Related papers: Factorized schemes of second-order accuracy for nu…
Efficient and energy stable high order time marching schemes are very important but not easy to construct for the study of nonlinear phase dynamics. In this paper, we propose and study two linearly stabilized second order semi-implicit…
We present a numerical scheme for solving a sixth-order Cahn-Hilliard type equation that captures the dynamics of phase transitions in a ternary mixture consisting of two immiscible fluids and a surface active molecule that is amphiphilic.…
We present a second-order-in-time finite difference scheme for the Cahn-Hilliard-Hele-Shaw equations. This numerical method is uniquely solvable and unconditionally energy stable. At each time step, this scheme leads to a system of…
An implicit Euler--Maruyama method with non-uniform step-size applied to a class of stochastic partial differential equations is studied. A spectral method is used for the spatial discretization and the truncation of the Wiener process. A…
A second order accurate, linear numerical method is analyzed for the Landau-Lifshitz equation with large damping parameters. This equation describes the dynamics of magnetization, with a non-convexity constraint of unit length of the…
In this paper, we present a parallel numerical algorithm for solving the phase field crystal equation. In the algorithm, a semi-implicit finite difference scheme is derived based on the discrete variational derivative method. Theoretical…
Parareal algorithms are studied for semilinear parabolic stochastic partial differential equations. These algorithms proceed as two-level integrators, with fine and coarse schemes, and have been designed to achieve a `parallel in real time'…
A numerical solution to the Fokker-Planck equation using a two-level scheme is presented. The Fokker-Planck (FP) equation is of parabolic type equation govern the time evolution of probability density function of the stochastic processes.…
This work presents arbitrary high order well balanced finite volume schemes for the Euler equations with a prescribed gravitational field. It is assumed that the desired equilibrium solution is known, and we construct a scheme which is…
This paper presents a sequence of deferred correction (DC) schemes built recursively from the implicit midpoint scheme for the numerical solution of general first order ordinary differential equations (ODEs). It is proven that each scheme…
A common way to numerically solve Fokker-Planck equations is the Chang-Cooper method in space combined with one of the Euler methods in time. However, the explicit Euler method is only conditionally positive, leading to severe restrictions…
This report presents a low computational and cognitive complexity, stable, time accurate and adaptive method for the Navier-Stokes equations. The improved method requires a minimally intrusive modification to an existing program based on…
The stability of classical semi-implicit scheme, and some more advanced iterative schemes recently proposed for Numerical Weather Prediction (NWP) purpose is examined. In all these schemes, the solution of the centred-implicit non-linear…
The equation with the time fractional substantial derivative and space fractional derivative describes the distribution of the functionals of the L\'evy flights; and the equation is derived as the macroscopic limit of the continuous time…
High order schemes are known to be unstable in the presence of shock discontinuities or under-resolved solution features for nonlinear conservation laws. Entropy stable schemes address this instability by ensuring that physically relevant…
We consider the study of a numerical scheme for an initial- and Dirichlet boundary- value problem for a nonlinear Schr\"odinger equation. We approximate the solution using a, local (non-uniform) two level scheme in time (see C. Besse [6]…
Various classes of stable finite difference schemes can be constructed to obtain a numerical solution. It is important to select among all stable schemes such a scheme that is optimal in terms of certain additional criteria. In this study,…
We consider a mechanical system with impact and n degrees of freedom, written in generalized coordinates. The system is not necessarily Lagrangian. The representative point of the system must remain inside a set of constraints K; the…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
The Allen-Cahn equation is a fundamental model for phase transitions, offering critical insights into the dynamics of interface evolution in various physical systems. This paper investigates the stability and robustness of frequently…