A rational spectral collocation method with adaptively transformed Chebyshev grid points

A spectral collocation method based on rational interpolants and adaptive grid points is presented. The rational interpolants approximate analytic functions with exponential accuracy by using prescribed barycentric weights and transformed Chebyshev points. The locations of the grid points are adapted to singularities of the underlying solution, and the locations of these singularities are approximated by the locations of poles of Chebyshev-Padé approximants. Numerical experiments on two time-dependent problems, one with finite time blow-up and one with a moving front, indicate that the method far outperforms the standard Chebyshev spectral collocation method for problems whose solutions have singularities in the complex plan close to [-1,1]

Efficient Uncertainty Quantification with the Polynomial Chaos Method for Stiff Systems

The polynomial chaos method has been widely adopted as a computationally feasible approach for uncertainty quantification. Most studies to date have focused on non-stiff systems. When stiff systems are considered, implicit numerical integration requires the solution of a nonlinear system of equations at every time step. Using the Galerkin approach, the size of the system state increases from $n$ to $S \times n$, where $S$ is the number of the polynomial chaos basis functions. Solving such systems with full linear algebra causes the computational cost to increase from $O(n^3)$ to $O(S^3n^3)$. The $S^3$-fold increase can make the computational cost prohibitive. This paper explores computationally efficient uncertainty quantification techniques for stiff systems using the Galerkin, collocation and collocation least-squares formulations of polynomial chaos. In the Galerkin approach, we propose a modification in the implicit time stepping process using an approximation of the Jacobian matrix to reduce the computational cost. The numerical results show a run time reduction with a small impact on accuracy. In the stochastic collocation formulation, we propose a least-squares approach based on collocation at a low-discrepancy set of points. Numerical experiments illustrate that the collocation least-squares approach for uncertainty quantification has similar accuracy with the Galerkin approach, is more efficient, and does not require any modifications of the original code

Convergence Analysis of A- Stable Implicit Runge- Kutta Methods of Reformulated Block Hybrid Multistep Methods

A number of techniques and solvers have been suggested, developed, and described, but a clear definition of stiffness has not been provided. In this paper, an analysis of the A-stable implicit Runge-Kutta methods is undertaken using stability function and the region of absolute stability, which shows region of each of the methods. Also to compare various stiff solvers based on their region of absolute stability. It helps a stiff ordinary differential equation solver, to identify appropriate numerical methods with unbounded region of absolute, appropriate for stiff problems solving

Numerical Methods -- Lecture Notes 2014-2015

In these notes some basic numerical methods will be described. The following topics are addressed: 1. Nonlinear Equations, 2. Linear Systems, 3. Polynomial Interpolation and Approximation, 4. Trigonometric Interpolation with DFT and FFT, 5. Numerical Integration, 6. Initial Value Problems for ODEs, 7. Stiff Initial Value Problems, 8. Two-Point Boundary Value Problems

IMEX evolution of scalar fields on curved backgrounds

Inspiral of binary black holes occurs over a time-scale of many orbits, far longer than the dynamical time-scale of the individual black holes. Explicit evolutions of a binary system therefore require excessively many time steps to capture interesting dynamics. We present a strategy to overcome the Courant-Friedrichs-Lewy condition in such evolutions, one relying on modern implicit-explicit ODE solvers and multidomain spectral methods for elliptic equations. Our analysis considers the model problem of a forced scalar field propagating on a generic curved background. Nevertheless, we encounter and address a number of issues pertinent to the binary black hole problem in full general relativity. Specializing to the Schwarzschild geometry in Kerr-Schild coordinates, we document the results of several numerical experiments testing our strategy.Comment: 28 pages, uses revtex4. Revised in response to referee's report. One numerical experiment added which incorporates perturbed initial data and adaptive time-steppin

Robust Spectral Methods for Solving Option Pricing Problems

Doctor Scientiae - DScRobust Spectral Methods for Solving Option Pricing Problems by Edson Pindza PhD thesis, Department of Mathematics and Applied Mathematics, Faculty of Natural Sciences, University of the Western Cape Ever since the invention of the classical Black-Scholes formula to price the financial derivatives, a number of mathematical models have been proposed by numerous researchers in this direction. Many of these models are in general very complex, thus closed form analytical solutions are rarely obtainable. In view of this, we present a class of efficient spectral methods to numerically solve several mathematical models of pricing options. We begin with solving European options. Then we move to solve their American counterparts which involve a free boundary and therefore normally difficult to price by other conventional numerical methods. We obtain very promising results for the above two types of options and therefore we extend this approach to solve some more difficult problems for pricing options, viz., jump-diffusion models and local volatility models. The numerical methods involve solving partial differential equations, partial integro-differential equations and associated complementary problems which are used to model the financial derivatives. In order to retain their exponential accuracy, we discuss the necessary modification of the spectral methods. Finally, we present several comparative numerical results showing the superiority of our spectral methods

Differential-Algebraic Equations

Differential-Algebraic Equations (DAE) are today an independent field of research, which is gaining in importance and becoming of increasing interest for applications and mathematics itself. This workshop has drawn the balance after about 25 years investigations of DAEs and the research aims of the future were intensively discussed