We consider a nonlocal scalar conservation law proposed by Andrew C. Fowler to describe the dynamics of dunes, and we develop a numerical procedure based on splitting methods to approximate its solutions. We begin by proving the convergence of the well-known Lie formula, which is an approximation of the exact solution of order one in time. We next use the split-step Fourier method to approximat...

We consider the 3D time dependent Stokes equation on a finite time interval and on a uniform rectangular mesh, approached in terms of velocity and pressure. In a parallel algorithm, based on a novel direction splitting approach, the pressure equation is derived from a perturbed form of the continuity equation, in which the incompressibility constraint is penalized in a negative norm induced by ...

We present a semi discrete method for constructing approximate solutions to the initial value problem for the m dimensional convection di usion equation ut r f u u The method is based on the use of operator splitting to isolate the convection part and the di usion part of the equation In the casem dimensional splitting is used to reduce the m dimensional convection problem to a series of one di...

Rapid processes of heat transfer are not described by the standard heat conduction equation. To take into account a finite velocity of heat transfer, we use the hyperbolic model of heat conduction, which is connected with the relaxation of heat fluxes. In this case, the mathematical model is based on a hyperbolic equation of second order or a system of equations for the temperature and heat flu...

We consider the time dependent Stokes equation on a finite time interval and on a uniform rectangular mesh, written in terms of velocity and pressure. A parallel algorithm based on a new direction splitting approach is developed. Here, the pressure equation is derived from a perturbed form of the continuity equation in which the incompressibility constraint is penalized in a negative norm induc...

In this paper we propose a fast, robust, and accurate operator splitting method for phase-field simulations of dendritic growth in both twoand three-dimensional space. The proposed method is based on operator splitting techniques. We split the governing phase-field equation into three parts: the first equation is calculated by using an explicit Euler’s method. The second is a heat equation with...

Operator splitting (OS) is a popular and convenient technique used to numerically solve reactive transport problems such as Fisher’s equation. Although OS has been widely used to solve Fisher’s equation, no characterization of the innate OS error has been presented. Here the exact characteristics of the OS error for travelling wave solutions of Fisher’s equation are revealed and explored. The a...

We apply the method of operator splitting on the generalized Korteweg{de Vries (KdV) equation ut +f(u)x+"uxxx = 0, by solving the nonlinear conservation law ut +f(u)x = 0 and the linear dispersive equation ut + "uxxx = 0 sequentially. We prove that if the approximation obtained by operator splitting converges, then the limit function is a weak solution of the generalized KdV equation. Convergen...

Here we consider the product of varieties with n-blocks collections . We give some cohomological splitting conditions for rank 2 bundles. A cohomological characterization for vector bundles is also provided. The tools are Beilinson’s type spectral sequences generalized by Costa and Miró-Roig. Moreover we introduce a notion of CastelnuovoMumford regularity on a product of finitely many projectiv...

We provide a new analytical approach to operator splitting for equations of the type ut = Au + B(u) where A is a linear operator and B is quadratic. A particular example is the Korteweg–de Vries (KdV) equation ut−uux +uxxx = 0. We show that the Godunov and Strang splitting methods converge with the expected rates if the initial data are sufficiently regular.