Related papers: Energy-preserving continuous-stage Runge-Kutta-Nys…

It is a classical theorem of Liouville that Hamiltonian systems preserve volume in phase space. Any symplectic Runge-Kutta method will respect this property for such systems, but it has been shown that no B-Series method can be volume…

The analytic form of a new class of factorized Runge-Kutta-Chebyshev (FRKC) stability polynomials of arbitrary order $N$ is presented. Roots of FRKC stability polynomials of degree $L=MN$ are used to construct explicit schemes comprising…

A semi-implicit-explicit (semi-IMEX) Runge-Kutta (RK) method is proposed for the numerical integration of ordinary differential equations (ODEs) of the form $\mathbf{u}' = \mathbf{f}(t,\mathbf{u}) + G(t,\mathbf{u}) \mathbf{u}$, where…

In this paper, exponential energy-preserving methods are formulated and analysed for solving charged-particle dynamics in a strong and constant magnetic field. The resulting method can exactly preserve the energy of the dynamics. Moreover,…

We introduce a new class of Runge-Kutta type methods suitable for time stepping to propagate hyperbolic solutions within tent-shaped spacetime regions. Unlike standard Runge-Kutta methods, the new methods yield expected convergence…

Recently a new class of nonlinearly partitioned Runge--Kutta (NPRK) methods was proposed for nonlinearly partitioned systems of autonomous ordinary differential equations, $y' = F(y,y)$. The target class of problems are ones in which…

Stabilized methods (also called Chebyshev methods) are explicit methods with extended stability domains along the negative real axis. These methods are intended for large mildly stiff problems, originating mainly from parabolic PDEs. In…

A third-order accurate implicit-explicit Runge-Kutta time marching numerical scheme is proposed and implemented for the Landau-Lifshitz-Gilbert equation, which models magnetization dynamics in ferromagnetic materials, with arbitrary damping…

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…

In this paper, we develop a higher order symmetric partitioned Runge-Kutta method for a coupled system of differential equations on Lie groups. We start with a discussion on partitioned Runge-Kutta methods on Lie groups of arbitrary order.…

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…

Phase-field model is a powerful mathematical tool to study the dynamics of interface and morphology changes in fluid mechanics and material sciences. However, numerically solving a phase field model for a real problem is a challenge task…

In this paper, we develop a provably energy stable and conservative discontinuous spectral element method for the shifted wave equation in second order form. The proposed method combines the advantages and central ideas of very successful…

This paper introduces a robust reformulation of the incompressible Navier-Stokes equations, establishing a foundational framework for designing efficient, structure-preserving algorithms that strictly conserve the original energy…

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…

This article extends the theory of dual-consistent summation-by-parts (SBP) and generalized SBP (GSBP) time-marching methods by showing that they are implicit Runge-Kutta schemes. Through this connection, the accuracy theory for the…

Many important applications are modelled by differential equations with positive solutions. However, it remains an outstanding open problem to develop numerical methods that are both (i) of a high order of accuracy and (ii) capable of…

The global-in-time energy estimate is derived for the second-order accurate exponential time differencing Runge-Kutta (ETDRK2) numerical scheme to the phase field crystal (PFC) equation, a sixth-order parabolic equation modeling crystal…

Energy preserving numerical methods for a certain class of PDEs are derived, applying the partition of unity method. The methods are extended to also be applicable in combination with moving mesh methods by the rezoning approach. These…

Compact Runge-Kutta (cRK) Discontinuous Galerkin (DG) methods, recently introduced in [Q. Chen, Z. Sun, and Y. Xing, SIAM J. Sci. Comput. SIAM J. Sci. Comput., 46: A1327-A1351, 2024], are a variant of RKDG methods for solving hyperbolic…