A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the disc...

We derive high-order compact finite difference schemes for option pricing in stochastic volatility models on non-uniform grids. The schemes are fourth-order accurate in space and secondorder accurate in time for vanishing correlation. In our numerical study we obtain highorder numerical convergence also for non-zero correlation and non-smooth payoffs which are typical in option pricing. In all ...

This work overcomes the difficulty of dealing with large curvatures in a high order matched interface and boundary (MIB) method proposed for solving elliptic interface problems. The MIB method smoothly extends the solution across the interface so that standard high order central finite difference schemes can be used without the loss of accuracy. One feature of the MIB is that it disassociates t...

We analyze the discretization of an initial-boundary value problem for the cubic Schrödinger equation in one space dimension by a Crank–Nicolson–type finite difference scheme. We then linearize the corresponding equations at each time level by Newton’s method and discuss an iterative modification of the linearized scheme which requires solving linear systems with the same tridiagonal matrix. We...

This paper illustrates the use of the differentiation matrix technique for solving differential equations in finance. The technique provides a compact and unified formulation for a variety of discretisation and time-stepping algorithms for solving problems in one and two dimensions. Using differentiation matrix models, we compare time-stepping algorithms for option pricing computations and pres...

We consider a model initial and boundary value problem for the wide-angle ‘parabolic’ equation Lur = icu of underwater acoustics, where L is a second-order differential operator in the depth variable z with depthand range-dependent coefficients. We discretize the problem by the Crank–Nicolson finite difference scheme and also by the forward Euler method using nonuniform partitions both in depth...

This paper introduces a novel high order interface scheme, the matched interface and boundary (MIB) method, for solving elliptic equations with discontinuous coefficients and singular sources on Cartesian grids. By appropriate use of auxiliary line and/or fictitious points, physical jump conditions are enforced at the interface. Unlike other existing interface schemes, the proposed method disas...

We investigate an explicit finite difference scheme for a BeelerReuter based model of cardiac electrical activity. As our main result, we prove that the finite difference solutions are bounded in the L∞-norm. We also prove the existence of a weak solution by showing convergence to the solutions of the underlying model as the discretization parameters tend to zero. The convergence proof is based...

Navier-Stokes equation. The expression is the nonlinear part of the equation and features in the approximation of the flow’s Reynold’s number (see Remarks 5.2(3), on page 721 of [2]). In other words, we discretize the linearized , non-homogeneous Navier-Stokes problem, representing the 3-D slow flow of a fluid. A typical example of a slow Navier-Stokes fluid flow is ground water through an aqui...

A family of inexpensive discretization schemes for diffusion problems on unstructured polygonal and polyhedral meshes is introduced. The material properties are described by a full tensor. The theoretical results are confirmed with numerical experiments.