Related papers: Exponential Runge-Kutta methods for delay equation…
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…
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 study symmetric integrators for solving second-order ordinary differential equations on the basis of the notion of continuous-stage Runge-Kutta-Nystrom methods. The construction of such methods heavily relies on the…
In this paper, two new families of fourth-order explicit exponential Runge--Kutta (ERK) methods with four stages are studied for solving first-order differential systems $y'(t)+My(t)=f(y(t))$. By comparing the Taylor series of the exact…
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…
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…
In this paper, we present error estimates of fully discrete Runge--Kutta discontinuous Galerkin (DG) schemes for linear time-dependent partial differential equations. The analysis applies to explicit Runge--Kutta time discretizations of any…
This paper introduces a novel framework for the solution of (large-scale) Lyapunov and Sylvester equations derived from numerical integration methods. Suitable systems of ordinary differential equations are introduced. Low-rank…
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 propose a new method that extends conservative explicit multirate methods to implicit explicit-multirate methods. We develop extensions of order one and two with different stability properties on the implicit side. The method is suitable…
In this work we present a new class of Runge-Kutta (RK) methods for solving systems of hyperbolic equations with a particular structure, generalization of a wave-equation. The new methods are {\it partially implicit} in the sense that a…
Set differential equations are usually formulated in terms of the Hukuhara differential, which implies heavy restrictions for the nature of a solution. We propose to reformulate set differential equations as ordinary differential equations…
We combine the recent relaxation approach with multiderivative Runge-Kutta methods to preserve conservation or dissipation of entropy functionals for ordinary and partial differential equations. Relaxation methods are minor modifications of…
In this work, we present a general technique for establishing the strong convergence of numerical methods for stochastic delay differential equations (SDDEs) in the infinite horizon. This technique can also be extended to analyze certain…
Stabilized Runge-Kutta methods are especially efficient for the numerical solution of large systems of stiff nonlinear differential equations because they are fully explicit. For semi-discrete parabolic problems, for instance, stabilized…
Many time-dependent differential equations are equipped with invariants. Preserving such invariants under discretization can be important, e.g., to improve the qualitative and quantitative properties of numerical solutions. Recently,…
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…
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…
Exponential Runge-Kutta methods are a well-established tool for the numerical integration of parabolic evolution equations. However, these schemes are typically developed under the assumption of homogeneous boundary conditions. In this…
Rational methods are intended to time integrate linear homogeneous problems. However, their scope can be extended so as to cover linear nonhomogeneous problems. In this paper the integration of semilinear problems is considered. The…