultaneously, and their simulation leads to partitioned systems of differential equations. This paper studies the solution of partitioned systems of differential equations using…
In this paper, we study symmetric integrators for solving second-order ordinary differential equations on the basis of the notion of continuous-stage Runge-Kutta-Nystrom methods. The construction of such methods heavily relies on the…
An approach is treated for numerical integration of ordinary differential equations systems of the first order with choice of a computation scheme, ensuring the required local precision. The treatment is made on the basis of schemes of…
The evolution of a continuous time Markov process with a finite number of states is usually calculated by the Master equation - a linear differential equations with a singular generator matrix. We derive a general method for reducing the…