Related papers: Optimized explicit Runge-Kutta schemes for the spe…

Implicit-explicit (IMEX) Runge-Kutta methods play a major rule in the numerical treatment of differential systems governed by stiff and non-stiff terms. This paper discusses order conditions and symplecticity properties of a class of IMEX…

Many important initial value problems have the property that energy is non-increasing in time. Energy stable methods, also referred to as strongly stable methods, guarantee the same property discretely. We investigate requirements for…

We put forward the use of total-variation-diminishing (or more generally, strong stability preserving) implicit-explicit Runge-Kutta methods for the time integration of the equations of motion associated with the semiconvection problem in…

In the present paper, a class of stochastic Runge-Kutta methods containing the second order stochastic Runge-Kutta scheme due to E. Platen for the weak approximation of It\^o stochastic differential equation systems with a multi-dimensional…

A unified theoretical framework is suggested to examine the energy dissipation properties at all stages of additive implicit-explicit Runge-Kutta (IERK) methods up to fourth-order accuracy for gradient flow problems. We construct some…

A novel class of explicit high-order energy-preserving methods are proposed for general Hamiltonian partial differential equations with non-canonical structure matrix. When the energy is not quadratic, it is firstly done that the original…

In a previous paper, a technique was suggested to avoid order reduction with any explicit exponential Runge-Kutta method when integrating initial boundary value nonlinear problems with time-dependent boundary conditions. In this paper, we…

We provide a framework for high-order discretizations of nonlinear scalar convection-diffusion equations that satisfy a discrete maximum principle. The resulting schemes can have arbitrarily high order accuracy in time and space, and can be…

In this paper we consider time-dependent PDEs discretized by a special class of Physics Informed Neural Networks whose design is based on the framework of Runge--Kutta and related time-Galerkin discretizations. The primary motivation for…

We present the formulation and optimization of a Runge-Kutta-type time-stepping scheme for solving the shallow water equations, aimed at substantially increasing the effective allowable time-step over that of comparable methods. This…

We introduce a family of stochastic optimization methods based on the Runge-Kutta-Chebyshev (RKC) schemes. The RKC methods are explicit methods originally designed for solving stiff ordinary differential equations by ensuring that their…

We study the stability of explicit Runge-Kutta methods for high order Lagrangian finite element approximation of linear parabolic equations and establish bounds on the largest eigenvalue of the system matrix which determines the largest…

A wide range of physical phenomena exhibit auxiliary admissibility criteria, such as conservation of entropy or various energies, which arise implicitly under exact solution of their governing PDEs. However, standard temporal schemes, such…

We apply Runge-Kutta methods to linear partial differential-algebraic equations of the form $Au_t(t,x) + B(u_{xx}(t,x)+ru_x(t,x))+Cu(t,x) = f(t,x)$, where $A,B,C\in\R^{n,n}$ and the matrix $A$ is singular. We prove that under certain…

In practical computation with Runge--Kutta methods, the stage equations are not satisfied exactly, due to roundoff errors, algebraic solver errors, and so forth. We show by example that propagation of such errors within a single step can…

We consider Implicit-Explicit (IMEX) Runge-Kutta (R-K) schemes for hyperbolic systems with stiff relaxation in the so-called diffusion limit. In such regime the system relaxes towards a convection-diffusion equation. The first objective of…

We propose a unified theoretical framework to examine the energy dissipation properties at all stages of explicit exponential Runge-Kutta (EERK) methods for gradient flow problems. The main part of the novel framework is to construct the…

This work constructs and analyzes new efficient high-order two-derivative diagonally implicit Runge--Kutta (TDDIRK) schemes with optimized phase errors. Specifically, we present a convergence result for TDDIRK methods and investigate their…

The Butcher theory provides a powerful tool for analyzing order conditions of Runge-Kutta schemes for ordinary differential equations (ODEs); however, such a theory has not yet been well established for backward stochastic differential…

Strong stability preserving (SSP) Runge-Kutta methods are often desired when evolving in time problems that have two components that have very different time scales. Where the SSP property is needed, it has been shown that implicit and…