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The numerical evaluation of an individual Bessel or Hankel function of large order and large argument is a notoriously problematic issue in physics. Recurrence relations are inefficient when an individual function of high order and argument…

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…

A new approach for the construction of high order A-stable explicit integrators for ordinary differential equations (ODEs) is theoretically studied. Basically, the integrators are obtained by splitting, at each time step, the solution of…

We present a high-order, fully explicit, asymptotic-preserving projective integration scheme for the nonlinear BGK equation. The method first takes a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to…

We introduce economical versions of standard implicit ODE solvers that are specifically tailored for the efficient and accurate simulation of neural networks. These reformulations allow to achieve a significant increase in the efficiency of…

Inspired by path-integral solutions to the quantum relaxation problem, we develop a numerical method to solve classical stochastic differential equations with multiplicative noise that avoids averaging over trajectories. To test the method,…

The aim of this paper is to construct and analyze exponential Runge-Kutta methods for the temporal discretization of a class of semilinear parabolic problems with arbitrary state-dependent delay. First, the well-posedness of the problem is…

Approximating functions by a linear span of truncated basis sets is a standard procedure for the numerical solution of differential and integral equations. Commonly used concepts of approximation methods are well-posed and convergent, by…

We present high-order, fully explicit projective integration schemes for nonlinear collisional kinetic equations such as the BGK and Boltzmann equation. The methods first take a few small (inner) steps with a simple, explicit method (such…

When the eigenvalues of the coefficient matrix for a linear scalar ordinary differential equation are of large magnitude, its solutions exhibit complicated behaviour, such as high-frequency oscillations, rapid growth or rapid decay. The…

This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. The proposed methods are based on a specific subset of explicit one-step…

Tree tensor networks (TTNs) provide a compact and structured representation of high-dimensional data, making them valuable in various areas of computational mathematics and physics. In this paper, we present a rigorous mathematical…

In this paper, we provide a systematic methodology for calculating multi-order asymptotic expansion of blow-up solutions near blow-up for autonomous ordinary differential equations (ODEs). Under the specific form of the principal term of…

Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were proposed and analyzed in [8]. These specially designed methods use reduced precision or the implicit computations and full…

In this paper, we propose an efficient exponential integrator finite element method for solving a class of semilinear parabolic equations in rectangular domains. The proposed method first performs the spatial discretization of the model…

Many HPC applications that solve differential equations rely on the Runge-Kutta family of methods for time integration. Among these methods, the fourth-order accurate RK4 scheme is especially popular. This time integration scheme requires…

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…

Differential equations arise in mathematics, physics,medicine, pharmacology, communications, image processing and animation, etc. An Ordinary Differential Equation (ODE) is a differential equation if it involves derivatives with respect to…

We establish that it is possible to extract accurate blockwise and bitwise soft output from Guessing Codeword Decoding with minimal additional computational complexity by considering it as a variant of Guessing Random Additive Noise…

In this paper, we propose a deep learning-based method, deep Euler method (DEM) to solve ordinary differential equations. DEM significantly improves the accuracy of the Euler method by approximating the local truncation error with deep…