Related papers: SUNDIALS Time Integrators for Exascale Application…
Welcome to a beautiful subject in scientific computing: numerical solution of ordinary differential equations (ODEs) with initial conditions.
Starting with some fundamental concepts, in this article we present the essential aspects of spectral methods and their applications to the numerical solution of Partial Differential Equations (PDEs). We start by using Lagrange and…
Differentiable programming allows for derivatives of functions implemented via computer code to be calculated automatically. These derivatives are calculated using automatic differentiation (AD). This thesis explores two applications of…
The work in this paper is four-fold. Firstly, we introduce an alternative approach to solve fractional ordinary differential equations as an expected value of a random time process. Using the latter, we present an interesting numerical…
The explicit split-operator algorithm is often used for solving the linear and nonlinear time-dependent Schr\"{o}dinger equations. However, when applied to certain nonlinear time-dependent Schr\"{o}dinger equations, this algorithm loses…
This paper investigates a numerical probabilistic method for the solution of some semilinear stochastic partial differential equations (SPDEs in short). The numerical scheme is based on discrete time approximation for solutions of systems…
Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations (PDEs). Naturally, reduced-order modeling techniques come at the price of computational accuracy for a decrease in computation…
The combination of machine learning and physical laws has shown immense potential for solving scientific problems driven by partial differential equations (PDEs) with the promise of fast inference, zero-shot generalisation, and the ability…
A transient magneto-quasistatic vector potential formulation involving nonlinear material is spatially discretized using the finite element method of first and second polynomial order. By applying a generalized Schur complement the…
Solving partial differential equations (PDEs) within the framework of probabilistic numerics offers a principled approach to quantifying epistemic uncertainty arising from discretization. By leveraging Gaussian process regression and…
In this paper we consider the numerical approximation of a semilinear reaction-diffusion model problem (PDEs) by means of reduced order methods (ROMs) based on proper orthogonal decomposition (POD). We focus on the time integration of the…
Multirate behavior of ordinary differential equations (ODEs) and differential-algebraic equations (DAEs) is characterized by widely separated time constants in different components of the solution or different additive terms of the…
Rapidly developing machine learning methods has stimulated research interest in computationally reconstructing differential equations (DEs) from observational data which may provide additional insight into underlying causative mechanisms.…
Integration operational matrix methods based on Zernike polynomials are used to determine approximate solutions of a class of non-homogeneous partial differential equations (PDEs) of first and second order. Due to the nature of the Zernike…
Numerical solutions of partial differential equations enable a broad range of scientific research. The Dedalus Project is a flexible, open-source, parallelized computational framework for solving general partial differential equations using…
The obstacle problem is a class of free boundary problems which finds applications in many disciplines such as porous media, financial mathematics and optimal control. In this paper, we propose two operator-splitting methods to solve the…
We analyze the concepts of analytically weak solutions of stochastic differential equations (SDEs) in Hilbert spaces with time-dependent unbounded operators and give conditions for existence and uniqueness of such solutions. Our studies are…
We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- to three-dimensional problems. It has three…
We establish a discrete operator--theoretic framework for the analysis of implicit Euler and Lie--Trotter splitting schemes for delay differential equations (DDEs). Both schemes are formulated in terms of discrete resolvent operators acting…
Ordinary differential equations (ODEs) are the primary means to modelling dynamical systems in many natural and engineering sciences. The number of equations required to describe a system with high heterogeneity limits our capability of…