L-stable Simpson’s 3/8 rule and Burgers’ equation

In this paper we develop an unconditionally stable third order time integration formula for the diffusion equation with Neumann boundary condition. We use a suitable arithmetic average approximation and explicit backward Euler formula and then develop a third order L-stable Simpson’s 3/8 type formula. We also observe that the arithmetic average approximation is not unique. Then L-stable Simpson’s 3/8 type formula and Hopf–Cole transformation is used to solve Burger’s equation with Dirichlet boundary condition. It is also observed that this numerical method deals efficiently in case of inconsistencies in initial and boundary conditions.