In this paper a variant of nonlinear exponential Euler scheme is proposed for solving heat conduction problems. The method based on iterations where at each iteration linear initial-value problem has to be solved. We compare the backward combined with iterations. For both methods we show monotonicity and boundedness solutions give sufficient conditions convergence Numerical tests are presented ...

We present a convergence analysis for the implicit-explicit (IMEX) Euler discretization of nonlinear evolution equations. The governing vector field of such an equation is assumed to be the sum of an unbounded dissipative operator and a Lipschitz continuous perturbation. By employing the theory of dissipative operators on Banach spaces, we prove that the IMEX Euler and the implicit Euler scheme...

Steplength thresholds for invariance preserving of three types of discretization methods on a polyhedron are considered. For Taylor approximation type discretization methods we prove that a valid steplength threshold can be obtained by finding the first positive zeros of a finite number of polynomial functions. Further, a simple and efficient algorithm is proposed to numerically compute the ste...

Stochastic collocation method has proved to be an efficient method and been widely applied to solve various partial differential equations with random input data, including NavierStokes equations. However, up to now, rigorous convergence analyses are limited to linear elliptic and parabolic equations; its performance for Navier-Stokes equations was demonstrated mostly by numerical experiments. ...

We construct and analyze a numerical scheme for the two-dimensional Vlasov-Poisson system based on a backward-Euler (BE) approximation in time combined with a mixed finite element method for a discretization of the Poisson equation in the spatial domain and a discontinuous Galerkin (DG) finite element approximation in the phase-space variables for the Vlasov equation. We prove the stability est...

In this paper we analyze a fully practical piecewise linear finite element approximation involving numerical integration, backward Euler time discretization, and possibly regularization and relaxation of the following degenerate parabolic equation arising in a model of reactive solute transport in porous media: find u(x, t) such that ∂tu+ ∂t[φ(u)]−∆u = f in Ω× (0, T ], u = 0 on ∂Ω× (0, T ] u(·,...

In this paper, a new numerical scheme for the time dependent GinzburgLandau (GL) equations under the Lorentz gauge is proposed. We first rewrite the original GL equations into a new mixed formulation, which consists of three parabolic equations for the order parameter ψ, the magnetic field σ= curlA, the electric potential θ=divA and a vector ordinary differential equation for the magnetic poten...

Structure-preserving numerical schemes for a nonlinear parabolic fourthorder equation, modeling the electron transport in quantum semiconductors, with periodic boundary conditions are analyzed. First, a two-step backward differentiation formula (BDF) semi-discretization in time is investigated. The scheme preserves the nonnegativity of the solution, is entropy stable and dissipates a modified e...

This paper is concerned with finding numerical solutions of a flow of ODE solutions. It describes a new method, based on Euler Backward method and interpolation, for finding solutions of autonomous problems. The special aspect is that this method is explicit, and resolves the flow with similar accuracy as Euler Backward method, even when the problem is stiff. An analysis is given of both stabil...

We propose an energy-stable parametric finite element method (ES-PFEM) to discretize the motion of a closed curve under surface diffusion with anisotropic energy $\gamma(\theta)$ -- in two dimensions, while $\theta$ is angle between outward unit normal vector and vertical axis. By introducing positive definite (density) matrix $G(\theta)$, we present new simple variational formulation for prove...