A second-order accurate in time, positivity-preserving, and unconditionally energy stable operator splitting scheme is proposed analyzed for reaction-diffusion systems with the detailed balance condition. The designed based on an energetic variational formulation, which reaction part reformulated terms of trajectory, both diffusion parts dissipate same free energy. At stage, trajectory equation...

We present a parallel time-stepping method for fluid-structure interactions. The interaction between the incompressible Navier-Stokes equations and hyperelastic solid is formulated in fully monolithic framework. Discretization space based on equal order finite element all variables variant of Crank-Nicolson scheme used as second time integrator. To accelerate solution systems, we analyze parall...

An implicit energy-decaying modified Crank--Nicolson time-stepping method is constructed for the viscous rotating shallow water equation on plane. Existence, uniqueness, and convergence of semidiscrete solutions are proved by using Schaefer's fixed point theorem $H^2$ estimates discretized hyperbolic--parabolic system. For practical computation, further in space, resulting a fully discrete fini...

A weak singularity in the solution of time-fractional differential equations can degrade accuracy numerical methods when employing a uniform mesh, especially with schemes involving Caputo derivative (order α,), where time is order (2−α) or (1+α). To deal this problem, we present second-order scheme for nonlinear time–space fractional reaction–diffusion equations. For spatial resolution, employ ...

Nonlinear quantum graphs are metric equipped with a nonlinear Schrödinger equation. Whereas in the last ten years they have known considerable developments on theoretical side, their study from numerical point of view remains its early stages. The goal this paper is to present Grafidi library, Python library which has been developed simulation equations mind. We will show how, help one can impl...

Continuously operated clarifier-thickener units can be modeled by a non-linear, scalar conservation law with a flux that involves two parameters that depend discontinuously on the space variable. This paper presents two numerical schemes for the solution of this equation that have formal second-order accuracy in both the time and space variable. One of the schemes is a standard total variation ...

In [13], we had performed numerical comparisons for different time stepping schemes for the incompressible Navier-Stokes equations. In this paper, we present the numerical analysis in the context of the Navier-Stokes equations for a new time-stepping θ-scheme which has been recently proposed by Glowinski [5]. Like the well-known classical FractionalStep-θ-scheme which had been introduced by Glo...

In this article, we aim to discuss the formulation of two explicit group iterative finite difference methods for time-dependent two dimensional Burger’s problem on a variable mesh. For the non-linear problems, the discretization leads to a non-linear system whose Jacobian is a tridiagonal matrix. We discuss the Newton’s explicit group iterative methods for a general Burger’s equation. The propo...

In this paper we study the numerical solutions of a class of complex PDE systems with free boundary conditions. This problem arises naturally in pricing American options with regimeswitching, which adds significant complexity in the PDE systems due to regime coupling. Developing efficient numerical schemes will have important applications in computational finance. We propose a new method to sol...