Related papers: Efficient energy-preserving numerical approximatio…
This paper presents a new method to approximate the time-dependent convection-diffusion equations using conforming finite element methods, ensuring that the discrete solution respects the physical bounds imposed by the differential…
We present a class of non-standard numerical schemes which are modifications of the discrete gradient method. They preserve the energy integral exactly (up to the round-off error). The considered class contains locally exact discrete…
In this paper we consider a mass- and energy--conserving Crank-Nicolson time discretization for a general class of nonlinear Schr\"odinger equations. This scheme, which enjoys popularity in the physics community due to its conservation…
A method for constructing first integral preserving numerical schemes for time-dependent partial differential equations on non-uniform grids is presented. The method can be used with both finite difference and partition of unity approaches,…
This paper establishes and analyzes a second-order accurate numerical scheme for the nonlinear partial integrodifferential equation with a weakly singular kernel. In the time direction, we apply the Crank-Nicolson method for the time…
For a general class of nonlinear port-Hamiltonian systems we develop a high-order time discretization scheme with certain structure preservation properties. The finite or infinite-dimensional system under consideration possesses a…
We study the numerical solution of a Cahn-Hilliard/Allen-Cahn system with strong coupling through state and gradient dependent non-diagonal mobility matrices. A fully discrete approximation scheme in space and time is proposed which…
We propose and analyze a linearly stabilized semi-implicit diffusive Crank--Nicolson scheme for the Cahn--Hilliard gradient flow. In this scheme, the nonlinear bulk force is treated explicitly with two second-order stabilization terms. This…
We study relaxation-based approaches for conserving mass and energy in the numerical solution of Schr\"odinger-Poisson (SP) type systems. Relaxation-based methods offer a general approach that can be applied as post-time step processing to…
In this paper, we study the Schr\"{o}dinger equation in the semiclassical regime and with multiscale potential function. We develop the so-called constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM), in…
In this paper, an implicit nonsymplectic exact energy-preserving integrator is specifically designed for a ten-dimensional phase-space conservative Hamiltonian system with five degrees of freedom. It is based on a suitable…
We present a new temporal discretization paradigm for developing energy-production-rate preserving numerical approximations to thermodynamically consistent partial differential equation systems, called the supplementary variable method. The…
In this paper, a linear second order numerical scheme is developed and investigated for the Allen-Cahn equation with a general positive mobility. In particular, our fully discrete scheme is mainly constructed based on the Crank-Nicolson…
Discrete gradient methods are a powerful tool for the time discretization of dynamical systems, since they are structure-preserving regardless of the form of the total energy. In this work, we discuss the application of discrete gradient…
The parabolic-elliptic Keller-Segel equation with sensitivity saturation, because of its pattern formation ability, is a challenge for numerical simulations. We provide two finite-volume schemes whose goals are to preserve, at the discrete…
In this paper, we focus on constructing numerical schemes preserving the averaged energy evolution law for nonlinear stochastic wave equations driven by multiplicative noise. We first apply the compact finite difference method and the…
The numerical solution of a nonlinear and space-fractional anti-diffusive equation used to model dune morphodynamics is considered. Spatial discretization is effected using a finite element method whereas the Crank-Nicolson scheme is used…
We present a finite-difference integration algorithm for solution of a system of differential equations containing a diffusion equation with nonlinear terms. The approach is based on Crank-Nicolson method with predictor-corrector algorithm…
This paper introduces filtered finite difference methods for numerically solving a dispersive evolution equation with solutions that are highly oscillatory in both space and time. We consider a semiclassically scaled nonlinear Schr\"odinger…
We perform some simulations of the semilinear Klein--Gordon equation in the de Sitter spacetime. We reported the accurate numerical results of the equation with the structure-preserving scheme (SPS) in an earlier publication (Tsuchiya and…