Related papers: Optimal second order diagonally implicit SSP Runge…
In this paper, we propose Galerkin-Legendre spectral method with implicit Runge-Kutta method for solving the unsteady two-dimensional Schrodinger equation with nonhomogeneous Dirichlet boundary conditions and initial condition. We apply a…
Mixed precision Runge--Kutta methods have been recently developed and used for the time-evolution of partial differential equations. Two-derivative Runge--Kutta schemes may offer enhanced stability and accuracy properties compared to…
Explicit Runge-Kutta schemes with large stable step sizes are developed for integration of high order spectral difference spatial discretization on quadrilateral grids. The new schemes permit an effective time step that is substantially…
A methodology that can generate the optimal coefficients of a numerical method with the use of an artificial neural network is presented in this work. The network can be designed to produce a finite difference algorithm that solves a…
A new Chebyshev-type family of stabilized explicit methods for solving mildly stiff ODEs is presented. Besides conventional conditions of order and stability we impose an additional restriction on the methods: their stability function must…
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…
We study gradient-based optimization methods obtained by directly discretizing a second-order ordinary differential equation (ODE) related to the continuous limit of Nesterov's accelerated gradient method. When the function is smooth…
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…
The use of symplectic numerical schemes on Hamiltonian systems is widely known to lead to favorable long-time behaviour. While this phenomenon is thoroughly understood in the context of finite-dimensional Hamiltonian systems, much less is…
A time discretization method is called strongly stable, if the norm of its numerical solution is nonincreasing. It is known that, even for linear semi-negative problems, many explicit Runge--Kutta (RK) methods fail to preserve this…
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…
The aim of this paper is to construct and analyze explicit exponential Runge-Kutta methods for the temporal discretization of linear and semilinear integro-differential equations. By expanding the errors of the numerical method in terms of…
This work considers multirate generalized-structure additively partitioned Runge-Kutta (MrGARK) methods for solving stiff systems of ordinary differential equations (ODEs) with multiple time scales. These methods treat different partitions…
This paper continues to study the explicit two-stage fourth-order accurate time discretiza- tions [5, 7]. By introducing variable weights, we propose a class of more general explicit one-step two-stage time discretizations, which are…
This paper is devoted to examining the stability of Runge-Kutta methods for solving nonlinear Volterra delay-integro-differential-algebraic equations (DIDAEs) with constant delay. Hybrid numerical schemes combining Runge-Kutta methods and…
In this paper we define an efficient implementation of Runge-Kutta methods of Radau IIA type, which are commonly used when solving stiff ODE-IVPs problems. The proposed implementation relies on an alternative low-rank formulation of the…
Many practical problems can be described by second-order system $\ddot{q}=-M\nabla U(q)$, in which people give special emphasis to some invariants with explicit physical meaning, such as energy, momentum, angular momentum, etc. However,…
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…
In the paper explicit functional continuous Runge-Kutta and Runge-Kutta-Nystr\"om methods for retarded functional differential equations are considered. New methods for first order equations as well as for second order equations of the…
In this paper, we propose the diagonal implicit Runge-Kutta methods and transformed Runge-Kutta methods for stochastic Poisson systems with multiple noises. We prove that the first methods can preserve the Poisson structure, Casimir…