We consider a time-dependent and a steady linear convection-diffusion equation. These equations are approximately solved by a combined finite element – finite volume method: the diffusion term is discretized by Crouzeix-Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the unsteady case, the implicit Euler method is u...

This paper deals with the numerical approximation of the solution of 1D parabolic singularly perturbed problems of reaction–diffusion type. The numerical method combines the standard implicit Euler method on a uniform mesh to discretize in time and a HODIE compact fourth order finite difference scheme to discretize in space, which is defined on a priori special meshes condensing the grid points...

We describe the development of a flux-limited gray radiation solver for the compressible astrophysics code, CASTRO. CASTRO uses an Eulerian grid with block-structured adaptive mesh refinement based on a nested hierarchy of logically-rectangular variable-sized grids with simultaneous refinement in both space and time. The gray radiation solver is based on a mixed-frame formulation of radiation h...

In this paper, we study finite volume element (FVE) method for convection–diffusion–reaction equations in a two-dimensional convex polygonal domain. These types of equations arise in the modeling of a waste scenario of a radioactive contaminant transport and reaction in flowing groundwater. Both spatially discrete scheme and discrete-in-time scheme are analyzed in this paper. For the spatially ...

In this talk we introduce a family of numerical approximations for the stochastic differentialequations (SDEs) with, possibly, no-globally Lipschitz coefficients. We show that for a given Lyapunovfunction V : R → [1,∞) we can construct a suitably tamed Euler scheme that preserves so calledV-stability property of the original SDEs without imposing any restrictions on the time dis...

We consider differences between log Γ(x) and truncations of certain classical asymptotic expansions in inverse powers of x − λ whose coefficients are expressed in terms of Bernoulli polynomials Bn(λ), and we obtain conditions under which these differences are strictly completely monotonic. In the symmetric cases λ = 0 and λ = 1/2, we recover results of Sonin, Nörlund and Alzer. Also we show how...

The backward Euler method is employed to approximate the invariant measure of a class stochastic differential equations (SDEs) driven by ‐stable processes. existence and uniqueness numerical are proved. Then shown converge underlying measure. Numerical examples provided demonstrate theoretical results.

Conditions on the stabilization parameters are explored for different approaches in deriving error estimates for the SUPG finite element stabilization of time-dependent convectiondiffusion-reaction equations. Exemplarily, it is shown for the SUPG method combined with the backward Euler scheme that standard energy arguments lead to estimates for stabilization parameters that depend on the length...

A rather general semilinear parabolic problem is studied together with its spatially semidiscrete nite element approximation. Both problems are formulated within the framework of nonlinear semigroups in the Sobolev space H 1 ((). The main result is an error estimate for solutions with initial data in H 1 ((), valid during an arbitrary nite time interval. The proof is based on the semigroup form...

This paper is concerned with the time discretization of nonlinear evolution equations. We work in an abstract Banach space setting of analytic semigroups that covers fully nonlinear parabolic initial-boundary value problems with smooth coefficients. We prove convergence of variable stepsize backward Euler discretizations under various smoothness assumptions on the exact solution. We further sho...