We interpret the Cayley transform of linear (finiteor infinite-dimensional) 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 (disc...

Several nonlinear Black-Scholes models have been proposed to take transaction cost, large investor performance and illiquid markets into account. One of the most comprehensive models introduced by Barles and Soner in [4] considers transaction cost in the hedging strategy and risk from an illiquid market. In this paper, we compare several finite difference methods for the solution of this model ...

0021-9991/$ see front matter 2008 Elsevier Inc doi:10.1016/j.jcp.2008.06.032 * Corresponding author. Tel.: +1 609 243 2635; fa E-mail address: [email protected] (S.C. Jardin). In solving the 1D (flux surface averaged) transport equations for the temperatures, magnetic fields, and densities in the ‘‘evolving equilibrium” description of a tokamak [1], one increasingly encounters highly nonlinear th...

In this paper, a numerical scheme named alternating segment Crank-Nikolson is used for solving heat equation. This scheme can be used directly on parallel computations. Truncation error and stability of the presented method is analyzed. Comparison in accuracy with the fully implicit Crank-Nikolson scheme is presented in numerical experiment.

Results from applying a Crank-Nicholson parabolic equation method (CN-PE) are presented in situations with a thin screen on a hard ground in a turbulent atmosphere, and with the acoustic source at ground level. The results are evaluated by comparison with G. A. Daigle’s model, which uses the sound scattering cross-section by V. I. Tatarskii together with diffraction theory. The results show a f...

The numerical solution of a Rayleigh-Taylor instability problem where an inviscid liquid of finite depth is accelerated into a gas of semi-infinite extent is obtained by transforming the irregular flow domain into a rectangular domain by a coordinate transformation. The free surface equation is solved by a Crank-Nicolson procedure. The boundary condition at the free surface for the velocity pot...

In this paper we show that standard preconditioners for parabolic PDEs discretized by implicit Euler or Crank–Nicolson schemes can be reused for higher–order fully implicit Runge–Kutta time discretization schemes. We prove that the suggested block diagonal preconditioners are order–optimal for A–stable and irreducible Runge–Kutta schemes with invertible coefficient matrices. The theoretical inv...

In this work, after reviewing two different ways to solve Riccati systems, we are able to present an extensive list of families of integrable nonlinear Schrödinger (NLS) equations with variable coefficients. Using Riccati equations and similarity transformations, we are able to reduce them to the standard NLS models. Consequently, we can construct bright-, darkand Peregrine-type soliton solutio...

In this work we study Crank-Nicolson Leap-Frog (CNLF) methods with fast slow wave splittings for Navier-Stokes equation plus a Coriolis force term, which is a simplification of geophysical flows. We present a new stabilized CNLF method where the added stabilization completely removes the method’s CFL time step condition. We give a comprehensive stability and error analysis. We prove that for Os...

The preconditioned Crank-Nicolson (pCN) method is a MCMC algorithm for implementing the Bayesian inferences in function spaces. A remarkable feature of the algorithm is that, unlike many usual MCMC algorithms, which become arbitrary slow under the mesh refinement, the efficiency of the algorithm is dimension independent. In this work we develop an adaptive version of the pCN algorithm, where th...