Ashvin Gopaul

Analysis of a Fourth-Order Scheme for a Three-Dimensional Convection-Diffusion Model Problem

We derive closed form expressions for the eigenvalues and discrete solution arising from a 19-point compact discretization of a three-dimensional convection-diffusion problem. It is shown that the coefficient matrix is positive definite when the cell-Reynolds number is greater than some critical value. By analyzing the terms composing the discrete solution, we prove that an oscillation-free discrete solution is guaranteed whenever the cell-Reynolds number exceeds a value which is grid-size dependent. An interesting result is that as the mesh size is refined, this value approaches the Golden Mean.

A weighted ENO-flux limiter scheme for hyperbolic conservation laws

We present a new scheme that combines essentially non-oscillatory (ENO) reconstructions together with monotone upwind schemes for scalar conservation laws interpolants. We modify a second-order ENO polynomial by choosing an additional point inside the stencil in order to obtain the highest accuracy when combined with the Harten–Osher reconstruction-evolution method limiter. Numerical experiments are done in order to compare a weighted version of the hybrid scheme to weighted essentially non-oscillatory (WENO) schemes with constant Courant–Friedrichs–Lewy number under relaxed step size restrictions. Our results show that the new scheme reduces smearing near shocks and corners, and in some cases it is more accurate near discontinuities compared with higher-order WENO schemes. The hybrid scheme avoids spurious oscillations while using a simple componentwise extension for solving hyperbolic systems. The new scheme is less damped than WENO schemes of comparable accuracy and less oscillatory than higher-order WENO schemes. Further experiments are done on multi-dimensional problems to show that our scheme remains non-oscillatory while giving good resolution of discontinuities.

Preconditioned iterative methods for the nine-point approximation to the convection---diffusion equation

Iterative methods preconditioned by incomplete factorizations and sparse approximate inverses are considered for solving linear systems arising from fourth-order finite difference schemes for convection-diffusion problems. Simple recurrences for implementing the ILU(0) factorization of the nine-point scheme are derived. Different sparsity patterns are considered for computing approximate inverses for the coefficient matrix and the quality of the preconditioner is studied in terms of plots of the field of values of the preconditioned matrices. In terms of algebraic properties of the preconditioned matrices, our experimental results show that incomplete factorizations give a preconditioner of better quality than approximate inverses. Comparison of the convergence rates of GMRES applied to the preconditioned linear systems is done with respect to the field of values, Ritz and harmonic Ritz values of the preconditioned matrices. Numerical results show that the GMRES residual norm decreases rapidly when the difference between the Ritz and harmonic Ritz values becomes small. We also describe the results of experiments when some well-known Krylov subspace methods are used to solve the linear system arising from the compact fourth-order discretizations.

A General Efficient Framework for Pricing Options Using Exponential Time Integration Schemes

In numerical option pricing, spatial discretization of the pricing equation leads to semi-discrete systems of the form

On block-circulant preconditioners for high-order compact approximations of convection-diffusion problems

{V}^{\prime}\left( \tau \right)=AV\left( \tau \right)+b\left( \tau \right),

Hermitian and Skew-Hermitian splitting methods for streamline upwind Petrov-Galerkin approximations of a grid-aligned flow problem

(4.1) where A ∊ ℜ m×m is in general a negative semi-definite matrix and b(τ) generally represents boundary condition implementations, a penalty term for American option or approximation of integral terms on an unbounded domain in models with jumps. With advances in the efficient computation of the matrix exponential (Schmelzer and Trefethen 2007), exponential time integration (Cox and Matthews 2002) is likely to be a method of choice for the solution of ODE systems of the form (4.1). Duhamel’s principle states that the exact integration of (4.1) over one time step gives

A fast high-order finite difference algorithm for pricing American options

V\left( {{{\tau }_{{j+1}}}} \right)={{e}^{{A\Delta \tau }}}V\left( {{{\tau }_{j}}} \right)+{{e}^{{A{{\tau }_{{j+1}}}}}}\int\nolimits_{{{{\tau }_{j}}}}^{{{{\tau }_{{j+1}}}}} {{{e}^{{-At}}}b\left( t \right)dt} ,