In this paper, a high-order and conditionally stable stochastic difference scheme is proposed for the numerical solution of $rm Ithat{o}$ stochastic advection diffusion equation with one dimensional white noise process. We applied a finite difference approximation of fourth-order for discretizing space spatial derivative of this equation. The main properties of deterministic difference schemes,...

Nonstandard finite difference schemes for the Black-Scholes partial differential equation preserving the positivity property are proposed. Computationally simple schemes are derived by using a nonlocal approximation in the reaction term of the Black-Scholes equation. Unlike the standard methods, the solutions of new proposed schemes are positive and free of the spurious oscillations.

A singularly perturbed delay parabolic problem of convection-diffusion type with a discontinuous convection coefficient and source term is examined. In the problem, strong interior layers weak boundary are exhibited due to large in spatial variable discontinuity source. The discretized by nonstandard finite difference scheme for time derivative, we used Crank–Nicolson scheme. To enhance order c...