Related papers: Explicit schemes for parabolic and hyperbolic equa…
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…
In this study, we propose high-order implicit and semi-implicit schemes for solving ordinary differential equations (ODEs) based on Taylor series expansion. These methods are designed to handle stiff and non-stiff components within a…
This paper is devoted to the study of numerical approximation schemes for a class of parabolic equations on (0, 1) perturbed by a non-linear rough signal. It is the continuation of [8, 7], where the existence and uniqueness of a solution…
Stabilized explicit methods are particularly efficient for large systems of stiff stochastic differential equations (SDEs) due to their extended stability domain. However, they loose their efficiency when a severe stiffness is induced by…
The reduction of computational costs in the numerical solution of nonstationary problems is achieved through splitting schemes. In this case, solving a set of less computationally complex problems provides the transition to a new level in…
Parabolic optimal control problems with control constraints are generally challenging, from either theoretical analysis or algorithmic design perspectives. Conceptually, the well-known alternating direction method of multipliers (ADMM) can…
This manuscript contains some thoughts on the discretization of the classical heat equation. Namely, we discuss the advantages and disadvantages of explicit and implicit schemes. Then, we show how to overcome some disadvantages while…
The novel contribution of this paper relies in the proposal of a fully implicit numerical method designed for nonlinear degenerate parabolic equations, in its convergence/stability analysis, and in the study of the related computational…
Hyperbolic partial differential equations (PDEs) cover a wide range of interesting phenomena, from human and hearth-sciences up to astrophysics: this unavoidably requires the treatment of many space and time scales in order to describe at…
We present a parametric family of semi-implicit second order accurate numerical methods for non-conservative and conservative advection equation for which the numerical solutions can be obtained in a fixed number of forward and backward…
In this paper implicit and explicit exact difference schemes (EDS) for system $\textbf{x}' = A\textbf{x}$ of three linear differential equations with constant coefficients are constructed. Numerical simulations for stiff problem and for…
We propose a semi-discrete numerical scheme and establish well-posedness of a class of parabolic systems. Such systems naturally arise while studying the optimal control of grain boundary motions. The latter is typically described using a…
Explicit stabilized integrators are an efficient alternative to implicit or semi-implicit methods to avoid the severe timestep restriction faced by standard explicit integrators applied to stiff diffusion problems. In this paper, we provide…
We present an original study on the numerical stabiliy of explicit schemes solving the incompressible Euler equations on an open domain with slipping boundary conditions. Relying on the skewness property of the non-linear term, we…
We explore the performance and advantages/disadvantages of using unconditionally stable explicit super time-stepping (STS) algorithms versus implicit schemes with Krylov solvers for integrating parabolic operators in thermodynamic MHD…
A novel Douglas alternating direction implicit (ADI) method is proposed in this work to solve a two-dimensional (2D) heat equation with interfaces. The ADI scheme is a powerful finite difference method for solving parabolic equations, due…
Implicit time integration schemes are widely used in computational fluid dynamics numerical codes to speed-up computations. Indeed, implicit schemes usually allow for less stringent time-step stability constraints than their explicit…
We prove first-order convergence of the semi-explicit Euler scheme combined with a finite element discretization in space for elliptic-parabolic problems which are weakly coupled. This setting includes poroelasticity, thermoelasticity, as…
On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other…
In this article, we have developed a higher order compact numerical method for variable coefficient parabolic problems with mixed derivatives. The finite difference scheme, presented here for two-dimensional domains, is based on fourth…