Error in calculation of volume intgral flux

[Balavarun5]

commit hash : ba250f5

Issue : Error in calculation of the volume integral flux using Integrate() function.

Calculation of the volume integral flux in the code is done using an Integrate() function.

This function evaluates the integral using quadrature method by taking in the coefficients of the integrand and number of quadrature points.

The term F(u) in integrand represents the flux, when the integral is carried out using the quadrature method, F(u^n), the flux at the quadrature points needs to be calculated.

To do this, first the wave equation for an element is to be calculated and then the value of this function at the quadrature points mapped in the element needs to be calculated.

This is done by taking the amplitude at the LGL points in each element (known) and obtaining the amplitude as a function of xi by representing it as a linear combination of the Lagrange basis polynomials.

U(x) = a_i L_i(xi)

Where the coefficients a_i are the value of the function at the LGL points.

Error

The error in calculating the volume integral using this method is not trivial.

DG_Maxwell/code/unit_test/test_waveEqn.py

Line 209 in ba250f5

threshold = 8e-9

The error of calculating the volume integral flux for an initial wave amplitude e ** (- x ** 2 / 0.4 ** 2)
and Lagrange basis polynomials calculated using 8 LGL points is plotted against the number of quadrature points below.

Possible error

The amplitude equation e ** (- x ** 2 / 0.4 ** 2) is calculated using the Lagrange basis polynomials (polynomials of order 7).

[mchandra]

How are you computing the reference solution with which you are computing the error?

[amanabt]

I wrote a test code for finding the error between the integral , calculated using the volume_integral_flux function and analytical integral.
I used scipy.integrate.quad to calculate the analytical integral.
You may find the test python notebook at this address volume_integral_error_analysis.ipynb

I experimented with varying and and some relationships between and . Below are the plots for various conditions.

Clearly, for the case of the error goes below . So, we may try to use this condition to minimise our error.