A Finite Difference Approximation for Dynamic Calculation of Vertical Free Hanging Slender Risers in Re-Entry Application

The dynamic calculations of slender marine risers, such as Finite Element Method （FEM） or Modal Expansion Solution Method （MESM）, are mainly for the slender structures with their both ends hinged to the surface and bottom. However, for the re-entry operation, risers held by vessels are in vertical free hanging state, so the displacement and velocity of lower joint would not be zero. For the model of free hanging flexible marine risers, the paper proposed a Finite Difference Approximation （FDA） method for its dynamic calculation. The riser is divided into a reasonable number of rigid discrete segments. And the dynamic model is established based on simple Euler-Bemoulli Beam Theory concerning tension, shear forces and bending moments at each node along the cylindrical structures, which is extendible for different boundary conditions. The governing equations with specific boundary conditions for riser＇s free hanging state are simplified by Keller-box method and solved with Newton iteration algorithm for a stable dynamic solution. The calculation starts when the riser is vertical and still in calm water, and its behavior is obtained along time responding to the lateral forward motion at the top. The dynamic behavior in response to the lateral parametric excitation at the top is also proposed and discussed in this paper.

FINITE DIFFERENCE APPROXIMATION FOR PRICING THE AMERICAN LOOKBACK OPTION

In this paper we are concerned with the pricing of lookback options with American type constrains. Based on the differential linear complementary formula associated with the pricing problem, an implicit difference scheme is constructed and analyzed. We show that there exists a unique difference solution which is unconditionally stable. Using the notion of viscosity solutions, we also prove that the finite difference solution converges uniformly to the viscosity solution of the continuous problem. Furthermore, by means of the variational inequality analysis method, the O（△t ＋ △x^2）-order error estimate is derived in the discrete L2-norm provided that the continuous problem is sufficiently regular. In addition, a numerical example is provided to illustrate the theoretical results.

IMPLICIT DIFFERENCE APPROXIMATION FOR A TIME FRACTIONAL ADVECTION-DISPERSION EQUATION

In this paper, a time fractional advection-dispersion equation is considered. From the relationship between the Caputo derivative and the Griinwald derivative, the Caputo derivative is approximated by using the Griinwald derivative. An implicit difference approximation for this equation is proposed. We prove that this approximation is unconditionally stable and convergent. Finally, numerical examples are given.

Analysis of an Implicit Finite Difference Scheme for Time Fractional Diffusion Equation

Time fractional diffusion equation is usually used to describe the problems involving non-Markovian random walks. This kind of equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α∈(0, 1). In this paper, an implicit finite difference scheme for solving the time fractional diffusion equation with source term is presented and analyzed, where the fractional derivative is described in the Caputo sense. Stability and convergence of this scheme are rigorously established by a Fourier analysis. And using numerical experiments illustrates the accuracy and effectiveness of the scheme mentioned in this paper.

A Study on the Finite Difference Approach of the Surface Laplacian

Surface Laplacian map provides a better spatial resolution than surface potential distribution. Different order finite difference approximations are deduced and compared by simulations on a plane in this paper. The results show high order approximation is better than low order approximation for noiseless situation. However, low order approximation is better for noise suppression. Results also show Laplacian is more sensitive to shallow neural activities and the temporal course of neural activities can be correctly reconstructed by a finite difference Laplacian.

High accuracy compact finite difference methods and their applications

Numerical simulation of complex flow fields with multi-scale structures is one of the most important and challenging branches of computational fluid dynamics. From linear analysis and numerical experiments it has been discovered that the higher-order accurate method can give reliable and efficient computational results, as well as better resolution of the complex flow fields with multi-scale structures. Compact finite difference schemes, which feature higher-order accuracy and spectral-like resolution with smaller stencils and easier application of boundary conditions, has attracted more and more interest and attention.

NUMERICAL SOLUTION OF THE SPACE FRACTIONAL DIFFERENTIAL EQUATION

In this paper, a space fractional differential equation is considered. The equation is obtained from the parabolic equation containing advection, diffusion and reaction terms by replacing the second order derivative in space by a fractional derivative in space of order. An implicit finite difference approximation for this equation is presented. The stability and convergence of the finite difference approximation are proved. A fractional-order method of lines is also presented. Finally, some numerical results are given.

Explicit High Order One-Step Methods for Decoupled Forward Backward Stochastic Differential Equations

By using the Feynman-Kac formula and combining with Itˆo-Taylor expansion and finite difference approximation,we first develop an explicit third order onestep method for solving decoupled forward backward stochastic differential equations.Then based on the third order one,an explicit fourth order method is further proposed.Several numerical tests are also presented to illustrate the stability and high order accuracy of the proposed methods.

On the Navier-Stokes Equations for Exothermically Reacting Compressible Fluids

We analyze mathematical models governing planar flow of chemical reaction from unburnt gases to burnt gases in certain physical regimes in which diffusive effects such as viscosity and heat conduction are significant. These models can be then formulated as the Navier-Stokes equations for exothermically reacting compressible fluids. We first establish the existence and dynamic behavior, including stability, regularity, and large-time behavior, of global discontinuous solutions of large oscillation to the Navier-Stokes equations with constant adiabatic exponent γ and specific heat Cv. Our approach for the existence and regularity is to combine the difference approximation techniques with the energy methods, total variation estimates, and weak convergence arguments to deal with large jump discontinuities; and for large-time behavior is an a posteriori argument directly from the weak form of the equations. The approach and ideas we develop here can be applied to solving a more complicated model where γand cv vary as the phase changes; and we then describe this model in detail and contrast the results on the asymptotic behavior of the solutions of these two different models. We also discuss other physical models describing dynamic combustion.