A high order scheme for unsteady heat conduction equations

Abstract

In this work a high order scheme is proposed for solving the heat conduction problems. In this scheme, the governing equation of heat conduction is firstly written into a system of first-order partial differential equations which are integrated over the control volumes around each node point and a finite time step. The resulting time integrals are approximated by numerical quadrature with an undetermined weighting parameter. The error analysis allows us to determine the parameter and the ratio of the time step to the square of grid spacing by making the error of the scheme as small as possible. Both theoretical and numerical results show that the proposed high order scheme can achieve at least sixth order accuracy at the expense of the same computing cost as the traditional finite volume method. And the high order scheme demonstrates a better performance than the high order compact finite difference method and the Runge–Kutta discontinuous Galerkin method.