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This work focuses on the numerical study of a recently published class of Runge-Kutta methods designed for mixed-precision arithmetic. We employ the methods in solving partial differential equations on modern hardware. In particular we…
There exist many Runge-Kutta methods (explicit or implicit), more or less adapted to specific problems. Some of them have interesting properties, such as stability for stiff problems or symplectic capability for problems with energy…
The reliability and precision of numerically solving stochastic non-Markovian equations by standard numerical codes, more specifically, with the fourth-order Runge-Kutta routine for solving differential equations, is gauged by comparing the…
We present efficient quantum algorithms for simulating time-dependent Hamiltonian evolution of general input states using an oracular model of a quantum computer. Our algorithms use either constant or adaptively chosen time steps and are…
Exponential time differencing methods is a power tool for high-performance numerical simulation of computationally challenging problems in condensed matter physics, fluid dynamics, chemical and biological physics, where mathematical models…
In this paper, authors successfully construct a new algorithm for the new higher order scheme of weak approximation of SDEs. The algorithm presented here is based on [1][2]. Although this algorithm shares some features with the algorithm…
The result after $N$ steps of an implicit Runge-Kutta time discretization of an inhomogeneous linear parabolic differential equation is computed, up to accuracy $\epsilon$, by solving only $$O\Big(\log N \log \frac1\epsilon \Big) $$ linear…
Quantum algorithms for Hamiltonian simulation and linear differential equations more generally have provided promising exponential speed-ups over classical computers on a set of problems with high real-world interest. However, extending…
The recently-introduced relaxation approach for Runge-Kutta methods can be used to enforce conservation of energy in the integration of Hamiltonian systems. We study the behavior of implicit and explicit relaxation Runge-Kutta methods in…
Linear differential equations are ubiquitous in science and engineering. Quantum computers can simulate quantum systems, which are described by a restricted type of linear differential equations. Here we extend quantum simulation algorithms…
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…
We develop continuous-stage Runge-Kutta methods based on weighted orthogonal polynomials in this paper. There are two main highlighted merits for developing such methods: Firstly, we do not need to study the tedious solution of…
Differential equations are ubiquitous in models of physical phenomena. Applications like steady-state analysis of heat flow and deflection in elastic bars often admit to a second order differential equation. In this paper, we discuss the…
Runge-Kutta methods have an irreplaceable position among numerical methods designed to solve ordinary differential equations. Especially, implicit ones are suitable for approximating solutions of stiff initial value problems. We propose a…
The numerical efficiency of different schemes for solving the Liouville-von Neumann equation within multilevel Redfield theory has been studied. Among the tested algorithms are the well-known Runge-Kutta scheme in two different…
This paper contains an error analysis of two randomized explicit Runge-Kutta schemes for ordinary differential equations (ODEs) with time-irregular coefficient functions. In particular, the methods are applicable to ODEs of Carath\'eodory…
We consider the efficient numerical solution of coupled dynamical systems, consisting of a small nonlinear part and a large linear time invariant part, possibly stemming from spatial discretization of an underlying partial differential…
From the discovery that the template system of C++ forms a Turing complete language in 1994, a programming technique called Template Metaprogramming has emerged that allows for the creation of faster, more generic and better code. Here, we…
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…
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…