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The goal of partial-order methods is to accelerate the exploration of concurrent systems by examining only a representative subset of all possible runs. The stateful approach builds a transition system with representative runs, while the…

Implicit Runge--Kutta (IRK) methods are highly effective for solving stiff ordinary differential equations (ODEs) but can be computationally expensive for large-scale problems due to the need of solving coupled algebraic equations at each…

Splitting-based time integration approaches such as fractional steps, alternating direction implicit, operator splitting, and locally one-dimensional methods partition the system of interest into components and solve individual components…

This paper derives a unifying theorem establishing consistency results for a broad class of tree-based algorithms. It improves current results in two aspects. First of all, it can be applied to algorithms that vary from traditional Random…

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…

Order-preserving couplings are elegant tools for obtaining robust estimates of the time-dependent and stationary distributions of Markov processes that are too complex to be analyzed exactly. The starting point of this paper is to study…

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

In this paper a set of previous general results for the development of B--series for a broad class of stochastic differential equations has been collected. The applicability of these results is demonstrated by the derivation of B--series…

In this paper, we analyze any-order Runge-Kutta spectral volume schemes (RKSV(s,k)) for solving the one-dimensional scalar hyperbolic equation. The RKSV(s,k) was constructed by using the $s$-th explicit Runge-Kutta method in…

A stochastic ordering approach is applied with Stein's method for approximation by the equilibrium distribution of a birth-death process. The usual stochastic order and the more general s-convex orders are discussed. Attention is focused on…

The purpose of the article is to derive equations that determine the trajectory of a non-conservative natural system in configuration space in non-stationary external fields. A theorem on the change in the kinetic energy of the system is…

In this paper, we present a novel class of high-order Runge--Kutta (RK) discontinuous Galerkin (DG) schemes for hyperbolic conservation laws. The new method extends beyond the traditional method of lines framework and utilizes…

In this paper we consider splitting methods for nonlinear ordinary differential equations in which one of the (partial) flows that results from the splitting procedure can not be computed exactly. Instead, we insert a well-chosen state…

In this paper, we study the form over the minimum spanning tree problem (MST) from which we will derive an intuitively generalized model and new methods with the upper bound of runtimes of logarithm. The new pattern we made has taken…

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…

This paper investigates the competitiveness of semi-implicit Runge-Kutta (RK) and spectral deferred correction (SDC) time-integration methods up to order six for incompressible Navier-Stokes problems in conjunction with a high-order…

Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic…

In this paper a new Runge-Kutta type scheme is introduced for nonlinear stochastic partial differential equations (SPDEs) with multiplicative trace class noise. The proposed scheme converges with respect to the computational effort with a…

Splitting methods constitute a well-established class of numerical schemes for the time integration of partial differential equations. Their main advantages over more traditional schemes are computational efficiency and superior geometric…

An explicit numerical strategy that practically preserves invariants is derived for conservative systems by combining an explicit high-order Runge-Kutta (RK) scheme with a simple modification of the standard projection approach, which is…