23 research outputs found
Highly accurate quadrature-based Scharfetter--Gummel schemes for charge transport in degenerate semiconductors
We introduce a family of two point flux expressions for charge carrier transport described by drift-diffusion problems in degenerate semiconductors with non-Boltzmann statistics which can be used in Vorono"i finite volume discretizations. In the case of Boltzmann statistics, Scharfetter and Gummel derived such fluxes by solving a linear two point boundary value problem yielding a closed form expression for the flux. Instead, a generalization of this approach to the nonlinear case yields a flux value given implicitly as the solution of a nonlinear integral equation. We examine the solution of this integral equation numerically via quadrature rules to approximate the integral as well as Newton's method to solve the resulting approximate integral equation. This approach results into a family of quadrature-based Scharfetter-Gummel flux approximations. We focus on four quadrature rules and compare the resulting schemes with respect to execution time and accuracy. A convergence study reveals that the solution of the approximate integral equation converges exponentially in terms of the number of quadrature points. With very few integration nodes they are already more accurate than a state-of-the-art reference flux, especially in the challenging physical scenario of high nonlinear diffusion. Finally, we show that thermodynamic consistency is practically guaranteed
Highly accurate quadrature-based Scharfetter-Gummel schemes for charge transport in degenerate semiconductors
We introduce a family of two point flux expressions for charge carrier
transport described by drift-diffusion problems in degenerate semiconductors
with non-Boltzmann statistics which can be used in Voronoi finite volume
discretizations. In the case of Boltzmann statistics, Scharfetter and Gummel
derived such fluxes by solving a linear two point boundary value problem
yielding a closed form expression for the flux. Instead, a generalization of
this approach to the nonlinear case yields a flux value given implicitly as
the solution of a nonlinear integral equation. We examine the solution of
this integral equation numerically via quadrature rules to approximate the
integral as well as Newtons method to solve the resulting approximate
integral equation. This approach results into a family of quadrature-based
Scharfetter-Gummel flux approximations. We focus on four quadrature rules and
compare the resulting schemes with respect to execution time and accuracy. A
convergence study reveals that the solution of the approximate integral
equation converges exponentially in terms of the number of quadrature points.
With very few integration nodes they are already more accurate than a
state-of-the-art reference flux, especially in the challenging physical
scenario of high nonlinear diffusion. Finally, we show that thermodynamic
consistency is practically guaranteed
Gauss-Jacobi-type quadrature rules for fractional directional integrals
Fractional directional integrals are the extensions of the Riemann–Liouville fractional integrals from one- to multi-dimensional spaces and play an important role in extending the fractional differentiation to diverse applications. In numerical evaluation of these integrals, the weakly singular kernels often fail the conventional quadrature rules such as Newton–Cotes and Gauss–Legendre rules. It is noted that these kernels after simple transforms can be taken as the Jacobi weight functions which are related to the weight factors of Gauss–Jacobi and Gauss–Jacobi–Lobatto rules. These rules can evaluate the fractional integrals at high accuracy. Comparisons with the three typical adaptive quadrature rules are presented to illustrate the efficacy of the Gauss–Jacobi-type rules in handling weakly singular kernels of different strengths. Potential applications of the proposed rules in formulating and benchmarking new numerical schemes for generalized fractional diffusion problems are briefly discussed in the final remarking section.postprin
Quadrature Strategies for Constructing Polynomial Approximations
Finding suitable points for multivariate polynomial interpolation and
approximation is a challenging task. Yet, despite this challenge, there has
been tremendous research dedicated to this singular cause. In this paper, we
begin by reviewing classical methods for finding suitable quadrature points for
polynomial approximation in both the univariate and multivariate setting. Then,
we categorize recent advances into those that propose a new sampling approach
and those centered on an optimization strategy. The sampling approaches yield a
favorable discretization of the domain, while the optimization methods pick a
subset of the discretized samples that minimize certain objectives. While not
all strategies follow this two-stage approach, most do. Sampling techniques
covered include subsampling quadratures, Christoffel, induced and Monte Carlo
methods. Optimization methods discussed range from linear programming ideas and
Newton's method to greedy procedures from numerical linear algebra. Our
exposition is aided by examples that implement some of the aforementioned
strategies
Increasing the Reliability of Adaptive Quadrature Using Explicit Interpolants
We present two new adaptive quadrature routines. Both routines differ from
previously published algorithms in many aspects, most significantly in how they
represent the integrand, how they treat non-numerical values of the integrand,
how they deal with improper divergent integrals and how they estimate the
integration error. The main focus of these improvements is to increase the
reliability of the algorithms without significantly impacting their efficiency.
Both algorithms are implemented in Matlab and tested using both the "families"
suggested by Lyness and Kaganove and the battery test used by Gander and
Gautschi and Kahaner. They are shown to be more reliable, albeit in some cases
less efficient, than other commonly-used adaptive integrators.Comment: 32 pages, submitted to ACM Transactions on Mathematical Softwar
FX Smile in the Heston Model
The Heston model stands out from the class of stochastic volatility (SV) models mainly for two reasons. Firstly, the process for the volatility is nonnegative and mean-reverting, which is what we observe in the markets. Secondly, there exists a fast and easily implemented semi-analytical solution for European options. In this article we adapt the original work of Heston (1993) to a foreign exchange (FX) setting. We discuss the computational aspects of using the semi-analytical formulas, performing Monte Carlo simulations, checking the Feller condition, and option pricing with FFT. In an empirical study we show that the smile of vanilla options can be reproduced by suitably calibrating three out of five model parameters.Heston model; vanilla option; stochastic volatility; Monte Carlo simulation; Feller condition; option pricing with FFT;
Stability of step size control based on a posteriori error estimates
A posteriori error estimates based on residuals can be used for reliable
error control of numerical methods. Here, we consider them in the context of
ordinary differential equations and Runge-Kutta methods. In particular, we take
the approach of Dedner & Giesselmann (2016) and investigate it when used to
select the time step size. We focus on step size control stability when
combined with explicit Runge-Kutta methods and demonstrate that a standard I
controller is unstable while more advanced PI and PID controllers can be
designed to be stable. We compare the stability properties of residual-based
estimators and classical error estimators based on an embedded Runge-Kutta
method both analytically and in numerical experiments
The algebraic method in quadrature for uncertainty quantification
A general method of quadrature for uncertainty quantification (UQ) is introduced based on the algebraic method in experimental design. This is a method based on the theory of zero-dimensional algebraic varieties. It allows quadrature of polynomials or polynomial approximands for quite general sets of quadrature points, here called “designs.” The method goes some way to explaining when quadrature weights are nonnegative and gives exact quadrature for monomials in the quotient ring defined by the algebraic method. The relationship to the classical methods based on zeros of orthogonal polynomials is discussed, and numerical comparisons are made with methods such as Gaussian quadrature and Smolyak grids. Application to UQ is examined in the context of polynomial chaos expansion and the probabilistic collocation method, where solution statistics are estimated