Optimization of nodes of Newton-Cotes formulas in the presence of an exponential boundary layer

Abstract

The problem of numerical integration of a function of one variable with large gradients in the region of the exponential boundary layer it is studied. The problem is that the use of composite Newton-Cotes formulas on a uniform grid leads to significant errors when decreasing the small parameter ε, regardless of the number of nodes of the basic quadrature formula. In the paper it is proposed to choose nodes based on minimizing the error of the composite Newton-Cotes formula. It is proved that the minimum error is achieved on the Bakhvalov mesh, while the error of the quadrature formula becomes uniform in the small parameter ε.