We derive novel, fast, and effective preconditioned iterative methods for steady time-dependent Navier--Stokes control problems. Our approach may be applied to problems which are discretized using backward Euler or Crank--Nicolson is also a valuable candidate Stokes Crank--Nicolson. The key ingredients of the solver saddle-point type approximation linear systems, an inner iteration $(1,1)$-bloc...

We develop space-time adaptive and high-order methods for valuing American options using a partial differential equation (PDE) approach. The linear complementarity problem arising due to the free boundary is handled by a penalty method. Both finite difference and finite element methods are considered for the space discretization of the PDE, while classical finite differences, such as Crank-Nico...

The aim of this paper is to extend the split-step idea for the solution of fractional partial differential equations. We consider the multidimensional nonlinear Schr"{o}dinger equation with the Riesz space fractional derivative and propose an efficient numerical algorithm to obtain it's approximate solutions. To this end, we first discretize the Riesz fractional derivative then apply the Crank-...

In this paper, we give a fourth-order compact finite difference scheme for the general forms of two point boundary value problems and two dimensional elliptic partial differential equations (PDE’s). By decomposing the coefficient matrix into a sum of several matrixes after we discretize the original problem, we can obtain a lower bound for the smallest eigenvalue of the coefficient matrix. Thus...

We consider an initial boundary value problem for a one-dimensional fractional-order parabolic equation with a space fractional derivative of Riemann–Liouville type and order α ∈ (1, 2). We study a spatial semidiscrete scheme using the standard Galerkin finite element method with piecewise linear finite elements, as well as fully discrete schemes based on the backward Euler method and the Crank...

A linearized numerical scheme is proposed to solve the nonlinear time-fractional parabolic problems with time delay. The based on standard Galerkin finite element method in spatial direction, fractional Crank-Nicolson method, and extrapolation methods temporal direction. novel discrete Grönwall inequality established. Thanks inequality, error estimate of a fully obtained. Several examples are p...

We analyze the spatially semidiscrete piecewise linear finite element method for a nonlocal parabolic equation resulting from thermistor problem. Our approach is based on the properties of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite element method. We assume minimal regularity of the exact solution that yields optimal order erro...

We interpret the Cayley transform of linear ( niteor in nitedimensional) state space systems as a numerical integration scheme of Crank Nicolson type. The scheme is known as Tustin's method in the engineering literature, and it has the following important Hamiltonian integrator property: if Tustin's method is applied to a conservative (continuous time) linear system, then the resulting (discret...

A finite difference method which is second-order accurate in time and in space is proposed for two-dimensional fractional percolation equations. Using the Fourier transform, a general approximation for the mixed fractional derivatives is analyzed. An approach based on the classical Crank-Nicolson scheme combined with the Richardson extrapolation is used to obtain temporally and spatially second...