Numerical Solution of Burgers' Equation Using Discrete Symmetries

Abstract

In this thesis, we meticulously construct an invariant Modi ed-Crank-Nicolson method that fast convergent to the exact solution of a one-dimensional non-linear heat equation. This innovative construction can be faithfully done by preferentially using discrete symmetry groups. Burgers' equation is reduced to a one-dimensional heat equation by using Hopf-Cole transformation. Moreover, this new transformation function represents the exact solution of Burgers' equation. The innovative invariant numerical scheme is carefully constructed by the composition of continuous and discrete symmetry groups. Furthermore, with this numerical scheme, the convergence and e ciency of the standard Crank-Nicolson method is meaningfully improved for the exact solution of Burgers' equation. The notable performance of this numerical scheme is shown both graphically and in tabular form.