Zhongdi Cen

Parameter-uniform finite difference scheme for a system of coupled singularly perturbed convection–diffusion equations

In this paper, we discuss the parameter-uniform finite difference method for a coupled system of singularly perturbed convection–diffusion equations. The leading term of each equation is multiplied by a small but different magnitude positive parameter, which leads to the overlap and interact boundary layer. We analyze the boundary layer and construct a piecewise-uniform mesh on the variant of the Shishkin mesh. We prove that our schemes converge almost first-order uniformly with respect to small parameters. We present some numerical experiments to support our theoretical analysis.

A second-order finite difference scheme for a class of singularly perturbed delay differential equations

In this paper a class of delay differential equations with a perturbation parameter ϵ is examined. A hybrid finite difference scheme on an appropriate piecewise uniform mesh of Shishkin-type is derived. We show that the scheme is almost second-order convergent, in the discrete maximum norm, independent of singular perturbation parameter. Numerical experiments support these theoretical results.

Finite difference scheme with a moving mesh for pricing Asian options

In this paper we propose a stable numerical method for pricing Asian call options, which is based on a central difference scheme with a moving mesh in the spatial discretization and the Rannacher time stepping scheme in the time discretization. At each time mesh point we make a piecewise uniform mesh to discretise the space interval, which ensures that the matrix associated with the discrete operator is an M-matrix. Hence the spatial discretization scheme is maximum-norm stable for arbitrary volatility and arbitrary interest rate. We show that the scheme is second-order convergent with respect to both time and spatial variables. Numerical results support the theoretical results.

Closed formulas for computing higher-order derivatives of functions involving exponential functions

For integers k ? 1 and n ? 0, the functions 1 / ( 1 - λ e α t ) k and the derivatives ( 1 / ( 1 - λ e α t ) ) ( n ) can be expressed each other by linear combinations. Based on this viewpoint, we find several new closed formulas for higher-order derivatives of trigonometric and hyperbolic functions, derive a higher-order convolution formula for the tangent numbers, and generalize a recurrence relation for the tangent numbers.

Numerical method for a class of singular non-linear boundary value problems using Green's functions

A numerical method based on Greens function is presented for solving a class of singular non-linear boundary value problems. By applying Greens function an equivalent integral equation, which can be solved by linear interpolation on a non-uniform mesh, can be derived from the singular non-linear boundary value problem. This equation is shown to be second-order convergent. The numerical method can be extended to more general singular boundary value problems. Numerical experiments support these theoretical results and indicate that the estimates are sharp.

A posteriori error analysis for a fractional differential equation

ABSTRACT Numerical treatment for a fractional differential equation (FDE) is proposed and analysed. The solution of the FDE may be singular near certain domain boundaries, which leads to numerical difficulty. We apply the upwind finite difference method to the FDE. The stability properties and a posteriori error analysis for the discrete scheme are given. Then, a posteriori adapted mesh based on a posteriori error analysis is established by equidistributing arc-length monitor function. Numerical experiments illustrate that the upwind finite difference method on a posteriori adapted mesh is more accurate than the method on uniform mesh.

A hybrid finite difference scheme for pricing Asian options

In this paper we apply a hybrid finite difference scheme to evaluate the prices of Asian call options with fixed strike price. We use the Crank-Nicolson method to discretize the time variable and a hybrid finite difference scheme to discretize the spatial variable. The hybrid difference scheme uses the central difference approximation whenever the mesh points are sufficiently away from the left-hand side of the domain to ensure the stability of the scheme; otherwise a midpoint upwind difference scheme is used. The matrix associated with the discrete operator is an M-matrix, which ensures that the spatial discretization scheme is maximum-norm stable. It is proved that the scheme is second-order convergent with respect to both time and spatial variables. Numerical experiments support these theoretical results.

A robust numerical method for a fractional differential equation

Abstract This paper is devoted to giving a rigorous numerical analysis for a fractional differential equation with order α  ∈ (0, 1). First the fractional differential equation is transformed into an equivalent Volterra integral equation of the second kind with a weakly singular kernel. Based on the a priori information about the exact solution, an integral discretization scheme on an a priori chosen adapted mesh is proposed. By applying the truncation error estimate techniques and a discrete analogue of Gronwall’s inequality, it is proved that the numerical method is first-order convergent in the discrete maximum norm. Numerical results indicate that this method is more accurate and robust than finite difference methods when α is close to 0.

A robust upwind difference scheme for pricing perpetual American put options under stochastic volatility

In this paper, we present an upwind difference scheme for the valuation of perpetual American put options, using Hestons stochastic volatility model. The matrix associated with the discrete operator is an M-matrix, which ensure that the scheme is stable. We apply the maximum principle to the discrete linear complementarity problem in two mesh sets and derive the error estimates. Numerical results support the theoretical results.

Uniformly convergent second-order difference scheme for a singularly perturbed periodical boundary value problem

A periodic boundary value problem with a small parameter multiplying the first- and second-order derivatives is considered. The problem is discretized using a hybrid difference scheme on a Shishkin mesh. We show that the scheme is almost second-order convergent in the maximum norm, which is independent of a singular perturbation parameter. Numerical experiment supports these theoretical results.