Related papers: A unified formulation of splitting-based implicit …
While implicit Runge--Kutta methods possess high order accuracy and important stability properties, implementation difficulties and the high expense of solving the coupled algebraic system at each time step are frequently cited as…
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…
Exponential integrators are special time discretization methods where the traditional linear system solves used by implicit schemes are replaced with computing the action of matrix exponential-like functions on a vector. A very general…
The structural flexibility of the exponential propagation iterative methods of Runge-Kutta type (EPIRK) enables construction of particularly efficient exponential time integrators. While the EPIRK methods have been shown to perform well on…
In this paper we consider various splitting schemes for unsteady problems containing the grad-div operator. The fully implicit discretization of such problems would yield at each time step a linear problem that couples all components of the…
For many systems of differential equations modeling problems in science and engineering, there are natural splittings of the right hand side into two parts, one non-stiff or mildly stiff, and the other one stiff. For such systems…
Computer simulations in QCD are based on the discretization of the theory on a Euclidean lattice. To compute the mean value of an observable, usually the Hybrid Monte Carlo method is applied. Here equations of motion, derived from an…
We reconsider the variational derivation of symplectic partitioned Runge-Kutta schemes. Such type of variational integrators are of great importance since they integrate mechanical systems with high order accuracy while preserving the…
Efficient high order numerical methods for evolving the solution of an ordinary differential equation are widely used. The popular Runge--Kutta methods, linear multi-step methods, and more broadly general linear methods, all have a global…
In this paper, two novel classes of implicit exponential Runge-Kutta (ERK) methods are studied for solving highly oscillatory systems. First of all, we analyze the symplectic conditions of two kinds of exponential integrators, and present a…
This paper introduces a novel paradigm for constructing linearly implicit and high-order unconditionally energy-stable schemes for general gradient flows, utilizing the scalar auxiliary variable (SAV) approach and the additive Runge-Kutta…
We propose a new probabilistic scheme which combines deep learning techniques with high order schemes for backward stochastic differential equations belonging to the class of Runge-Kutta methods to solve high-dimensional semi-linear…
A component-splitting method is proposed to improve convergence characteristics for implicit time integration of compressible multicomponent reactive flows. The characteristic decomposition of flux jacobian of multicomponent Navier-Stokes…
Some properties of numerical time integration methods using summation by parts operators and simultaneous approximation terms are studied. These schemes can be interpreted as implicit Runge-Kutta methods with desirable stability properties…
This work introduces a general framework for constructing high-order, linearly stable, partitioned solvers for multiphysics problems from a monolithic implicit-explicit Runge-Kutta (IMEX-RK) discretization of the semi-discrete equations.…
Irksome is a library based on the Unified Form Language (UFL) that automates the application of Runge-Kutta time-stepping methods for finite element spatial discretizations of partial differential equations (PDEs). This paper describes…
We consider a Lagrangian system $L(q,\dot q) = \sum_{l=1}^{N}L^{\{l\}}(q,\dot q)$, where the $q$-variable is treated by a Generalized Additive Runge--Kutta (GARK) method. Applying the technique of discrete variations, we show how to…
Integrable systems are usually given in terms of functions of continuous variables (on ${\mathbb R}$), functions of discrete variables (on ${\mathbb Z}$) and recently in terms of functions of $q$-variables (on ${\mathbb K}_{q}$). We…
A single-step high-order implicit time integration scheme for the solution of transient and wave propagation problems is presented. It is constructed from the Pad\'e expansions of the matrix exponential solution of a system of first-order…
The integration of aerodynamic drag is a fundamental step in simulating dust dynamics in hydrodynamical simulations. We propose a novel integration scheme, designed to be compatible with Strang splitting techniques, which allows for the…