Week 6 - Multivariate Newton Rhapson Solver

Aim: To solve the set of non-linear differential equations using Multivariate Newton Rhapson method

Given:

• Equations:

• Initial condition: y1=y2=y3=0
• Numerical Analysis: To solve using the Backward difference method & Jacobian method
• Solver: Newton Rhapson Method is used to iteratively approximate the roots of the given function by selecting a optimum relaxation factor. This univariate method can be extended for coupled systems of ODEs.

Algorithm:

BDF:

Equations discretized using BDF

Jacobian matrix:

F=[f1,f2,f3] is the function matrix

The problem is solved using python. First solve the non-linear quadratic functions. Calculate the jacobian matrix numerically, by discretization using the BDF scheme. An arbirary dy value is assigned for discretization of PDE. In this case,dy=1e-10.

Dummy values are assigned in an array using a timeloop. The simulation is run for 600s(10 mins)

code: https://drive.google.com/file/d/1rTPk5juAunYsZ-jAGfYsetzxwbJ6KZ_i/view?usp=sharing

The error decrease with each timestep. The ODEs coverge and the error decreases to zero.