Related papers: Higher order operator splitting Fourier spectral m…
Nonlinear elliptic problems arise in many fields, including plasma physics, astrophysics, and optimal transport. In this article, we propose a novel operator-splitting/finite element method for solving such problems. We begin by introducing…
The use of operator-splitting methods to solve differential equations is widespread, but the methods are generally only defined for a given number of operators, most commonly two. Most operator-splitting methods are not generalizable to…
Current generations of graphics processing units have turned into highly parallel devices with general computing capabilities. Thus, graphics processing units may be utilized, for example, to solve time dependent partial differential…
In this paper, we propose and analyze a linear second-order numerical method for solving the Allen-Cahn equation with a general mobility. The proposed fully-discrete scheme is carefully constructed based on the combination of first and…
Splitting methods for the numerical integration of differential equations of order greater than two involve necessarily negative coefficients. This order barrier can be overcome by considering complex coefficients with positive real part.…
In [1], we inaugurated a new area of optimal control (OC) theory that we called "periodic fractional OC theory," which was developed to find optimal ways to periodically control a fractional dynamic system. The typical mathematical…
A method is described to solve the nonlinear Langevin equations arising from quadratic interactions in quantum mechanics. While, the zeroth order linearization approximation to the operators is normally used, here first and second order…
The generalized-$\alpha$ method encompasses a wide range of time integrators. The method possesses high-frequency dissipation while minimizing unwanted low-frequency dissipation and the numerical dissipation can be controlled by the user.…
We present two algorithms for computing what we call the absolute factorization of a difference operator. We also give an algorithm to solve third order difference equations in terms of second order equations, together with applications to…
This paper provides a summary of the fractal calculus framework. It presents higher-order homogeneous and nonhomogeneous linear fractal differential equations with $\alpha$-order. Solutions for these equations with constant coefficients are…
We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when the vector field associated with the ODE…
In this paper we present a novel multiscale splitting approach to solve multiscale Schroedinger equation, which have large different time-scales. The energy potential is based on highly oscillating functions, which are magnitudes faster…
In this work, we show high order splitting methods of integration without negative steps, allowing us to solve numerically irreversible problems, like reaction-diffusion equations. The methods consist in a suitable affine combinations of…
In this paper, we develop high-order splitting methods for linear port-Hamiltonian systems, focusing on preserving their intrinsic structure, particularly the dissipation inequality. Port-Hamiltonian systems are characterized by their…
In this paper, we propose Fourier pseudospectral methods to solve the variable-order space fractional wave equation and develop an accelerated matrix-free approach for its effective implementation. In constant-order cases, our methods can…
Integral deferred correction (IDC) methods have been shown to be an efficient way to achieve arbitrary high order accuracy and possess good stability properties. In this paper, we construct high order operator splitting schemes using the…
We propose two efficient energetic spectral-element methods in time for marching nonlinear gradient systems with the phase-field Allen--Cahn equation as an example: one fully implicit nonlinear method and one semi-implicit linear method.…
Splitting algorithms for finding a zero of sum of operators often involve multiple steps which are referred to as forward or backward steps. Forward steps are the explicit use of the operators and backward steps involve the operators…
An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves the square root of an elliptic operator of second order. Finite element approximation in space is employed.…
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…