We derive a posteriori error estimates for the discretization of the heat equation in a unified and fully discrete setting comprising the discontinuous Galerkin, finite volume, mixed finite element, and conforming and nonconforming finite element methods in space and the backward Euler scheme in time. Our estimates are based on a H-conforming reconstruction of the potential, continuous and piec...

In this note we reconsider the continuous time limit of the GARCH(1, 1) process. Let > k and p2 k denote, respectively, the cumulative returns and the volatility processes. We consider the continuous time approximation of the couple (> k , p2 k ). We show that, by choosing di!erent parameterizations, as a function of the discrete interval h, we can obtain either a degenerate or a non-degenerate...

We show that there are ordinary differential equations in R with unbounded solutions, for which the difference equations obtained by using the forward Euler method have all solutions bounded.

We consider the symplectic Euler method for Hamiltonian systems with holonomic constraints and its generalization to a class of overdetermined differential-algebraic equations (ODAEs). It is shown that a natural extension of the symplectic Euler method as a SPARK method is inconsistent for ODAEs which are nonlinear in the algebraic variables. A different non-trivial extension is given and shown...

Abstract. Convergence of a full discretization is shown for a general class of nonlinear parabolic November 10, 2011 problems. The numerical method combines the backward Euler method for the time discretization with a generalized internal approximation scheme for the spatial discretization. The governing monotone elliptic differential operator is described by a nonlinearity that may have anisot...

(2011) Convergence rate of numerical solutions to SFDEs with jumps. Strathprints is designed to allow users to access the research output of the University of Strathclyde. Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. You may not engage in further distribution of the material for any profitmaking activities or any ...

The exponential stability of continuous-time Hopfield neural networks is not preserved when implemented on digital computers by means of explicit numerical methods, whereas the implicit (or backward) Euler method preserves this exponential stability under exactly the same sufficient conditions as those previously obtained for the continuous model. The proof is based on the nonlinear measure app...

In this work we are interested in the numerical approximation of 1D parabolic singularly perturbed problems of reaction–diffusion type. To approximate the multiscale solution of this problem we use a numerical scheme combining the classical backward Euler method and central differencing. The scheme is defined on some special meshes which are the tensor product of a uniform mesh in time and a sp...

For the simulation of one-dimensional ame conngurations reliable numerical tools are needed which have to be both highly eecient (large number of para-metric calculations) and at the same time accurate (in order to avoid numerical errors). This can only be accomplished using fully adaptive discretization techniques both in space and time together with a control of the discretization error. We p...