Related papers: Runge-Kutta methods for rough differential equatio…
We demonstrate the effectiveness of a novel scheme for numerically solving linear differential equations whose solutions exhibit extreme oscillation. We take a standard Runge-Kutta approach, but replace the Taylor expansion formula with a…
The use of high order fully implicit Runge-Kutta methods is of significant importance in the context of the numerical solution of transient partial differential equations, in particular when solving large scale problems due to fine space…
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…
In this paper we investigate the existence, uniqueness and approximation of solutions of delay differential equations (DDEs) with the right-hand side functions $f=f(t,x,z)$ that are Lipschitz continuous with respect to $x$ but only H\"older…
We consider Hamiltonian systems driven by multi-dimensional Gaussian processes in rough path sense, which include fractional Brownian motions with Hurst parameter $H\in(1/4,1/2]$. We indicate that the phase flow preserves the symplectic…
The recently-introduced relaxation approach for Runge-Kutta methods can be used to enforce conservation of energy in the integration of Hamiltonian systems. We study the behavior of implicit and explicit relaxation Runge-Kutta methods in…
We present a C++ implementation of a fifth order semi-implicit Runge-Kutta algorithm for solving Ordinary Differential Equations. This algorithm can be used for studying many different problems and in particular it can be applied for…
Runge--Kutta (RK) methods are widely used techniques for solving a class of initial value problems. In this article, we introduce an adaptive multiquadratic (MQ) radial basis function (RBF)-based method to develop enhanced explicit RK…
Finite element methods provide accurate and efficient methods for the numerical solution of partial differential equations by means of restricting variational problems to finite-dimensional approximating spaces. However, they do not…
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…
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…
We study gradient-based optimization methods obtained by direct Runge-Kutta discretization of the ordinary differential equation (ODE) describing the movement of a heavy-ball under constant friction coefficient. When the function is high…
We present a numerical method for the approximation of solutions for the class of stochastic differential equations driven by Brownian motions which induce stochastic variation in fixed directions. This class of equations arises naturally…
In this master thesis we have compared different second order stabilized explicit Runge-Kutta methods when applied to the incompressible Navier-Stokes equations by means of a projection method and a differential algebraic approach. We…
Finite element discretization of time dependent problems also require effective time-stepping schemes. While implicit Runge-Kutta methods provide favorable accuracy and stability problems, they give rise to large and complicated systems of…
We study the approximation of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H>1/2$. For the mean-square error at a single point we derive the optimal rate of convergence that can be achieved…
We consider the rough differential equations driven by tempered fractional Brownian motion with Hurst index $H\in (\frac{1}{4}, \frac{1}{3})$ and tempered parameter $\lambda>0$. First, by means of piecewise linear approximation, we…
We propose a kernel compression method for solving Distributed-Order (DO) Fractional Partial Differential Equations (DOFPDEs) at the cost of solving corresponding local-in-time PDEs. The key concepts are (1) discretization of the integral…
A new method for the numerical solution of ODEs is presented. This approach is based on an approximate formulation of the Taylor methods that has a much easier implementation than the original Taylor methods, since only the functions in the…
This work introduces a new class of Runge-Kutta methods for solving nonlinearly partitioned initial value problems. These new methods, named nonlinearly partitioned Runge-Kutta (NPRK), generalize existing additive and component-partitioned…