Inconsistencies in calculation of A-matrix using Gauss-Lobatto quadrature

[Balavarun5]

Commit hash: 08b38d7

Description

The calculation of A matrix involves integral of product of Lagrange basis polynomials evaluated over Gauss-Lobatto nodes, This results in some inconsistencies when compared to values obtained by solvers (which use Gaussian quadrature)

Report

A report explaining this issue in detail can be found here

[Balavarun5]

The problem with the existing approach of integrating the product of Lagrange basis polynomials using Lobatto quadrature was that the number of Lobatto points used for integration was same as the number of LGL points used to calculate the Lagrange basis polynomials.

As a test, the integral of lagrange basis polynomials L_0 and L_1 are calculated using 8 LGL points is calculated using both Gaussian and Lobatto quadrature.

The number of points used in each quadrature (N_quad) is varied and the error is plotted against N_quad.

As can be seen, The error while calculating the Lobatto quadrature using 8 (N_quad) points is non-trivial. The error drastically reduces for N_quad = 9.

The solution then is to use a higher number of Lobatto points to integrate than the number of LGL points used to evaluate the Lagrange basis polynomials.

[Balavarun5]

The existing PR #16 is to be updated with the necessary changes and approved before closing this issue.

[Balavarun5]

The PR #16 has been merged and the unit test for A matrix is passing.