The discrete orthogonal polynomial least squares method for approximation and solving partial differential equations

We investigate numerical approximations based on polynomials that are orthogonal with respect to a weighted discrete inner product and develop an algorithm for solving time dependent differential equations. We focus on the family of super Gaussian weight functions and derive a criterion for the choice of parameters that provides good accuracy and stability for the time evolution of partial differential equations. Our results show that this approach circumvents the problems related to the Runge phenomenon on equally spaced nodes and provides high accuracy in space. For time stability, small corrections near the ends of the interval are computed using local polynomial interpolation. Several numerical experiments illustrate the performance of the method.