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 ...

We study a generalized Crank–Nicolson scheme for the time discretization of a fractional wave equation, in combination with a space discretization by linear finite elements. The schemeuses a non-uniformgrid in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show th...

We analyze finite difference methods for the Gross-Pitaevskii equation with an angular momentum rotation term in two and three dimensions and obtain the optimal convergence rate, for the conservative Crank-Nicolson finite difference (CNFD) method and semi-implicit finite difference (SIFD) method, at the order of O(h2 + τ2) in the l2-norm and discrete H1-norm with time step τ and mesh size h. Be...

Black-Scholes model plays a very significant role in the world of quantitative finance. In this paper, focus are on both nonlinear and linear (BS) equations with numerical approximations. We aim to find an effective approximations for model. Several models from most relevant class European option analyzed study. The problem is approached by transforming into convection-diffusion equation later ...

Thermal interaction of fluids and solids, or conjugate heat transfer (CHT), is encountered in many engineering applications. Since time-accurate computations of such coupled problems can be computationally expensive, we consider loosely-coupled and stronglycoupled solution algorithms in which higher order multi-stage Runge-Kutta schemes are employed for time integration. The higher order time i...

This paper provides an error analysis for the Crank–Nicolson extrapolation scheme of time discretization applied to the spatially discrete stabilized finite element approximation of the two-dimensional time-dependent Navier–Stokes problem, where the finite element space pair (Xh,Mh) for the approximation (uh, p n h) of the velocity u and the pressure p is constructed by the low-order finite ele...

The study of convective heat and mass transfer flow over a vertical plate with nth order chemical reaction in a porous medium is analyzed both analytically and numerically. The resulting governing boundary layer equations are highly non-linear and coupled form of partial differential equations and have been solved by using implicit finite difference method of Crank-Nicolson. To check the accura...

In this work we study a preconditioned iterative method for some higher–order time discretizations of linear parabolic partial differential equations. We use the Padé approximations of the exponential function to discretize in time and show that standard solution algorithms for lower–order time discretization schemes, such as Crank–Nicolson and implicit Euler, can be reused as preconditioners f...

A semilinear second-order parabolic equation is considered in a regular and a singularly perturbed regime. For this equation, we give computable a posteriori error estimates in the maximum norm. Semidiscrete and fully discrete versions of the backward Euler, Crank–Nicolson, and discontinuous Galerkin dG(r) methods are addressed. For their full discretizations, we employ elliptic reconstructions...