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.
