We consider fully discrete schemes for the one dimensional linear Schrödinger equation and analyze whether the classical dispersive properties of the continuous model are presented in these approximations. In particular Strichartz estimates and the local smoothing of the numerical solutions are analyzed. Using a backward Euler approximation of the linear semigroup we introduce a convergent sche...

A second-order singularly perturbed parabolic equation in one space dimension is considered. For this equation, we give computable a posteriori error estimates in the maximum norm for two semidiscretisations in time and a full discretisation using P1 FEM in space. Both the Backward-Euler method and the Crank-Nicolson method are considered, and certain critical details of the implementation are ...

This paper discusses the relevant theoretical problem of the numerical derivative estimation of noisy signals. In this paper, a comparative study of some different schemes of the differentiators is given: Kalman filter, the well-known Super Twisting algorithm, Super Twisting with dynamic gains and Euler backward difference method. The analysis of the study results can focus on the strengths and...

This paper deals with the numerical analysis of nonlinear Black-Scholes equation with transaction costs. An unconditionally stable and monotone splitting method, ensuring positive numerical solution and avoiding unstable oscillations, is proposed. This numerical method is based on the LOD-Backward Euler method which allows us to solve the discrete equation explicitly. The numerical results for ...

Abstract — A semilinear second-order singularly perturbed parabolic equation in one space dimension is considered. For this equation, we give computable a posteriori error estimates in the maximum norm for a difference scheme that uses Backward-Euler in time and central differencing in space. Sharp L1-norm bounds for the Green’s function of the parabolic operator and its derivatives are derived...

Differential equations of the type ∂ttu − ∇ · a(∇∂tu) − ∆u = f are studied, where the nonlinear mapping a is continuous, monotone and satisfies a coercivity condition in terms of a (generalised) N -function. The class of problems thus includes the case of anisotropic and nonpolynomial growth. Global existence of solutions in the sense of distributions is shown via convergence of the backward Eu...

In this paper, we study the qualitative behaviour of approximation schemes for Backward Stochastic Differential Equations (BSDEs) by introducing a new notion of numerical stability. For the Euler scheme, we provide sufficient conditions in the one-dimensional and multidimensional case to guarantee the numerical stability. We then perform a classical Von Neumann stability analysis in the case of...

Temperature decay in an aluminium plate is observed using Galerkin finite element method for 2D transient heat conduction equation. Taking ∆t of 0.001 hr , temperature variation is studied using unconditionally stable first order and second order accurate schemes backward Euler and modified CrankNicholson respectively. A comparative study has been made taking different combinations of meshes an...

Abstract. We derive optimal order, residual-based a posteriori error estimates for time discretizations by the two–step BDF method for linear parabolic equations. Appropriate reconstructions of the approximate solution play a key role in the analysis. To utilize the BDF method we employ one step by both the trapezoidal method or the backward Euler scheme. Our a posteriori error estimates are of...

It is shown that for a parabolic problem with maximal Lp-regularity (for 1 < p < ∞), the time discretization by a linear multistep method or Runge–Kutta method has maximal `p-regularity uniformly in the stepsize if the method is A-stable (and satisfies minor additional conditions). In particular, the implicit Euler method, the Crank–Nicolson method, the second-order backward difference formula ...