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 ...

We develop an algorithm to solve tridiagonal systems of linear equations, which appear in implicit finite-difference schemes partial differential equations (PDEs), being the time-dependent Schr\"{o}dinger equation (TDSE) ideal candidate benefit from it. Our N-shaped partition method optimizes implementation numerical calculation on parallel architectures, without memory size constraints. Specif...

In this paper, we propose and analyze a fully discrete finite element projection method for the magnetohydrodynamic (MHD) equations. A modified Crank–Nicolson Galerkin are used to discretize model in time space, respectively, appropriate semi-implicit treatments applied fluid convection term two coupling terms. These approximations result linear system with variable coefficients which unique so...

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...

We study a Crank–Nicolson type time discretisation (known as Tustin’s method in engineering literature) for a conservative, infinite-dimensional linear dynamical system whose transfer function is scalar and inner. We show that this discretisation approximates the state trajectory at any given time. We first prove the result for canonical Hankel range realisations, and the general case is then o...

Implicit schemes have been extensively used in building physics to compute the solution of moisture diffusion problems in porous materials for improving stability conditions. Nevertheless, these schemes require important sub-iterations when treating nonlinear problems. To overcome this disadvantage, this paper explores the use of improved explicit schemes, such as Dufort–Frankel, Crank–Nicolson...