Steady and Stable: Numerical Investigations of Nonlinear Partial Differential Equations

Excerpt: Mathematics is a language which can describe patterns in everyday life as well as abstract concepts existing only in our minds. Patterns exist in data, functions, and sets constructed around a common theme, but the most tangible patterns are visual. Visual demonstrations can help undergraduate students connect to abstract concepts in advanced mathematical courses. The study of partial differential equations, in particular, benefits from numerical analysis and simulation

New analytic approach to address Put - Call parity violation due to discrete dividends

The issue of developing simple Black-Scholes type approximations for pricing European options with large discrete dividends was popular since early 2000's with a few different approaches reported during the last 10 years. Moreover, it has been claimed that at least some of the resulting expressions represent high-quality approximations which closely match results obtained by the use of numerics. In this paper we review, on the one hand, these previously suggested Black-Scholes type approximations and, on the other hand, different versions of the corresponding Crank-Nicolson numerical schemes with a primary focus on their boundary condition variations. Unexpectedly we often observe substantial deviations between the analytical and numerical results which may be especially pronounced for European Puts. Moreover, our analysis demonstrates that any Black-Scholes type approximation which adjusts Put parameters identically to Call parameters has an inherent problem of failing to detect a little known Put-Call Parity violation phenomenon. To address this issue we derive a new analytic approximation which is in a better agreement with the corresponding numerical results in comparison with any of the previously known analytic approaches for European Calls and Puts with large discrete dividends

Fourth order real space solver for the time-dependent Schr\"odinger equation with singular Coulomb potential

We present a novel numerical method and algorithm for the solution of the 3D axially symmetric time-dependent Schr\"odinger equation in cylindrical coordinates, involving singular Coulomb potential terms besides a smooth time-dependent potential. We use fourth order finite difference real space discretization, with special formulae for the arising Neumann and Robin boundary conditions along the symmetry axis. Our propagation algorithm is based on merging the method of the split-operator approximation of the exponential operator with the implicit equations of second order cylindrical 2D Crank-Nicolson scheme. We call this method hybrid splitting scheme because it inherits both the speed of the split step finite difference schemes and the robustness of the full Crank-Nicolson scheme. Based on a thorough error analysis, we verified both the fourth order accuracy of the spatial discretization in the optimal spatial step size range, and the fourth order scaling with the time step in the case of proper high order expressions of the split-operator. We demonstrate the performance and high accuracy of our hybrid splitting scheme by simulating optical tunneling from a hydrogen atom due to a few-cycle laser pulse with linear polarization

A machine learning framework for data driven acceleration of computations of differential equations

We propose a machine learning framework to accelerate numerical computations of time-dependent ODEs and PDEs. Our method is based on recasting (generalizations of) existing numerical methods as artificial neural networks, with a set of trainable parameters. These parameters are determined in an offline training process by (approximately) minimizing suitable (possibly non-convex) loss functions by (stochastic) gradient descent methods. The proposed algorithm is designed to be always consistent with the underlying differential equation. Numerical experiments involving both linear and non-linear ODE and PDE model problems demonstrate a significant gain in computational efficiency over standard numerical methods