A method of differential iteration for boundary value problems in extended thermodynamics

The classical well-posed boundary conditions in Navier–Stokes–Fourier (NSF) theory are usually insufficient for the corresponding problems in extended theories of thermodynamics. Some additional boundary data may be needed for the uniqueness of solutions. Owing to the specific structure of systems of balance equations in extended thermodynamics, no such data will be needed in the proposed iterative method by decoupling the system into two subsystems and solving them alternatively with an iterative procedure. One of them can be solved uniquely with the classical boundary conditions, and the other determines the remaining non-equilibrium field variables by direct evaluation. The method does not rely on any criterion for uniqueness, in contrast to various physical criteria proposed for such problems recently. In Part I, the shearing flow with heat conduction is considered as an illustrative numerical example for the proposed method. The condition for convergence based on the estimated error that can easily be checked in numerical iterations is proved in Part II. Furthermore, stability and uniqueness of the numerical iterative solution are also considered. Additional examples are given to substantiate the main results on convergence, stability and uniqueness.