High-order finite difference methods for a second order dual-phase-lagging models of microscale heat transfer

Two ADI solvers are given for 2D and 3D heat conduct equations at microscale.They are uniquely solvable.It is shown that they are of order 2 in time and order 4 in space.Numerical results exhibit the efficiency of the solvers. In this paper, a compact alternating direction implicit (ADI) method, which combines the fourth-order compact difference for the approximations of the second spatial derivatives and the approximation factorizations of difference operators, is firstly presented for solving two-dimensional (2D) second order dual-phase-lagging models of microscale heat transfer. By the discrete energy method, it is shown that it is second-order accurate in time and fourth-order accurate in space with respect to L2-norms. Additionally, the compact ADI method is successfully generalized to solve corresponding three-dimensional (3D) problem. Also, the convergence result of the solver for 3D case is given rigorously. Finally, numerical examples are carried out to testify the computational efficiency of the algorithms and exhibit the correctness of theoretical results.

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