Related papers: On the stability of strong-stability-preserving mo…
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…
New time integration methods are proposed for simulating incompressible multiphase flow in pipelines described by the one-dimensional two-fluid model. The methodology is based on 'half-explicit' Runge-Kutta methods, being explicit for the…
We develop a family of second-order implicit-explicit (IMEX) schemes for the stiff BGK kinetic equation. The method is asymptotic-preserving (can capture the Euler limit without numerically resolving the small Knudsen number) as well as…
The paper concerns semidiscretizations in time of stochastic Maxwell equations driven by additive noise. We show that the equations admit physical properties and mathematical structures, including regularity, energy and divergence evolution…
The unified gas-kinetic scheme (UGKS) is becoming increasingly popular for multiscale simulations in all flow regimes. This paper provides the first analytical study on the stability of the UGKS applied to a linear kinetic model, which is…
In this paper, we present a comprehensive long-time stability analysis of a second-order explicit exponential Runge--Kutta (ERK2) method for the Cahn--Hilliard (CH) equation. By employing Fourier spectral collocation in space and a…
For many nonlinear control systems, the chosen equilibrium determines both the steady-state efficiency and the dynamic performance. This paper addresses the issue of obtaining an optimal equilibrium in terms of some steady-state operation…
We propose a new Lagrange multiplier approach to construct positivity preserving schemes for parabolic type equations. The new approach introduces a space-time Lagrange multiplier to enforce the positivity with the Karush-Kuhn-Tucker (KKT)…
This work introduces a stabilised finite element formulation for the Stokes flow problem with a nonlinear slip boundary condition of friction type. The boundary condition is enforced with the help of an additional Lagrange multiplier and…
We study spatially partitioned embedded Runge--Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient…
Many time-dependent partial differential equations (PDEs) can be transformed into an ordinary differential equations (ODEs) containing moderately stiff and non-stiff terms after spatial semi-discretization. In the present paper, we…
Linearized numerical stability bounds for solving the nonlinear time-dependent Schr\"odinger equation (NLSE) using explicit finite-differencing are shown. The bounds are computed for the fourth-order Runge-Kutta scheme in time and both…
This paper focuses on the strong convergence rate of both Runge--Kutta methods and simplified step-$N$ Euler schemes for stochastic differential equations driven by multi-dimensional fractional Brownian motions with $H\in(\frac12,1)$. Based…
We study the strong approximation of the solutions to singular stochastic kinetic equations (also referred to as second-order SDEs) driven by $\alpha$-stable processes, using an Euler-type scheme inspired by [11]. For these equations, the…
We report the development of a discontinuous spectral element flow solver that includes the implementation of both spectral difference and flux reconstruction formulations. With this high order framework, we have constructed a foundation…
We put forward the use of total-variation-diminishing (or more generally, strong stability preserving) implicit-explicit Runge-Kutta methods for the time integration of the equations of motion associated with the semiconvection problem in…
We study the local discretization error of Patankar-type Runge-Kutta methods applied to semi-discrete PDEs. For a known two-stage Patankar-type scheme the local error in PDE sense for linear advection or diffusion is shown to be of the…
The shallow water flow model is widely used to describe water flows in rivers, lakes, and coastal areas. Accounting for uncertainty in the corresponding transport-dominated nonlinear PDE models presents theoretical and numerical challenges…
Explicit Runge-Kutta schemes with large stable step sizes are developed for integration of high order spectral difference spatial discretization on quadrilateral grids. The new schemes permit an effective time step that is substantially…
In this paper, we investigate the orbital stability of peakons for a modified Camassa-Holm equation with cubic nonlinearity derived from the two-dimensional Euler equation. By overcoming the difficulties caused by one of the complicated…