Related papers: Symplecticity-preserving continuous-stage Runge-Ku…
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…
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…
Based on reasonable testing model problems, we study the preservation by symplectic Runge-Kutta method (SRK) and symplectic partitioned Runge-Kutta method (SPRK) of structures for fixed points of linear Hamiltonian systems. The…
For the approximation of solutions for It\^o and Stratonovich stochastic differential equations (SDEs)a new class of efficient stochastic Runge-Kutta (SRK) methods is developed. As the main novelty only two stages are necessary for the…
We are concerned with the efficient implementation of symplectic implicit Runge-Kutta (IRK) methods applied to systems of (non-necessarily Hamiltonian) ordinary differential equations by means of Newton-like iterations. We pay particular…
This study computes the gradient of a function of numerical solutions of ordinary differential equations (ODEs) with respect to the initial condition. The adjoint method computes the gradient approximately by solving the corresponding…
Explicit Runge--Kutta (RK) methods are susceptible to a reduction in the observed order of convergence when applied to initial-boundary value problem with time-dependent boundary conditions. We study conditions on explicit RK methods that…
In this paper, we consider stochastic Runge-Kutta methods for stochastic Hamiltonian partial differential equations and present some sufficient conditions for multisymplecticity of stochastic Runge-Kutta methods of stochastic Hamiltonian…
Runge-Kutta methods are a popular class of numerical methods for solving ordinary differential equations. Every Runge-Kutta method is characterized by two basic parameters: its order, which measures the accuracy of the solution it produces,…
Using simplifying assumptions that are related to the time reversal symmetry, a 1-dimensional family of 8-stage pseudo-symplectic Runge-Kutta methods of order (4, 8), i.e., methods of order 4 that preserve symplectic structure up to order…
In this paper, we construct stochastic symplectic Runge--Kutta (SSRK) methods of high strong order for Hamiltonian systems with additive noise. By means of colored rooted tree theory, we combine conditions of mean-square order 1.5 and…
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…
A new class of third order Runge-Kutta methods for stochastic differential equations with additive noise is introduced. In contrast to Platen's method, which to the knowledge of the author has been up to now the only known third order…
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…
In this paper, we are concerned with arbitrarily high-order momentum-preserving and energy-preserving schemes for solving the generalized Rosenau-type equation, respectively. The derivation of the momentum-preserving schemes is made within…
Classical convergence theory of Runge-Kutta methods assumes that the time step is small relative to the Lipschitz constant of the ordinary differential equation (ODE). For stiff problems, that assumption is often violated, and a problematic…
We explore a novel way to numerically resolve the scaling behavior of finite-time singularities in solutions of nonlinear parabolic PDEs. The Runge--Kutta--Legendre (RKL) and Runge--Kutta--Gegenbauer (RKG) super-time-stepping methods were…
High order energy-preserving methods for Hamiltonian systems are presented. For this aim, an energy-preserving condition of continuous stage Runge--Kutta methods is proved. Order conditions are simplified and parallelizable conditions are…
We propose an extended framework for continuous-stage Runge-Kutta methods which enables us to treat more complicated cases especially for the case weighting on infinite intervals. By doing this, various types of weighted orthogonal…
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…