Computing Integrals Involved the Gaussian Function with a Small Standard Deviation
This work addresses the challenge of accurate numerical integration for rapidly varying Gaussian-weighted integrals, which is relevant for applications in physics and engineering.
The paper develops numerical integration methods for integrals involving a Gaussian function with small standard deviation, achieving polynomial or exponential accuracy depending on the smoothness of the integrand's factor.
We develop efficient numerical integration methods for computing an integral whose integrand is a product of a smooth function and the Gaussian function with a small standard deviation. Traditional numerical integration methods applied to the integral normally lead to poor accuracy due to the rapid change in high order derivatives of its integrand when the standard deviation is small. The proposed quadrature schemes are based on graded meshes designed according to the standard deviation so that the quadrature errors on the resulting subintervals are approximately equal. The integral in each subinterval is then computed by considering the Gaussian function as a weight function and interpolating the smooth factor of the integrand at the Chebyshev points of the first kind. For a finite order differentiable factor, we design a quadrature scheme having accuracy of a polynomial order and for an infinitely differentiable factor of the integrand, we design a quadrature scheme having accuracy of an exponential order. Numerical results are presented to confirm the accuracy of these proposed quadrature schemes.