Related papers: Local time-stepping for adaptive multiresolution u…
Many important differential equations model quantities whose value must remain positive or stay in some bounded interval. These bounds may not be preserved when the model is solved numerically. We propose to ensure positivity or other…
The minimization of the loss function is of paramount importance in deep neural networks. On the other hand, many popular optimization algorithms have been shown to correspond to some evolution equation of gradient flow type. Inspired by…
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 paper aims at developing low-storage implicit Runge-Kutta methods which are easy to implement and achieve higher-order of convergence for both the velocity and pressure in the finite volume formulation of the incompressible…
The non-differentiability of the singular nonlinearity (such as $f=\ln|u|^2$) at $u=0$ presents significant challenges in devising accurate and efficient numerical schemes for the logarithmic Schr\"{o}dinger equation (LogSE). To address…
Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were proposed and analyzed in 4. These specially designed methods use reduced precision for the implicit computations and full…
The accuracy of solving partial differential equations (PDEs) on coarse grids is greatly affected by the choice of discretization schemes. In this work, we propose to learn time integration schemes based on neural networks which satisfy…
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 we continue our work on adaptive timestep control for weakly non- stationary problems. The core of the method is a space-time splitting of adjoint error representations for target functionals due to S\"uli and Hartmann. The…
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…
The paper concerns semidiscretizations in time of stochastic Maxwell equations driven by additive noise. We show that the equations admit physical properties and mathematical structures, including regularity, energy and divergence evolution…
In this paper we present and analyze a general framework for constructing high order explicit local time stepping (LTS) methods for hyperbolic conservation laws. In particular, we consider the model problem discretized by Runge-Kutta…
The design of numerical integrators for solving stochastic dynamics with high weak order relies on tedious calculations and is subject to a high number of order conditions. The original approaches from the literature consider strong…
Irksome is a library based on the Unified Form Language (UFL) that enables automated generation of Runge--Kutta methods for time-stepping finite element spatial discretizations of partial differential equations (PDE). Allowing users to…
This paper focuses on the strong convergence rate of both Runge--Kutta methods and simplified step-$N$ Euler schemes for stochastic differential equations driven by multi-dimensional fractional Brownian motions with $H\in(\frac12,1)$. Based…
Applied to the master equation, the usual numerical integration methods, such as Runge-Kutta, become inefficient when the rates associated with various transitions differ by several orders of magnitude. We introduce an integration scheme…
In this paper a new Runge-Kutta type scheme is introduced for nonlinear stochastic partial differential equations (SPDEs) with multiplicative trace class noise. The proposed scheme converges with respect to the computational effort with a…
We study the connections between ordinary differential equations and optimization algorithms in a non-Euclidean setting. We propose a novel accelerated algorithm for minimising convex functions over a convex constrained set. This algorithm…
This paper deals with the numerical computations of two space dimensional time dependent parabolic partial differential equations by adopting adopting an optimal five stage fourth-order strong stability preserving Runge Kutta (SSP-RK54)…
We propose a new Eulerian-Lagrangian Runge-Kutta finite volume method for numerically solving convection and convection-diffusion equations. Eulerian-Lagrangian and semi-Lagrangian methods have grown in popularity mostly due to their…