High accuracy stable numerical solution of ID microscale heat transport equation

We investigate the use of a fourth-order compact finite difference scheme for solving a one-dimensional heat transport equation at the microscale. The fourth-order compact scheme is used with a Crank-Nicholson type integrator by introducing an intermediate function for the heat transport equation. The new scheme is proved to be unconditionally stable with respect to initial values. Numerical experiments are conducted to compare the new scheme with the existing scheme based on second-order spatial discretization. It is shown that the new scheme is computationally more efficient and more accurate than the second-order scheme.